On 6/5/22 2:35 PM, olcott wrote:
> On 6/5/2022 1:31 PM, Richard Damon wrote:
>> On 6/5/22 1:02 PM, olcott wrote:
>>> On 6/5/2022 11:59 AM, Richard Damon wrote:
>>>> On 6/5/22 12:39 PM, olcott wrote:
>>>>> On 6/5/2022 11:31 AM, Richard Damon wrote:
>>>>>> On 6/5/22 11:57 AM, olcott wrote:
>>>>>>> On 6/5/2022 7:38 AM, Ben wrote:
>>>>>>>> Alan Mackenzie <acm@muc.de> writes:
>>>>>>>>
>>>>>>>>> olcott <NoOne@nowhere.com> wrote:
>>>>>>>>>> On 6/5/2022 6:12 AM, Alan Mackenzie wrote:
>>>>>>>>
>>>>>>>>>>> ...  What would be the turing machine equivalent of an
>>>>>>>>>>> "abnormal termination" in C?
>>>>>>>>>
>>>>>>>>>> An aborted simulation.
>>>>>>>>>
>>>>>>>>> There's no such thing on a turing machine.  It either runs and
>>>>>>>>> halts, or
>>>>>>>>> it runs forever.
>>>>>>>>>
>>>>>>>>> Your aborted simulation is just one final state of a turing
>>>>>>>>> machine,
>>>>>>>>> which has thus halted.
>>>>>>>>
>>>>>>>> A year ago I tried to get PO to accept a few basic facts about the
>>>>>>>> topic.  One of these was
>>>>>>>>
>>>>>>>> (B) Every computation that halts, for whatever reason, is a halting
>>>>>>>>      computation.
>>>>>>>>
>>>>>>>> After much ducking a diving, PO replied "OK".
>>>>>>>>
>>>>>>>
>>>>>>> Whatever I said years ago has been superseded by my current
>>>>>>> understanding:
>>>>>>>
>>>>>>> Computation that halts ... the Turing machine will halt whenever
>>>>>>> it enters a final state. (Linz:1990:234)
>>>>>>>
>>>>>>>
>>>>>>
>>>>>> And H^ applied to <H^> will Halt if H applied to <H^> <H^> rejects
>>>>>> its input as non-halting, thus showing that the H was wrong.
>>>>>
>>>>> You are a brain dead moron on this point.
>>>>>
>>>>>
>>>>
>>>> Why, for speaking Truth that you can't counter?
>>>
>>> The actual truth is beyond your intellectual capacity on this point.
>>>
>>
>> I cam say the same about you, but MY logic is actually correct, and
>> veriviable by others.
> Everyone that claims that the correctly simulated input to H(P,P)
> reaches its final state is woefully incompetent even at basic software
> engineering.
>
>

Why do you say that?

Since the DEFINITION of a correct simulation of something means it
matchs the behavior of what it is simulating, the correct simulation of
the input to H(P,P) is what P(P) does, since H claims to be a Halt
Decider, and the input to a Halt Decider is the representation of a
computation, and the computation for the input P,P would be P(P).

Since even YOU agree that P(P) will halt if H(P,P) returns 0, then the
ONLY possible result for a correct simulation of the input to H(P,P),
for an H(P,P) that returns 0, is to Halt because it reaches its final state.

And in fact, even you have posted traces of a simulation that shows this.

Unless you can point out the error IN YOUR OWN SIMULATION that shows
that the correct simulation of the input to H(P,P) reached its final
halt instruction, then you claim is obviously incorrect.

For your reference, here is your trace below. Please show where YOU made
the error:

On 4/27/21 12:55 AM, olcott wrote:
Message-ID: <Teudndbu59GVBBr9nZ2dnUU7-V2dnZ2d@giganews.com>
> void H_Hat(u32 P)
> {
> u32 Input_Halts = Halts(P, P);
> if (Input_Halts)
> HERE: goto HERE;
> }
>
>
> int main()
> {
> H_Hat((u32)H_Hat);
> }
>
>
> _H_Hat()
> [00000b98](01) 55 push ebp
> [00000b99](02) 8bec mov ebp,esp
>
[00000b9b](01) 51 push ecx
> [00000b9c](03) 8b4508 mov eax,[ebp+08]
> [00000b9f](01) 50 push eax
> [00000ba0](03) 8b4d08 mov ecx,[ebp+08]
> [00000ba3](01) 51 push ecx
> [00000ba4](05) e88ffdffff call 00000938
> [00000ba9](03) 83c408 add esp,+08
> [00000bac](03) 8945fc mov [ebp-04],eax
> [00000baf](04) 837dfc00 cmp dword [ebp-04],+00
> [00000bb3](02) 7402 jz 00000bb7
> [00000bb5](02) ebfe jmp 00000bb5
> [00000bb7](02) 8be5 mov esp,ebp
> [00000bb9](01) 5d pop ebp
> [00000bba](01) c3 ret
> Size in bytes:(0035) [00000bba]
>
> _main()
> [00000bc8](01) 55 push ebp
> [00000bc9](02) 8bec mov ebp,esp
> [00000bcb](05) 68980b0000 push 00000b98
> [00000bd0](05) e8c3ffffff call 00000b98
> [00000bd5](03) 83c404 add esp,+04
> [00000bd8](02) 33c0 xor eax,eax
> [00000bda](01) 5d pop ebp
> [00000bdb](01) c3 ret
> Size in bytes:(0020) [00000bdb]
>
> ===============================
> ...[00000bc8][001015d4][00000000](01) 55 push ebp
> ...[00000bc9][001015d4][00000000](02) 8bec mov ebp,esp
> ...[00000bcb][001015d0][00000b98](05) 68980b0000 push 00000b98
> ...[00000bd0][001015cc][00000bd5](05) e8c3ffffff call 00000b98
> ...[00000b98][001015c8][001015d4](01) 55 push ebp
> ...[00000b99][001015c8][001015d4](02) 8bec mov ebp,esp
> ...[00000b9b][001015c4][00000000](01) 51 push ecx
> ...[00000b9c][001015c4][00000000](03) 8b4508 mov eax,[ebp+08]
> ...[00000b9f][001015c0][00000b98](01) 50 push eax
> ...[00000ba0][001015c0][00000b98](03) 8b4d08 mov ecx,[ebp+08]
> ...[00000ba3][001015bc][00000b98](01) 51 push ecx
> ...[00000ba4][001015b8][00000ba9](05) e88ffdffff call 00000938
> Begin Local Halt Decider Simulation at Machine Address:b98
> ...[00000b98][00211674][00211678](01) 55 push ebp
> ...[00000b99][00211674][00211678](02) 8bec mov ebp,esp
> ...[00000b9b][00211670][00201644](01) 51 push ecx
> ...[00000b9c][00211670][00201644](03) 8b4508 mov eax,[ebp+08]
> ...[00000b9f][0021166c][00000b98](01) 50 push eax
> ...[00000ba0][0021166c][00000b98](03) 8b4d08 mov ecx,[ebp+08]
> ...[00000ba3][00211668][00000b98](01) 51 push ecx
> ...[00000ba4][00211664][00000ba9](05) e88ffdffff call 00000938
> ...[00000b98][0025c09c][0025c0a0](01) 55 push ebp
> ...[00000b99][0025c09c][0025c0a0](02) 8bec mov ebp,esp
> ...[00000b9b][0025c098][0024c06c](01) 51 push ecx
> ...[00000b9c][0025c098][0024c06c](03) 8b4508 mov eax,[ebp+08]
> ...[00000b9f][0025c094][00000b98](01) 50 push eax
> ...[00000ba0][0025c094][00000b98](03) 8b4d08 mov ecx,[ebp+08]
> ...[00000ba3][0025c090][00000b98](01) 51 push ecx
> ...[00000ba4][0025c08c][00000ba9](05) e88ffdffff call 00000938
> Local Halt Decider: Infinite Recursion Detected Simulation Stopped

Above decision was from the call the Halts inside H_Hat, deciding that
H_Hat(H_Hat) seems to be non-halting, it then returns that answer and is
processed below:

> ...[00000ba9][001015c4][00000000](03) 83c408 add esp,+08
> ...[00000bac][001015c4][00000000](03) 8945fc mov [ebp-04],eax
> ...[00000baf][001015c4][00000000](04) 837dfc00 cmp dword [ebp-04],+00
> ...[00000bb3][001015c4][00000000](02) 7402 jz 00000bb7
> ...[00000bb7][001015c8][001015d4](02) 8be5 mov esp,ebp
> ...[00000bb9][001015cc][00000bd5](01) 5d pop ebp
> ...[00000bba][001015d0][00000b98](01) c3 ret
> ...[00000bd5][001015d4][00000000](03) 83c404 add esp,+04
> ...[00000bd8][001015d4][00000000](02) 33c0 xor eax,eax
> ...[00000bda][001015d8][00100000](01) 5d pop ebp
> ...[00000bdb][001015dc][00000098](01) c3 ret

SEE IT HALTED!

> Number_of_User_Instructions(39)
> Number of Instructions Executed(26567)

On 6/5/2022 1:54 PM, Richard Damon wrote:
> On 6/5/22 2:35 PM, olcott wrote:
>> On 6/5/2022 1:31 PM, Richard Damon wrote:
>>> On 6/5/22 1:02 PM, olcott wrote:
>>>> On 6/5/2022 11:59 AM, Richard Damon wrote:
>>>>> On 6/5/22 12:39 PM, olcott wrote:
>>>>>> On 6/5/2022 11:31 AM, Richard Damon wrote:
>>>>>>> On 6/5/22 11:57 AM, olcott wrote:
>>>>>>>> On 6/5/2022 7:38 AM, Ben wrote:
>>>>>>>>> Alan Mackenzie <acm@muc.de> writes:
>>>>>>>>>
>>>>>>>>>> olcott <NoOne@nowhere.com> wrote:
>>>>>>>>>>> On 6/5/2022 6:12 AM, Alan Mackenzie wrote:
>>>>>>>>>
>>>>>>>>>>>> ...  What would be the turing machine equivalent of an
>>>>>>>>>>>> "abnormal termination" in C?
>>>>>>>>>>
>>>>>>>>>>> An aborted simulation.
>>>>>>>>>>
>>>>>>>>>> There's no such thing on a turing machine.  It either runs and
>>>>>>>>>> halts, or
>>>>>>>>>> it runs forever.
>>>>>>>>>>
>>>>>>>>>> Your aborted simulation is just one final state of a turing
>>>>>>>>>> machine,
>>>>>>>>>> which has thus halted.
>>>>>>>>>
>>>>>>>>> A year ago I tried to get PO to accept a few basic facts about the
>>>>>>>>> topic.  One of these was
>>>>>>>>>
>>>>>>>>> (B) Every computation that halts, for whatever reason, is a
>>>>>>>>> halting
>>>>>>>>>      computation.
>>>>>>>>>
>>>>>>>>> After much ducking a diving, PO replied "OK".
>>>>>>>>>
>>>>>>>>
>>>>>>>> Whatever I said years ago has been superseded by my current
>>>>>>>> understanding:
>>>>>>>>
>>>>>>>> Computation that halts ... the Turing machine will halt whenever
>>>>>>>> it enters a final state. (Linz:1990:234)
>>>>>>>>
>>>>>>>>
>>>>>>>
>>>>>>> And H^ applied to <H^> will Halt if H applied to <H^> <H^>
>>>>>>> rejects its input as non-halting, thus showing that the H was wrong.
>>>>>>
>>>>>> You are a brain dead moron on this point.
>>>>>>
>>>>>>
>>>>>
>>>>> Why, for speaking Truth that you can't counter?
>>>>
>>>> The actual truth is beyond your intellectual capacity on this point.
>>>>
>>>
>>> I cam say the same about you, but MY logic is actually correct, and
>>> veriviable by others.
>> Everyone that claims that the correctly simulated input to H(P,P)
>> reaches its final state is woefully incompetent even at basic software
>> engineering.
>>
>>
>
> Why do you say that?
>
> Since the DEFINITION of a correct simulation of something means it
> matchs the behavior of what it is simulating, the correct simulation of
> the input to H(P,P) is what P(P) does

YOU GOD DAMNED LIAR

--
Copyright 2022 Pete Olcott

"Talent hits a target no one else can hit;
Genius hits a target no one else can see."
Arthur Schopenhauer

On 6/5/2022 1:54 PM, Richard Damon wrote:
> On 6/5/22 2:35 PM, olcott wrote:
>> On 6/5/2022 1:31 PM, Richard Damon wrote:
>>> On 6/5/22 1:02 PM, olcott wrote:
>>>> On 6/5/2022 11:59 AM, Richard Damon wrote:
>>>>> On 6/5/22 12:39 PM, olcott wrote:
>>>>>> On 6/5/2022 11:31 AM, Richard Damon wrote:
>>>>>>> On 6/5/22 11:57 AM, olcott wrote:
>>>>>>>> On 6/5/2022 7:38 AM, Ben wrote:
>>>>>>>>> Alan Mackenzie <acm@muc.de> writes:
>>>>>>>>>
>>>>>>>>>> olcott <NoOne@nowhere.com> wrote:
>>>>>>>>>>> On 6/5/2022 6:12 AM, Alan Mackenzie wrote:
>>>>>>>>>
>>>>>>>>>>>> ...  What would be the turing machine equivalent of an
>>>>>>>>>>>> "abnormal termination" in C?
>>>>>>>>>>
>>>>>>>>>>> An aborted simulation.
>>>>>>>>>>
>>>>>>>>>> There's no such thing on a turing machine.  It either runs and
>>>>>>>>>> halts, or
>>>>>>>>>> it runs forever.
>>>>>>>>>>
>>>>>>>>>> Your aborted simulation is just one final state of a turing
>>>>>>>>>> machine,
>>>>>>>>>> which has thus halted.
>>>>>>>>>
>>>>>>>>> A year ago I tried to get PO to accept a few basic facts about the
>>>>>>>>> topic.  One of these was
>>>>>>>>>
>>>>>>>>> (B) Every computation that halts, for whatever reason, is a
>>>>>>>>> halting
>>>>>>>>>      computation.
>>>>>>>>>
>>>>>>>>> After much ducking a diving, PO replied "OK".
>>>>>>>>>
>>>>>>>>
>>>>>>>> Whatever I said years ago has been superseded by my current
>>>>>>>> understanding:
>>>>>>>>
>>>>>>>> Computation that halts ... the Turing machine will halt whenever
>>>>>>>> it enters a final state. (Linz:1990:234)
>>>>>>>>
>>>>>>>>
>>>>>>>
>>>>>>> And H^ applied to <H^> will Halt if H applied to <H^> <H^>
>>>>>>> rejects its input as non-halting, thus showing that the H was wrong.
>>>>>>
>>>>>> You are a brain dead moron on this point.
>>>>>>
>>>>>>
>>>>>
>>>>> Why, for speaking Truth that you can't counter?
>>>>
>>>> The actual truth is beyond your intellectual capacity on this point.
>>>>
>>>
>>> I cam say the same about you, but MY logic is actually correct, and
>>> veriviable by others.
>> Everyone that claims that the correctly simulated input to H(P,P)
>> reaches its final state is woefully incompetent even at basic software
>> engineering.
>>
>>
>
> Why do you say that?
>
> Since the DEFINITION of a correct simulation of something means it
> matchs the behavior of what it is simulating,

This part is correct.

_P()
[00001352](01) 55 push ebp
[00001353](02) 8bec mov ebp,esp
[00001355](03) 8b4508 mov eax,[ebp+08]
[00001358](01) 50 push eax // push P
[00001359](03) 8b4d08 mov ecx,[ebp+08]
[0000135c](01) 51 push ecx // push P
[0000135d](05) e840feffff call 000011a2 // call H
[00001362](03) 83c408 add esp,+08
[00001365](02) 85c0 test eax,eax
[00001367](02) 7402 jz 0000136b
[00001369](02) ebfe jmp 00001369
[0000136b](01) 5d pop ebp
[0000136c](01) c3 ret
Size in bytes:(0027) [0000136c]

It is completely obvious that when H(P,P) correctly emulates its input
that it must emulate the first seven instructions of P. Because the
seventh instruction repeats this process we can know with complete
certainty that the emulated P never reaches its final “ret” instruction,
thus never halts.

--
Copyright 2022 Pete Olcott

"Talent hits a target no one else can hit;
Genius hits a target no one else can see."
Arthur Schopenhauer

On Sun, 5 Jun 2022 14:13:55 -0400
Richard Damon <Richard@Damon-Family.org> wrote:

> On 6/5/22 1:19 PM, Mr Flibble wrote:
> > On Sun, 5 Jun 2022 12:01:32 -0500
> > olcott <NoOne@NoWhere.com> wrote:
> >
> >> On 6/5/2022 11:56 AM, Mr Flibble wrote:
> >>> On Sun, 5 Jun 2022 12:50:13 -0400
> >>> Richard Damon <Richard@Damon-Family.org> wrote:
> >>>
> >>>> On 6/5/22 12:28 PM, Mr Flibble wrote:
> >>>>> On Sun, 5 Jun 2022 12:22:45 -0400
> >>>>> Richard Damon <Richard@Damon-Family.org> wrote:
> >>>>>
> >>>>>> On 6/5/22 11:49 AM, Mr Flibble wrote:
> >>>>>>> On Sun, 5 Jun 2022 15:44:32 -0000 (UTC)
> >>>>>>> Alan Mackenzie <acm@muc.de> wrote:
> >>>>>>>
> >>>>>>>> Mr Flibble <flibble@reddwarf.jmc> wrote:
> >>>>>>>>> On Sun, 5 Jun 2022 16:28:05 +0100
> >>>>>>>>> Andy Walker <anw@cuboid.co.uk> wrote:
> >>>>>>>>
> >>>>>>>>>> On 05/06/2022 14:47, Mr Flibble wrote:
> >>>>>>>>>>> On Sun, 5 Jun 2022 07:58:42 -0400
> >>>>>>>>>>> Richard Damon <Richard@Damon-Family.org> wrote:
> >>>>>>>>>>>> [...] Sort of like how the number Pi has an
> >>>>>>>>>>>> exact value, but you can never actually express it
> >>>>>>>>>>>> (because it takes an infinite number of digits).
> >>>>>>>>>>> PI does not have an exact value; no irrational number has
> >>>>>>>>>>> an exact value.
> >>>>>>>>
> >>>>>>>>>> Of course "pi" has an exact value; as do [eg]
> >>>>>>>>>> "sqrt(2)", "e", and all the other computable real [and
> >>>>>>>>>> complex] numbers. Whether that value can be expressed in
> >>>>>>>>>> finite terms in some particular representation is quite
> >>>>>>>>>> another matter. That in turn depends on the
> >>>>>>>>>> representation; standard decimals is merely one [common]
> >>>>>>>>>> choice. Note that in symbolic computer systems, those
> >>>>>>>>>> computable reals are typically written "pi" [or whatever],
> >>>>>>>>>> and the computer works with that exactly, so that [eg]
> >>>>>>>>>> "sin^2 (pi/3) == 3/4", not 0.7499...; and also that in
> >>>>>>>>>> decimal-type notations most rationals equally have no
> >>>>>>>>>> terminating expansion. Numbers such as "pi" and "sqrt(2)"
> >>>>>>>>>> are not defined as decimal expansions but via their
> >>>>>>>>>> properties [eg that "sqrt(2)" is the unique positive real
> >>>>>>>>>> whose square is 2, or equivalently that it is the ratio of
> >>>>>>>>>> the diagonal of a square to its side, and "pi" is the
> >>>>>>>>>> least positive real whose sine is zero]. Those properties
> >>>>>>>>>> are exact, and tell you all you ever need to know about
> >>>>>>>>>> those numbers.
> >>>>>>>>
> >>>>>>>> [ .... ]
> >>>>>>>>
> >>>>>>>>> What has decimal (base 10) expansion got to do with
> >>>>>>>>> anything? An irrational number has a non-terminating
> >>>>>>>>> sequence in ANY base. I am sorry but you are simply
> >>>>>>>>> mistaken: irrational numbers do NOT have an exact value;
> >>>>>>>>> this is obvious to anyone who understands logic and uses a
> >>>>>>>>> sane definition for infinity.
> >>>>>>>>
> >>>>>>>> That irrational numbers are exact values is clear to anybody
> >>>>>>>> with a degree in maths. Definitions of "infinity" (of which
> >>>>>>>> there are many) have nothing to do with this.
> >>>>>>>
> >>>>>>> You are wrong and fractally so so your degree in maths appears
> >>>>>>> to be worthless. An irrational number's sequence is
> >>>>>>> statistically random, has no fixed point on the number line
> >>>>>>> ergo has no exact representation. Any number with no exact
> >>>>>>> representation has, by definition, no exact value, only an
> >>>>>>> approximation. Infinity has everything to do with this as an
> >>>>>>> irrational's sequence ("digits") never terminates (i.e. it is
> >>>>>>> an INFINITELY long sequence).
> >>>>>>>
> >>>>>>> /Flibble
> >>>>>>>
> >>>>>>
> >>>>>> Nope. Irrational numbers DO have exact points on the number
> >>>>>> line.
> >>>>>>
> >>>>>> And what does representation have to do with exact value?
> >>>>>>
> >>>>>> Also, irrational numbers sequence of digits are not necessarily
> >>>>>> statistically random, in some representations, they can be VERY
> >>>>>> predictible for some numbers.
> >>>>>>
> >>>>>> One simple construction to show exact position, draw a box with
> >>>>>> sides exactly 1.
> >>>>>>
> >>>>>> Draw a line though opposite corners and make one point the
> >>>>>> value 0.
> >>>>>>
> >>>>>> The other corner will be EXACTLY at the point sqrt(2), so that
> >>>>>> irrational number has an exact point on the number line.
> >>>>>>
> >>>>>> You just don't understand what an exact value means, likely
> >>>>>> because you can't understand things that are somewhat
> >>>>>> abstract.
> >>>>>
> >>>>> An irrational number does not have an exact point on the number
> >>>>> line as it will move about as you "zoom in", you can keep
> >>>>> "zooming in" forever (i.e. infinitely) and it will keep moving
> >>>>> about because the number never terminates.
> >>>>>
> >>>>> If I couldn't understand things that are somewhat abstract then
> >>>>> I wouldn't have a computer science degree (BSc Hons) and 30
> >>>>> years of industry experience.
> >>>>>
> >>>>> /Flibble
> >>>>>
> >>>>
> >>>> Then why do you think irrational numbers don't have an exact
> >>>> location?
> >>>>
> >>>> I know people with degrees (even with honors) and industry
> >>>> experiance that still show that they don't really understand what
> >>>> they are talking about.
> >>>>
> >>>> The "width" of the point representing the location of an
> >>>> irrational number is just as much "0" as that of a rational
> >>>> number, so specifies just as exact of a location.
> >>>>
> >>>> The fact that we can't write it in a rational base with a finite
> >>>> number of digits doesn't actally mean anything.
> >>>
> >>> 3.1415xxxxxxxxx (a)
> >>> 3.14159xxxxxxxx (b)
> >>>
> >>> (b) is nearer to 3.14160 than 3.14150 that (a) indicates hence its
> >>> value "moves about" as accuracy increases.
> >>>
> >>> /Flibble
> >>>
> >>
> >> You are confusing the representation of the number in decimal
> >> digits with the actual number itself.
> >
> > No, you are. I am merely pointing out that the number changes up to
> > a factor of 1/base as you evaluate it at increasing accuracy.
> >
> > /Flibble
> >
>
> But APPROXIMATIONS to the number are not the number itself.
>
> The irrational number all have an EXACT location on the number line.

Click here to read the complete article

On Sun, 5 Jun 2022 14:17:29 -0400
Richard Damon <Richard@Damon-Family.org> wrote:

> On 6/5/22 1:22 PM, Mr Flibble wrote:
> > On Sun, 5 Jun 2022 13:05:09 -0400
> > Richard Damon <Richard@Damon-Family.org> wrote:
> >
> >> On 6/5/22 12:56 PM, Mr Flibble wrote:
> >>> On Sun, 5 Jun 2022 12:50:13 -0400
> >>> Richard Damon <Richard@Damon-Family.org> wrote:
> >>>
> >>>> On 6/5/22 12:28 PM, Mr Flibble wrote:
> >>>>> On Sun, 5 Jun 2022 12:22:45 -0400
> >>>>> Richard Damon <Richard@Damon-Family.org> wrote:
> >>>>>
> >>>>>> On 6/5/22 11:49 AM, Mr Flibble wrote:
> >>>>>>> On Sun, 5 Jun 2022 15:44:32 -0000 (UTC)
> >>>>>>> Alan Mackenzie <acm@muc.de> wrote:
> >>>>>>>
> >>>>>>>> Mr Flibble <flibble@reddwarf.jmc> wrote:
> >>>>>>>>> On Sun, 5 Jun 2022 16:28:05 +0100
> >>>>>>>>> Andy Walker <anw@cuboid.co.uk> wrote:
> >>>>>>>>
> >>>>>>>>>> On 05/06/2022 14:47, Mr Flibble wrote:
> >>>>>>>>>>> On Sun, 5 Jun 2022 07:58:42 -0400
> >>>>>>>>>>> Richard Damon <Richard@Damon-Family.org> wrote:
> >>>>>>>>>>>> [...] Sort of like how the number Pi has an
> >>>>>>>>>>>> exact value, but you can never actually express it
> >>>>>>>>>>>> (because it takes an infinite number of digits).
> >>>>>>>>>>> PI does not have an exact value; no irrational number has
> >>>>>>>>>>> an exact value.
> >>>>>>>>
> >>>>>>>>>> Of course "pi" has an exact value; as do [eg]
> >>>>>>>>>> "sqrt(2)", "e", and all the other computable real [and
> >>>>>>>>>> complex] numbers. Whether that value can be expressed in
> >>>>>>>>>> finite terms in some particular representation is quite
> >>>>>>>>>> another matter. That in turn depends on the
> >>>>>>>>>> representation; standard decimals is merely one [common]
> >>>>>>>>>> choice. Note that in symbolic computer systems, those
> >>>>>>>>>> computable reals are typically written "pi" [or whatever],
> >>>>>>>>>> and the computer works with that exactly, so that [eg]
> >>>>>>>>>> "sin^2 (pi/3) == 3/4", not 0.7499...; and also that in
> >>>>>>>>>> decimal-type notations most rationals equally have no
> >>>>>>>>>> terminating expansion. Numbers such as "pi" and "sqrt(2)"
> >>>>>>>>>> are not defined as decimal expansions but via their
> >>>>>>>>>> properties [eg that "sqrt(2)" is the unique positive real
> >>>>>>>>>> whose square is 2, or equivalently that it is the ratio of
> >>>>>>>>>> the diagonal of a square to its side, and "pi" is the
> >>>>>>>>>> least positive real whose sine is zero]. Those properties
> >>>>>>>>>> are exact, and tell you all you ever need to know about
> >>>>>>>>>> those numbers.
> >>>>>>>>
> >>>>>>>> [ .... ]
> >>>>>>>>
> >>>>>>>>> What has decimal (base 10) expansion got to do with
> >>>>>>>>> anything? An irrational number has a non-terminating
> >>>>>>>>> sequence in ANY base. I am sorry but you are simply
> >>>>>>>>> mistaken: irrational numbers do NOT have an exact value;
> >>>>>>>>> this is obvious to anyone who understands logic and uses a
> >>>>>>>>> sane definition for infinity.
> >>>>>>>>
> >>>>>>>> That irrational numbers are exact values is clear to anybody
> >>>>>>>> with a degree in maths. Definitions of "infinity" (of which
> >>>>>>>> there are many) have nothing to do with this.
> >>>>>>>
> >>>>>>> You are wrong and fractally so so your degree in maths appears
> >>>>>>> to be worthless. An irrational number's sequence is
> >>>>>>> statistically random, has no fixed point on the number line
> >>>>>>> ergo has no exact representation. Any number with no exact
> >>>>>>> representation has, by definition, no exact value, only an
> >>>>>>> approximation. Infinity has everything to do with this as an
> >>>>>>> irrational's sequence ("digits") never terminates (i.e. it is
> >>>>>>> an INFINITELY long sequence).
> >>>>>>>
> >>>>>>> /Flibble
> >>>>>>>
> >>>>>>
> >>>>>> Nope. Irrational numbers DO have exact points on the number
> >>>>>> line.
> >>>>>>
> >>>>>> And what does representation have to do with exact value?
> >>>>>>
> >>>>>> Also, irrational numbers sequence of digits are not necessarily
> >>>>>> statistically random, in some representations, they can be VERY
> >>>>>> predictible for some numbers.
> >>>>>>
> >>>>>> One simple construction to show exact position, draw a box with
> >>>>>> sides exactly 1.
> >>>>>>
> >>>>>> Draw a line though opposite corners and make one point the
> >>>>>> value 0.
> >>>>>>
> >>>>>> The other corner will be EXACTLY at the point sqrt(2), so that
> >>>>>> irrational number has an exact point on the number line.
> >>>>>>
> >>>>>> You just don't understand what an exact value means, likely
> >>>>>> because you can't understand things that are somewhat
> >>>>>> abstract.
> >>>>>
> >>>>> An irrational number does not have an exact point on the number
> >>>>> line as it will move about as you "zoom in", you can keep
> >>>>> "zooming in" forever (i.e. infinitely) and it will keep moving
> >>>>> about because the number never terminates.
> >>>>>
> >>>>> If I couldn't understand things that are somewhat abstract then
> >>>>> I wouldn't have a computer science degree (BSc Hons) and 30
> >>>>> years of industry experience.
> >>>>>
> >>>>> /Flibble
> >>>>>
> >>>>
> >>>> Then why do you think irrational numbers don't have an exact
> >>>> location?
> >>>>
> >>>> I know people with degrees (even with honors) and industry
> >>>> experiance that still show that they don't really understand what
> >>>> they are talking about.
> >>>>
> >>>> The "width" of the point representing the location of an
> >>>> irrational number is just as much "0" as that of a rational
> >>>> number, so specifies just as exact of a location.
> >>>>
> >>>> The fact that we can't write it in a rational base with a finite
> >>>> number of digits doesn't actally mean anything.
> >>>
> >>> 3.1415xxxxxxxxx (a)
> >>> 3.14159xxxxxxxx (b)
> >>>
> >>> (b) is nearer to 3.14160 than 3.14150 that (a) indicates hence its
> >>> value "moves about" as accuracy increases.
> >>>
> >>> /Flibble
> >>>
> >>
> >> But (a) and (b) aren't "pi"
> >
> > No, they are approximations of pi. The value of the approximation
> > changes up to a factor of 1/base as you evaluate it at increasing
> > accuracy.
> >
> >>
> >> All you are showing is that approximations to numbers get better as
> >> they get better, which is just a strange tautology.
> >
> > Yes and therefor they jump about on the number line as accuracy
> > increases.
> >
> >>
> >> The number PI, has only one precise value, the exact ratio of the
> >> circumference of a circle to its diameter on a flat plane (which
> >> will always be the same).
> >
> > pi cannot have a precise value as it neither terminates nor has a
> > repetend in any base.
> >
> >>
> >> The fact that it can't be expressed, isn't an issue on exactness,
> >> but of finite representation.
> >
> > We can only ever have a finite representation.
> >
> >>
> >> Note, that the set of numbers with finite representation is a
> >> countable set, so it isn't surprising that the uncountable infinity
> >> of the reals (that includes the irrationals) is not all finitely
> >> expressible.
> >
> > Stating the obvious.
> >
> > /Flibble
> >
>
> So you DO have problems with abstractions.
>
> The problem is, we don't actually need the finite numerical
> representation of a number if we have the definition of what the
> number is (which is another sort of abstract finite representation).
>
> The "finite representation" of pi is the ratio of the circumference
> to the diameter of a circle on a plane.

On Sun, 5 Jun 2022 14:24:17 -0400
Richard Damon <Richard@Damon-Family.org> wrote:

> On 6/5/22 1:17 PM, Mr Flibble wrote:
> > On Sun, 5 Jun 2022 12:57:56 -0400
> > Richard Damon <Richard@Damon-Family.org> wrote:
> >
> >> On 6/5/22 12:37 PM, Mr Flibble wrote:
> >>> On Sun, 5 Jun 2022 12:17:48 -0400
> >>> Richard Damon <Richard@Damon-Family.org> wrote:
> >>>
> >>>> On 6/5/22 11:34 AM, Mr Flibble wrote:
> >>>>> On Sun, 5 Jun 2022 16:28:05 +0100
> >>>>> Andy Walker <anw@cuboid.co.uk> wrote:
> >>>>>
> >>>>>> On 05/06/2022 14:47, Mr Flibble wrote:
> >>>>>>> On Sun, 5 Jun 2022 07:58:42 -0400
> >>>>>>> Richard Damon <Richard@Damon-Family.org> wrote:
> >>>>>>>> [...] Sort of like how the number Pi has an
> >>>>>>>> exact value, but you can never actually express it (because
> >>>>>>>> it takes an infinite number of digits).
> >>>>>>> PI does not have an exact value; no irrational number has an
> >>>>>>> exact value.
> >>>>>>
> >>>>>> Of course "pi" has an exact value; as do [eg]
> >>>>>> "sqrt(2)", "e", and all the other computable real [and
> >>>>>> complex] numbers. Whether that value can be expressed in
> >>>>>> finite terms in some particular representation is quite
> >>>>>> another matter. That in turn depends on the representation;
> >>>>>> standard decimals is merely one [common] choice. Note that in
> >>>>>> symbolic computer systems, those computable reals are
> >>>>>> typically written "pi" [or whatever], and the computer works
> >>>>>> with that exactly, so that [eg] "sin^2 (pi/3) == 3/4", not
> >>>>>> 0.7499...; and also that in decimal-type notations most
> >>>>>> rationals equally have no terminating expansion. Numbers such
> >>>>>> as "pi" and "sqrt(2)" are not defined as decimal expansions
> >>>>>> but via their properties [eg that "sqrt(2)" is the unique
> >>>>>> positive real whose square is 2, or equivalently that it is
> >>>>>> the ratio of the diagonal of a square to its side, and "pi" is
> >>>>>> the least positive real whose sine is zero]. Those properties
> >>>>>> are exact, and tell you all you ever need to know about those
> >>>>>> numbers.
> >>>>>>
> >>>>>> [I have removed my name from the "Subject:"; I don't
> >>>>>> know why anyone saw fit to attach it to this debate, such as it
> >>>>>> is, on the HP.]
> >>>>>
> >>>>> What has decimal (base 10) expansion got to do with anything? An
> >>>>> irrational number has a non-terminating sequence in ANY base. I
> >>>>> am sorry but you are simply mistaken: irrational numbers do NOT
> >>>>> have an exact value; this is obvious to anyone who understands
> >>>>> logic and uses a sane definition for infinity.
> >>>>>
> >>>>> /Flibble
> >>>>>
> >>>>
> >>>> How about in base pi? then it is the number 10
> >>>
> >>> how about base banana? then it is the number 10.
> >>>
> >>> PI, like banana, is just a symbol representing an irrational
> >>> number that has no exact value. To use it here is circular and
> >>> therefor erroneous.
> >>>
> >>>>
> >>>> Base pi is an interesting base for some problems.
> >>>>
> >>>> What is your definition of "an exact value"?
> >>>>
> >>>> Maybe the problem is you don't quite understand the meaning of
> >>>> that term.
> >>>
> >>> Of course I understand the fucking term. For the purposes of this
> >>> discussion an exact value is a real number (non-integer) that
> >>> terminates in a base that is not a multiple of itself.
> >>>
> >>> /Flibble
> >>>
> >>
> >> Where do you get that definition from?
> >>
> >> So 1/3 isn't an exact value?
> >
> > 1/3 is 0.1 in base 3 so does have an exact value.
>
> And base 3 is a multiple of 1/3, which you said wasn't allowed.
>
> >
> > Let me rephrase: for the purposes of this discussion an exact value
> > is a real number that either terminates in some base or has a
> > repetend in other (non-irrational) bases.
> >
> > /Flibble
> >
>
> No, that is NOT a correct definition. That isn't a bad definition of
> a RATIONAL number, as any number that can be written as a finite
> string, or a string with a repetend can be also expressed as a ratio
> of two numbers.
>
> There is nothing in the actual meaning of "exact value" that needs
> the value to be expressible as a finite string of digits.

You are wrong, and fractally so which is ironic given the topic under
discussion.

/Flibble

On Sun, 5 Jun 2022 18:07:46 -0000 (UTC)
Alan Mackenzie <acm@muc.de> wrote:

> Mr Flibble <flibble@reddwarf.jmc> wrote:
> > On Sun, 5 Jun 2022 12:57:56 -0400
> > Richard Damon <Richard@Damon-Family.org> wrote:
>
> [ .... ]
>
> >> So 1/3 isn't an exact value?
>
> > 1/3 is 0.1 in base 3 so does have an exact value.
>
> > Let me rephrase: for the purposes of this discussion an exact value
> > is a real number that either terminates in some base or has a
> > repetend in other (non-irrational) bases.
>
> So what you seem to be saying is that an exact value is a rational
> number. That, somehow, irrational numbers are inexact. There is no
> basis in modern maths for that last assertion. But for that rider -
> "for the purposes of this discussion" shows that you wish to have a
> discussion based on falsehood and superstition.

That would be a fair conclusion: irrational numbers are inexact as it
is impossible to evaluate them to infinite precision as infinite
precision is meaningless.

/Flibble

On 6/5/22 3:09 PM, olcott wrote:
> On 6/5/2022 1:54 PM, Richard Damon wrote:
>> On 6/5/22 2:35 PM, olcott wrote:
>>> On 6/5/2022 1:31 PM, Richard Damon wrote:
>>>> On 6/5/22 1:02 PM, olcott wrote:
>>>>> On 6/5/2022 11:59 AM, Richard Damon wrote:
>>>>>> On 6/5/22 12:39 PM, olcott wrote:
>>>>>>> On 6/5/2022 11:31 AM, Richard Damon wrote:
>>>>>>>> On 6/5/22 11:57 AM, olcott wrote:
>>>>>>>>> On 6/5/2022 7:38 AM, Ben wrote:
>>>>>>>>>> Alan Mackenzie <acm@muc.de> writes:
>>>>>>>>>>
>>>>>>>>>>> olcott <NoOne@nowhere.com> wrote:
>>>>>>>>>>>> On 6/5/2022 6:12 AM, Alan Mackenzie wrote:
>>>>>>>>>>
>>>>>>>>>>>>> ...  What would be the turing machine equivalent of an
>>>>>>>>>>>>> "abnormal termination" in C?
>>>>>>>>>>>
>>>>>>>>>>>> An aborted simulation.
>>>>>>>>>>>
>>>>>>>>>>> There's no such thing on a turing machine.  It either runs
>>>>>>>>>>> and halts, or
>>>>>>>>>>> it runs forever.
>>>>>>>>>>>
>>>>>>>>>>> Your aborted simulation is just one final state of a turing
>>>>>>>>>>> machine,
>>>>>>>>>>> which has thus halted.
>>>>>>>>>>
>>>>>>>>>> A year ago I tried to get PO to accept a few basic facts about
>>>>>>>>>> the
>>>>>>>>>> topic.  One of these was
>>>>>>>>>>
>>>>>>>>>> (B) Every computation that halts, for whatever reason, is a
>>>>>>>>>> halting
>>>>>>>>>>      computation.
>>>>>>>>>>
>>>>>>>>>> After much ducking a diving, PO replied "OK".
>>>>>>>>>>
>>>>>>>>>
>>>>>>>>> Whatever I said years ago has been superseded by my current
>>>>>>>>> understanding:
>>>>>>>>>
>>>>>>>>> Computation that halts ... the Turing machine will halt
>>>>>>>>> whenever it enters a final state. (Linz:1990:234)
>>>>>>>>>
>>>>>>>>>
>>>>>>>>
>>>>>>>> And H^ applied to <H^> will Halt if H applied to <H^> <H^>
>>>>>>>> rejects its input as non-halting, thus showing that the H was
>>>>>>>> wrong.
>>>>>>>
>>>>>>> You are a brain dead moron on this point.
>>>>>>>
>>>>>>>
>>>>>>
>>>>>> Why, for speaking Truth that you can't counter?
>>>>>
>>>>> The actual truth is beyond your intellectual capacity on this point.
>>>>>
>>>>
>>>> I cam say the same about you, but MY logic is actually correct, and
>>>> veriviable by others.
>>> Everyone that claims that the correctly simulated input to H(P,P)
>>> reaches its final state is woefully incompetent even at basic
>>> software engineering.
>>>
>>>
>>
>> Why do you say that?
>>
>> Since the DEFINITION of a correct simulation of something means it
>> matchs the behavior of what it is simulating,
>
> This part is correct.
>
> _P()
> [00001352](01)  55              push ebp
> [00001353](02)  8bec            mov ebp,esp
> [00001355](03)  8b4508          mov eax,[ebp+08]
> [00001358](01)  50              push eax      // push P
> [00001359](03)  8b4d08          mov ecx,[ebp+08]
> [0000135c](01)  51              push ecx      // push P
> [0000135d](05)  e840feffff      call 000011a2 // call H
> [00001362](03)  83c408          add esp,+08
> [00001365](02)  85c0            test eax,eax
> [00001367](02)  7402            jz 0000136b
> [00001369](02)  ebfe            jmp 00001369
> [0000136b](01)  5d              pop ebp
> [0000136c](01)  c3              ret
> Size in bytes:(0027) [0000136c]
>
> It is completely obvious that when H(P,P) correctly emulates its input
> that it must emulate the first seven instructions of P. Because the
> seventh instruction repeats this process we can know with complete
> certainty that the emulated P never reaches its final “ret” instruction,
> thus never halts.
>

So, you can't find an error in the trace you snipped, that implies that
you need to accept it as correct.

Since you also claim that you can prove the exact opposite, what that
proves is that when we add your defintions to the rules of Computation
Theory, the field becomes inconsistent.

That means that there must be an inconsistency in your rules.

YOU FAIL.

One of the problems is your statement:

> Because the seventh instruction repeats this process

Because that isn't what the seventh instruction does (at least not if H
isn't just a call to its input).

The seventh instruction starts NOT just a "Simulation of the input to
H(P,P)" but a DECISION on whether or not this input will Halt. THAT
decision is a conditional operation, and thus we now have the
conditional in the loop that possibly breaks the infinite loop.

Either H DOES break the loop, to answer Non-Halting, and thus return
that answer to P and make P(P) Halting (and thus make H wrong), or H
does NOT break the loop, and thus fails to ever answer the question and
thus fail to be a decider.

Since H is required to be an actual computation, all copies of it given
the same input must do the same thing, so if the first H(P,P) is giving
this answer, we know the first case is the one in effect, and H is
proved incorrect.

If H IS just a call to its input, then we are in the second case, and
the outer H isn't able to answer either, or it is proved to not be a
computation, and thus not eligable to be a decider.

This has been explained to you many times, you failure to learn by know
is telling about your mental state.

On 6/5/22 3:17 PM, Mr Flibble wrote:
> On Sun, 5 Jun 2022 14:17:29 -0400
> Richard Damon <Richard@Damon-Family.org> wrote:
>
>> On 6/5/22 1:22 PM, Mr Flibble wrote:
>>> On Sun, 5 Jun 2022 13:05:09 -0400
>>> Richard Damon <Richard@Damon-Family.org> wrote:
>>>
>>>> On 6/5/22 12:56 PM, Mr Flibble wrote:
>>>>> On Sun, 5 Jun 2022 12:50:13 -0400
>>>>> Richard Damon <Richard@Damon-Family.org> wrote:
>>>>>
>>>>>> On 6/5/22 12:28 PM, Mr Flibble wrote:
>>>>>>> On Sun, 5 Jun 2022 12:22:45 -0400
>>>>>>> Richard Damon <Richard@Damon-Family.org> wrote:
>>>>>>>
>>>>>>>> On 6/5/22 11:49 AM, Mr Flibble wrote:
>>>>>>>>> On Sun, 5 Jun 2022 15:44:32 -0000 (UTC)
>>>>>>>>> Alan Mackenzie <acm@muc.de> wrote:
>>>>>>>>>
>>>>>>>>>> Mr Flibble <flibble@reddwarf.jmc> wrote:
>>>>>>>>>>> On Sun, 5 Jun 2022 16:28:05 +0100
>>>>>>>>>>> Andy Walker <anw@cuboid.co.uk> wrote:
>>>>>>>>>>
>>>>>>>>>>>> On 05/06/2022 14:47, Mr Flibble wrote:
>>>>>>>>>>>>> On Sun, 5 Jun 2022 07:58:42 -0400
>>>>>>>>>>>>> Richard Damon <Richard@Damon-Family.org> wrote:
>>>>>>>>>>>>>> [...] Sort of like how the number Pi has an
>>>>>>>>>>>>>> exact value, but you can never actually express it
>>>>>>>>>>>>>> (because it takes an infinite number of digits).
>>>>>>>>>>>>> PI does not have an exact value; no irrational number has
>>>>>>>>>>>>> an exact value.
>>>>>>>>>>
>>>>>>>>>>>> Of course "pi" has an exact value; as do [eg]
>>>>>>>>>>>> "sqrt(2)", "e", and all the other computable real [and
>>>>>>>>>>>> complex] numbers. Whether that value can be expressed in
>>>>>>>>>>>> finite terms in some particular representation is quite
>>>>>>>>>>>> another matter. That in turn depends on the
>>>>>>>>>>>> representation; standard decimals is merely one [common]
>>>>>>>>>>>> choice. Note that in symbolic computer systems, those
>>>>>>>>>>>> computable reals are typically written "pi" [or whatever],
>>>>>>>>>>>> and the computer works with that exactly, so that [eg]
>>>>>>>>>>>> "sin^2 (pi/3) == 3/4", not 0.7499...; and also that in
>>>>>>>>>>>> decimal-type notations most rationals equally have no
>>>>>>>>>>>> terminating expansion. Numbers such as "pi" and "sqrt(2)"
>>>>>>>>>>>> are not defined as decimal expansions but via their
>>>>>>>>>>>> properties [eg that "sqrt(2)" is the unique positive real
>>>>>>>>>>>> whose square is 2, or equivalently that it is the ratio of
>>>>>>>>>>>> the diagonal of a square to its side, and "pi" is the
>>>>>>>>>>>> least positive real whose sine is zero]. Those properties
>>>>>>>>>>>> are exact, and tell you all you ever need to know about
>>>>>>>>>>>> those numbers.
>>>>>>>>>>
>>>>>>>>>> [ .... ]
>>>>>>>>>>
>>>>>>>>>>> What has decimal (base 10) expansion got to do with
>>>>>>>>>>> anything? An irrational number has a non-terminating
>>>>>>>>>>> sequence in ANY base. I am sorry but you are simply
>>>>>>>>>>> mistaken: irrational numbers do NOT have an exact value;
>>>>>>>>>>> this is obvious to anyone who understands logic and uses a
>>>>>>>>>>> sane definition for infinity.
>>>>>>>>>>
>>>>>>>>>> That irrational numbers are exact values is clear to anybody
>>>>>>>>>> with a degree in maths. Definitions of "infinity" (of which
>>>>>>>>>> there are many) have nothing to do with this.
>>>>>>>>>
>>>>>>>>> You are wrong and fractally so so your degree in maths appears
>>>>>>>>> to be worthless. An irrational number's sequence is
>>>>>>>>> statistically random, has no fixed point on the number line
>>>>>>>>> ergo has no exact representation. Any number with no exact
>>>>>>>>> representation has, by definition, no exact value, only an
>>>>>>>>> approximation. Infinity has everything to do with this as an
>>>>>>>>> irrational's sequence ("digits") never terminates (i.e. it is
>>>>>>>>> an INFINITELY long sequence).
>>>>>>>>>
>>>>>>>>> /Flibble
>>>>>>>>>
>>>>>>>>
>>>>>>>> Nope. Irrational numbers DO have exact points on the number
>>>>>>>> line.
>>>>>>>>
>>>>>>>> And what does representation have to do with exact value?
>>>>>>>>
>>>>>>>> Also, irrational numbers sequence of digits are not necessarily
>>>>>>>> statistically random, in some representations, they can be VERY
>>>>>>>> predictible for some numbers.
>>>>>>>>
>>>>>>>> One simple construction to show exact position, draw a box with
>>>>>>>> sides exactly 1.
>>>>>>>>
>>>>>>>> Draw a line though opposite corners and make one point the
>>>>>>>> value 0.
>>>>>>>>
>>>>>>>> The other corner will be EXACTLY at the point sqrt(2), so that
>>>>>>>> irrational number has an exact point on the number line.
>>>>>>>>
>>>>>>>> You just don't understand what an exact value means, likely
>>>>>>>> because you can't understand things that are somewhat
>>>>>>>> abstract.
>>>>>>>
>>>>>>> An irrational number does not have an exact point on the number
>>>>>>> line as it will move about as you "zoom in", you can keep
>>>>>>> "zooming in" forever (i.e. infinitely) and it will keep moving
>>>>>>> about because the number never terminates.
>>>>>>>
>>>>>>> If I couldn't understand things that are somewhat abstract then
>>>>>>> I wouldn't have a computer science degree (BSc Hons) and 30
>>>>>>> years of industry experience.
>>>>>>>
>>>>>>> /Flibble
>>>>>>>
>>>>>>
>>>>>> Then why do you think irrational numbers don't have an exact
>>>>>> location?
>>>>>>
>>>>>> I know people with degrees (even with honors) and industry
>>>>>> experiance that still show that they don't really understand what
>>>>>> they are talking about.
>>>>>>
>>>>>> The "width" of the point representing the location of an
>>>>>> irrational number is just as much "0" as that of a rational
>>>>>> number, so specifies just as exact of a location.
>>>>>>
>>>>>> The fact that we can't write it in a rational base with a finite
>>>>>> number of digits doesn't actally mean anything.
>>>>>
>>>>> 3.1415xxxxxxxxx (a)
>>>>> 3.14159xxxxxxxx (b)
>>>>>
>>>>> (b) is nearer to 3.14160 than 3.14150 that (a) indicates hence its
>>>>> value "moves about" as accuracy increases.
>>>>>
>>>>> /Flibble
>>>>>
>>>>
>>>> But (a) and (b) aren't "pi"
>>>
>>> No, they are approximations of pi. The value of the approximation
>>> changes up to a factor of 1/base as you evaluate it at increasing
>>> accuracy.
>>>
>>>>
>>>> All you are showing is that approximations to numbers get better as
>>>> they get better, which is just a strange tautology.
>>>
>>> Yes and therefor they jump about on the number line as accuracy
>>> increases.
>>>
>>>>
>>>> The number PI, has only one precise value, the exact ratio of the
>>>> circumference of a circle to its diameter on a flat plane (which
>>>> will always be the same).
>>>
>>> pi cannot have a precise value as it neither terminates nor has a
>>> repetend in any base.
>>>
>>>>
>>>> The fact that it can't be expressed, isn't an issue on exactness,
>>>> but of finite representation.
>>>
>>> We can only ever have a finite representation.
>>>
>>>>
>>>> Note, that the set of numbers with finite representation is a
>>>> countable set, so it isn't surprising that the uncountable infinity
>>>> of the reals (that includes the irrationals) is not all finitely
>>>> expressible.
>>>
>>> Stating the obvious.
>>>
>>> /Flibble
>>>
>>
>> So you DO have problems with abstractions.
>>
>> The problem is, we don't actually need the finite numerical
>> representation of a number if we have the definition of what the
>> number is (which is another sort of abstract finite representation).
>>
>> The "finite representation" of pi is the ratio of the circumference
>> to the diameter of a circle on a plane.
>
> No, instead if you use logic you must come to the conclusion that there
> is no ratio of the circumference to the diameter of a circle: rational
> numbers describe ratios, irrational numbers do not. QED.
>
> /Flibble
>

On 6/5/22 3:19 PM, Mr Flibble wrote:
> On Sun, 5 Jun 2022 18:07:46 -0000 (UTC)
> Alan Mackenzie <acm@muc.de> wrote:
>
>> Mr Flibble <flibble@reddwarf.jmc> wrote:
>>> On Sun, 5 Jun 2022 12:57:56 -0400
>>> Richard Damon <Richard@Damon-Family.org> wrote:
>>
>> [ .... ]
>>
>>>> So 1/3 isn't an exact value?
>>
>>> 1/3 is 0.1 in base 3 so does have an exact value.
>>
>>> Let me rephrase: for the purposes of this discussion an exact value
>>> is a real number that either terminates in some base or has a
>>> repetend in other (non-irrational) bases.
>>
>> So what you seem to be saying is that an exact value is a rational
>> number. That, somehow, irrational numbers are inexact. There is no
>> basis in modern maths for that last assertion. But for that rider -
>> "for the purposes of this discussion" shows that you wish to have a
>> discussion based on falsehood and superstition.
>
> That would be a fair conclusion: irrational numbers are inexact as it
> is impossible to evaluate them to infinite precision as infinite
> precision is meaningless.
>
> /Flibble
>

So, you AGREE that it shows you wish to have a discusion based on
falsehood and superstition?

On Sun, 5 Jun 2022 15:30:45 -0400
Richard Damon <Richard@Damon-Family.org> wrote:

> On 6/5/22 3:17 PM, Mr Flibble wrote:
> > On Sun, 5 Jun 2022 14:17:29 -0400
> > Richard Damon <Richard@Damon-Family.org> wrote:
> >
> >> On 6/5/22 1:22 PM, Mr Flibble wrote:
> >>> On Sun, 5 Jun 2022 13:05:09 -0400
> >>> Richard Damon <Richard@Damon-Family.org> wrote:
> >>>
> >>>> On 6/5/22 12:56 PM, Mr Flibble wrote:
> >>>>> On Sun, 5 Jun 2022 12:50:13 -0400
> >>>>> Richard Damon <Richard@Damon-Family.org> wrote:
> >>>>>
> >>>>>> On 6/5/22 12:28 PM, Mr Flibble wrote:
> >>>>>>> On Sun, 5 Jun 2022 12:22:45 -0400
> >>>>>>> Richard Damon <Richard@Damon-Family.org> wrote:
> >>>>>>>
> >>>>>>>> On 6/5/22 11:49 AM, Mr Flibble wrote:
> >>>>>>>>> On Sun, 5 Jun 2022 15:44:32 -0000 (UTC)
> >>>>>>>>> Alan Mackenzie <acm@muc.de> wrote:
> >>>>>>>>>
> >>>>>>>>>> Mr Flibble <flibble@reddwarf.jmc> wrote:
> >>>>>>>>>>> On Sun, 5 Jun 2022 16:28:05 +0100
> >>>>>>>>>>> Andy Walker <anw@cuboid.co.uk> wrote:
> >>>>>>>>>>
> >>>>>>>>>>>> On 05/06/2022 14:47, Mr Flibble wrote:
> >>>>>>>>>>>>> On Sun, 5 Jun 2022 07:58:42 -0400
> >>>>>>>>>>>>> Richard Damon <Richard@Damon-Family.org> wrote:
> >>>>>>>>>>>>>> [...] Sort of like how the number Pi has an
> >>>>>>>>>>>>>> exact value, but you can never actually express it
> >>>>>>>>>>>>>> (because it takes an infinite number of digits).
> >>>>>>>>>>>>> PI does not have an exact value; no irrational number
> >>>>>>>>>>>>> has an exact value.
> >>>>>>>>>>
> >>>>>>>>>>>> Of course "pi" has an exact value; as do [eg]
> >>>>>>>>>>>> "sqrt(2)", "e", and all the other computable real [and
> >>>>>>>>>>>> complex] numbers. Whether that value can be expressed in
> >>>>>>>>>>>> finite terms in some particular representation is quite
> >>>>>>>>>>>> another matter. That in turn depends on the
> >>>>>>>>>>>> representation; standard decimals is merely one [common]
> >>>>>>>>>>>> choice. Note that in symbolic computer systems, those
> >>>>>>>>>>>> computable reals are typically written "pi" [or
> >>>>>>>>>>>> whatever], and the computer works with that exactly, so
> >>>>>>>>>>>> that [eg] "sin^2 (pi/3) == 3/4", not 0.7499...; and also
> >>>>>>>>>>>> that in decimal-type notations most rationals equally
> >>>>>>>>>>>> have no terminating expansion. Numbers such as "pi" and
> >>>>>>>>>>>> "sqrt(2)" are not defined as decimal expansions but via
> >>>>>>>>>>>> their properties [eg that "sqrt(2)" is the unique
> >>>>>>>>>>>> positive real whose square is 2, or equivalently that it
> >>>>>>>>>>>> is the ratio of the diagonal of a square to its side,
> >>>>>>>>>>>> and "pi" is the least positive real whose sine is zero].
> >>>>>>>>>>>> Those properties are exact, and tell you all you ever
> >>>>>>>>>>>> need to know about those numbers.
> >>>>>>>>>>
> >>>>>>>>>> [ .... ]
> >>>>>>>>>>
> >>>>>>>>>>> What has decimal (base 10) expansion got to do with
> >>>>>>>>>>> anything? An irrational number has a non-terminating
> >>>>>>>>>>> sequence in ANY base. I am sorry but you are simply
> >>>>>>>>>>> mistaken: irrational numbers do NOT have an exact value;
> >>>>>>>>>>> this is obvious to anyone who understands logic and uses a
> >>>>>>>>>>> sane definition for infinity.
> >>>>>>>>>>
> >>>>>>>>>> That irrational numbers are exact values is clear to
> >>>>>>>>>> anybody with a degree in maths. Definitions of "infinity"
> >>>>>>>>>> (of which there are many) have nothing to do with this.
> >>>>>>>>>
> >>>>>>>>> You are wrong and fractally so so your degree in maths
> >>>>>>>>> appears to be worthless. An irrational number's sequence is
> >>>>>>>>> statistically random, has no fixed point on the number line
> >>>>>>>>> ergo has no exact representation. Any number with no exact
> >>>>>>>>> representation has, by definition, no exact value, only an
> >>>>>>>>> approximation. Infinity has everything to do with this as
> >>>>>>>>> an irrational's sequence ("digits") never terminates (i.e.
> >>>>>>>>> it is an INFINITELY long sequence).
> >>>>>>>>>
> >>>>>>>>> /Flibble
> >>>>>>>>>
> >>>>>>>>
> >>>>>>>> Nope. Irrational numbers DO have exact points on the number
> >>>>>>>> line.
> >>>>>>>>
> >>>>>>>> And what does representation have to do with exact value?
> >>>>>>>>
> >>>>>>>> Also, irrational numbers sequence of digits are not
> >>>>>>>> necessarily statistically random, in some representations,
> >>>>>>>> they can be VERY predictible for some numbers.
> >>>>>>>>
> >>>>>>>> One simple construction to show exact position, draw a box
> >>>>>>>> with sides exactly 1.
> >>>>>>>>
> >>>>>>>> Draw a line though opposite corners and make one point the
> >>>>>>>> value 0.
> >>>>>>>>
> >>>>>>>> The other corner will be EXACTLY at the point sqrt(2), so
> >>>>>>>> that irrational number has an exact point on the number line.
> >>>>>>>>
> >>>>>>>> You just don't understand what an exact value means, likely
> >>>>>>>> because you can't understand things that are somewhat
> >>>>>>>> abstract.
> >>>>>>>
> >>>>>>> An irrational number does not have an exact point on the
> >>>>>>> number line as it will move about as you "zoom in", you can
> >>>>>>> keep "zooming in" forever (i.e. infinitely) and it will keep
> >>>>>>> moving about because the number never terminates.
> >>>>>>>
> >>>>>>> If I couldn't understand things that are somewhat abstract
> >>>>>>> then I wouldn't have a computer science degree (BSc Hons) and
> >>>>>>> 30 years of industry experience.
> >>>>>>>
> >>>>>>> /Flibble
> >>>>>>>
> >>>>>>
> >>>>>> Then why do you think irrational numbers don't have an exact
> >>>>>> location?
> >>>>>>
> >>>>>> I know people with degrees (even with honors) and industry
> >>>>>> experiance that still show that they don't really understand
> >>>>>> what they are talking about.
> >>>>>>
> >>>>>> The "width" of the point representing the location of an
> >>>>>> irrational number is just as much "0" as that of a rational
> >>>>>> number, so specifies just as exact of a location.
> >>>>>>
> >>>>>> The fact that we can't write it in a rational base with a
> >>>>>> finite number of digits doesn't actally mean anything.
> >>>>>
> >>>>> 3.1415xxxxxxxxx (a)
> >>>>> 3.14159xxxxxxxx (b)
> >>>>>
> >>>>> (b) is nearer to 3.14160 than 3.14150 that (a) indicates hence
> >>>>> its value "moves about" as accuracy increases.
> >>>>>
> >>>>> /Flibble
> >>>>>
> >>>>
> >>>> But (a) and (b) aren't "pi"
> >>>
> >>> No, they are approximations of pi. The value of the approximation
> >>> changes up to a factor of 1/base as you evaluate it at increasing
> >>> accuracy.
> >>>
> >>>>
> >>>> All you are showing is that approximations to numbers get better
> >>>> as they get better, which is just a strange tautology.
> >>>
> >>> Yes and therefor they jump about on the number line as accuracy
> >>> increases.
> >>>
> >>>>
> >>>> The number PI, has only one precise value, the exact ratio of the
> >>>> circumference of a circle to its diameter on a flat plane (which
> >>>> will always be the same).
> >>>
> >>> pi cannot have a precise value as it neither terminates nor has a
> >>> repetend in any base.
> >>>
> >>>>
> >>>> The fact that it can't be expressed, isn't an issue on exactness,
> >>>> but of finite representation.
> >>>
> >>> We can only ever have a finite representation.
> >>>
> >>>>
> >>>> Note, that the set of numbers with finite representation is a
> >>>> countable set, so it isn't surprising that the uncountable
> >>>> infinity of the reals (that includes the irrationals) is not all
> >>>> finitely expressible.
> >>>
> >>> Stating the obvious.
> >>>
> >>> /Flibble
> >>>
> >>
> >> So you DO have problems with abstractions.
> >>
> >> The problem is, we don't actually need the finite numerical
> >> representation of a number if we have the definition of what the
> >> number is (which is another sort of abstract finite
> >> representation).
> >>
> >> The "finite representation" of pi is the ratio of the circumference
> >> to the diameter of a circle on a plane.
> >
> > No, instead if you use logic you must come to the conclusion that
> > there is no ratio of the circumference to the diameter of a circle:
> > rational numbers describe ratios, irrational numbers do not. QED.
> >
> > /Flibble
> >
>
> Then you don't understand what is a ratio. Rational numbers are the
> ratio of INTEGERS, not all ratios. Ratios define the relative
> magnatude of one number to another, ANY number.
>
> Since it is clear you don't understand the meaning of the basic
> terms, your OPINION about the weightier things becomes suspect.

On 6/5/2022 2:25 PM, Richard Damon wrote:
>
> On 6/5/22 3:09 PM, olcott wrote:
>> On 6/5/2022 1:54 PM, Richard Damon wrote:
>>> On 6/5/22 2:35 PM, olcott wrote:
>>>> On 6/5/2022 1:31 PM, Richard Damon wrote:
>>>>> On 6/5/22 1:02 PM, olcott wrote:
>>>>>> On 6/5/2022 11:59 AM, Richard Damon wrote:
>>>>>>> On 6/5/22 12:39 PM, olcott wrote:
>>>>>>>> On 6/5/2022 11:31 AM, Richard Damon wrote:
>>>>>>>>> On 6/5/22 11:57 AM, olcott wrote:
>>>>>>>>>> On 6/5/2022 7:38 AM, Ben wrote:
>>>>>>>>>>> Alan Mackenzie <acm@muc.de> writes:
>>>>>>>>>>>
>>>>>>>>>>>> olcott <NoOne@nowhere.com> wrote:
>>>>>>>>>>>>> On 6/5/2022 6:12 AM, Alan Mackenzie wrote:
>>>>>>>>>>>
>>>>>>>>>>>>>> ...  What would be the turing machine equivalent of an
>>>>>>>>>>>>>> "abnormal termination" in C?
>>>>>>>>>>>>
>>>>>>>>>>>>> An aborted simulation.
>>>>>>>>>>>>
>>>>>>>>>>>> There's no such thing on a turing machine.  It either runs
>>>>>>>>>>>> and halts, or
>>>>>>>>>>>> it runs forever.
>>>>>>>>>>>>
>>>>>>>>>>>> Your aborted simulation is just one final state of a turing
>>>>>>>>>>>> machine,
>>>>>>>>>>>> which has thus halted.
>>>>>>>>>>>
>>>>>>>>>>> A year ago I tried to get PO to accept a few basic facts
>>>>>>>>>>> about the
>>>>>>>>>>> topic.  One of these was
>>>>>>>>>>>
>>>>>>>>>>> (B) Every computation that halts, for whatever reason, is a
>>>>>>>>>>> halting
>>>>>>>>>>>      computation.
>>>>>>>>>>>
>>>>>>>>>>> After much ducking a diving, PO replied "OK".
>>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> Whatever I said years ago has been superseded by my current
>>>>>>>>>> understanding:
>>>>>>>>>>
>>>>>>>>>> Computation that halts ... the Turing machine will halt
>>>>>>>>>> whenever it enters a final state. (Linz:1990:234)
>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>
>>>>>>>>> And H^ applied to <H^> will Halt if H applied to <H^> <H^>
>>>>>>>>> rejects its input as non-halting, thus showing that the H was
>>>>>>>>> wrong.
>>>>>>>>
>>>>>>>> You are a brain dead moron on this point.
>>>>>>>>
>>>>>>>>
>>>>>>>
>>>>>>> Why, for speaking Truth that you can't counter?
>>>>>>
>>>>>> The actual truth is beyond your intellectual capacity on this point.
>>>>>>
>>>>>
>>>>> I cam say the same about you, but MY logic is actually correct, and
>>>>> veriviable by others.
>>>> Everyone that claims that the correctly simulated input to H(P,P)
>>>> reaches its final state is woefully incompetent even at basic
>>>> software engineering.
>>>>
>>>>
>>>
>>> Why do you say that?
>>>
>>> Since the DEFINITION of a correct simulation of something means it
>>> matchs the behavior of what it is simulating,
>>
>> This part is correct.
>>
>> _P()
>> [00001352](01)  55              push ebp
>> [00001353](02)  8bec            mov ebp,esp
>> [00001355](03)  8b4508          mov eax,[ebp+08]
>> [00001358](01)  50              push eax      // push P
>> [00001359](03)  8b4d08          mov ecx,[ebp+08]
>> [0000135c](01)  51              push ecx      // push P
>> [0000135d](05)  e840feffff      call 000011a2 // call H
>> [00001362](03)  83c408          add esp,+08
>> [00001365](02)  85c0            test eax,eax
>> [00001367](02)  7402            jz 0000136b
>> [00001369](02)  ebfe            jmp 00001369
>> [0000136b](01)  5d              pop ebp
>> [0000136c](01)  c3              ret
>> Size in bytes:(0027) [0000136c]
>>
>> It is completely obvious that when H(P,P) correctly emulates its input
>> that it must emulate the first seven instructions of P. Because the
>> seventh instruction repeats this process we can know with complete
>> certainty that the emulated P never reaches its final “ret”
>> instruction, thus never halts.
>>
>
> So, you can't find an error in the trace you snipped, that implies that
> you need to accept it as correct.
>
> Since you also claim that you can prove the exact opposite, what that
> proves is that when we add your defintions to the rules of Computation
> Theory, the field becomes inconsistent.
>
> That means that there must be an inconsistency in your rules.
>
> YOU FAIL.
>
> One of the problems is your statement:
>
>> Because the seventh instruction repeats this process
>
> Because that isn't what the seventh instruction does (at least not if H
> isn't just a call to its input).
>
> The seventh instruction starts NOT just a "Simulation of the input to
> H(P,P)" but a DECISION on whether or not this input will Halt. THAT
> decision is a conditional operation, and thus we now have the
> conditional in the loop that possibly breaks the infinite loop.

When-so-ever the halt decider must abort its simulation to prevent
infinite simulation it is always correct to reject its input as non
halting.

void Infinite_Loop()
{ HERE: goto HERE;
}

int main()
{ Output("Input_Halts = ", H0(Infinite_Loop));
}

_Infinite_Loop()
[00001342](01) 55 push ebp
[00001343](02) 8bec mov ebp,esp
[00001345](02) ebfe jmp 00001345
[00001347](01) 5d pop ebp
[00001348](01) c3 ret
Size in bytes:(0007) [00001348]

--
Copyright 2022 Pete Olcott

"Talent hits a target no one else can hit;
Genius hits a target no one else can see."
Arthur Schopenhauer

On Sun, 5 Jun 2022 15:32:01 -0400
Richard Damon <Richard@Damon-Family.org> wrote:

> On 6/5/22 3:19 PM, Mr Flibble wrote:
> > On Sun, 5 Jun 2022 18:07:46 -0000 (UTC)
> > Alan Mackenzie <acm@muc.de> wrote:
> >
> >> Mr Flibble <flibble@reddwarf.jmc> wrote:
> >>> On Sun, 5 Jun 2022 12:57:56 -0400
> >>> Richard Damon <Richard@Damon-Family.org> wrote:
> >>
> >> [ .... ]
> >>
> >>>> So 1/3 isn't an exact value?
> >>
> >>> 1/3 is 0.1 in base 3 so does have an exact value.
> >>
> >>> Let me rephrase: for the purposes of this discussion an exact
> >>> value is a real number that either terminates in some base or has
> >>> a repetend in other (non-irrational) bases.
> >>
> >> So what you seem to be saying is that an exact value is a rational
> >> number. That, somehow, irrational numbers are inexact. There is
> >> no basis in modern maths for that last assertion. But for that
> >> rider - "for the purposes of this discussion" shows that you wish
> >> to have a discussion based on falsehood and superstition.
> >
> > That would be a fair conclusion: irrational numbers are inexact as
> > it is impossible to evaluate them to infinite precision as infinite
> > precision is meaningless.
> >
> > /Flibble
> >
>
> So, you AGREE that it shows you wish to have a discusion based on
> falsehood and superstition?

I agree that irrational numbers are inexact.

/Flibble

On 6/5/22 3:18 PM, Mr Flibble wrote:
> On Sun, 5 Jun 2022 14:24:17 -0400
> Richard Damon <Richard@Damon-Family.org> wrote:
>
>> On 6/5/22 1:17 PM, Mr Flibble wrote:
>>> On Sun, 5 Jun 2022 12:57:56 -0400
>>> Richard Damon <Richard@Damon-Family.org> wrote:
>>>
>>>> On 6/5/22 12:37 PM, Mr Flibble wrote:
>>>>> On Sun, 5 Jun 2022 12:17:48 -0400
>>>>> Richard Damon <Richard@Damon-Family.org> wrote:
>>>>>
>>>>>> On 6/5/22 11:34 AM, Mr Flibble wrote:
>>>>>>> On Sun, 5 Jun 2022 16:28:05 +0100
>>>>>>> Andy Walker <anw@cuboid.co.uk> wrote:
>>>>>>>
>>>>>>>> On 05/06/2022 14:47, Mr Flibble wrote:
>>>>>>>>> On Sun, 5 Jun 2022 07:58:42 -0400
>>>>>>>>> Richard Damon <Richard@Damon-Family.org> wrote:
>>>>>>>>>> [...] Sort of like how the number Pi has an
>>>>>>>>>> exact value, but you can never actually express it (because
>>>>>>>>>> it takes an infinite number of digits).
>>>>>>>>> PI does not have an exact value; no irrational number has an
>>>>>>>>> exact value.
>>>>>>>>
>>>>>>>> Of course "pi" has an exact value; as do [eg]
>>>>>>>> "sqrt(2)", "e", and all the other computable real [and
>>>>>>>> complex] numbers. Whether that value can be expressed in
>>>>>>>> finite terms in some particular representation is quite
>>>>>>>> another matter. That in turn depends on the representation;
>>>>>>>> standard decimals is merely one [common] choice. Note that in
>>>>>>>> symbolic computer systems, those computable reals are
>>>>>>>> typically written "pi" [or whatever], and the computer works
>>>>>>>> with that exactly, so that [eg] "sin^2 (pi/3) == 3/4", not
>>>>>>>> 0.7499...; and also that in decimal-type notations most
>>>>>>>> rationals equally have no terminating expansion. Numbers such
>>>>>>>> as "pi" and "sqrt(2)" are not defined as decimal expansions
>>>>>>>> but via their properties [eg that "sqrt(2)" is the unique
>>>>>>>> positive real whose square is 2, or equivalently that it is
>>>>>>>> the ratio of the diagonal of a square to its side, and "pi" is
>>>>>>>> the least positive real whose sine is zero]. Those properties
>>>>>>>> are exact, and tell you all you ever need to know about those
>>>>>>>> numbers.
>>>>>>>>
>>>>>>>> [I have removed my name from the "Subject:"; I don't
>>>>>>>> know why anyone saw fit to attach it to this debate, such as it
>>>>>>>> is, on the HP.]
>>>>>>>
>>>>>>> What has decimal (base 10) expansion got to do with anything? An
>>>>>>> irrational number has a non-terminating sequence in ANY base. I
>>>>>>> am sorry but you are simply mistaken: irrational numbers do NOT
>>>>>>> have an exact value; this is obvious to anyone who understands
>>>>>>> logic and uses a sane definition for infinity.
>>>>>>>
>>>>>>> /Flibble
>>>>>>>
>>>>>>
>>>>>> How about in base pi? then it is the number 10
>>>>>
>>>>> how about base banana? then it is the number 10.
>>>>>
>>>>> PI, like banana, is just a symbol representing an irrational
>>>>> number that has no exact value. To use it here is circular and
>>>>> therefor erroneous.
>>>>>
>>>>>>
>>>>>> Base pi is an interesting base for some problems.
>>>>>>
>>>>>> What is your definition of "an exact value"?
>>>>>>
>>>>>> Maybe the problem is you don't quite understand the meaning of
>>>>>> that term.
>>>>>
>>>>> Of course I understand the fucking term. For the purposes of this
>>>>> discussion an exact value is a real number (non-integer) that
>>>>> terminates in a base that is not a multiple of itself.
>>>>>
>>>>> /Flibble
>>>>>
>>>>
>>>> Where do you get that definition from?
>>>>
>>>> So 1/3 isn't an exact value?
>>>
>>> 1/3 is 0.1 in base 3 so does have an exact value.
>>
>> And base 3 is a multiple of 1/3, which you said wasn't allowed.
>>
>>>
>>> Let me rephrase: for the purposes of this discussion an exact value
>>> is a real number that either terminates in some base or has a
>>> repetend in other (non-irrational) bases.
>>>
>>> /Flibble
>>>
>>
>> No, that is NOT a correct definition. That isn't a bad definition of
>> a RATIONAL number, as any number that can be written as a finite
>> string, or a string with a repetend can be also expressed as a ratio
>> of two numbers.
>>
>> There is nothing in the actual meaning of "exact value" that needs
>> the value to be expressible as a finite string of digits.
>
> You are wrong, and fractally so which is ironic given the topic under
> discussion.
>
> /Flibble
>

By what reference?

Exact means without approximation

Value, in this context, means the numerical amount.

The number pi, and sqrt(2) met that definition.

There is NO approximation in the definition of either value, pi is exact
the ratio of the circumferance and diameter of a circle on a plane (the
circumferance / diameter). This is an exact number.

They represent a numerical amount.

Thus, they meet the definition of an exact value.

You seem to be stuck in logic milleniums old where things that couldn't
be converted into counting numbers were beyond understanding.

Mr Flibble <flibble@reddwarf.jmc> wrote:
> On Sun, 5 Jun 2022 18:07:46 -0000 (UTC)
> Alan Mackenzie <acm@muc.de> wrote:

>> Mr Flibble <flibble@reddwarf.jmc> wrote:
>> > On Sun, 5 Jun 2022 12:57:56 -0400
>> > Richard Damon <Richard@Damon-Family.org> wrote:

>> [ .... ]

>> >> So 1/3 isn't an exact value?

>> > 1/3 is 0.1 in base 3 so does have an exact value.

>> > Let me rephrase: for the purposes of this discussion an exact value
>> > is a real number that either terminates in some base or has a
>> > repetend in other (non-irrational) bases.

>> So what you seem to be saying is that an exact value is a rational
>> number. That, somehow, irrational numbers are inexact. There is no
>> basis in modern maths for that last assertion. But for that rider -
>> "for the purposes of this discussion" shows that you wish to have a
>> discussion based on falsehood and superstition.

> That would be a fair conclusion: irrational numbers are inexact ....

Wrong. Numbers just are. There is no such thing as an "inexact number".

> .... as it is impossible to evaluate them ....

It is indeed. One evaluates expressions, not numbers. Think about it,
how can you possibly evaluate 2?

> .... to infinite precision as infinite precision is meaningless.

Numbers don't have precision, any more than they have the colour blue.
Approximations have precision. You're confusing numbers with
approximations to them.

> /Flibble

--
Alan Mackenzie (Nuremberg, Germany).

On 6/5/22 3:33 PM, olcott wrote:
> On 6/5/2022 2:25 PM, Richard Damon wrote:
>>
>> On 6/5/22 3:09 PM, olcott wrote:
>>> On 6/5/2022 1:54 PM, Richard Damon wrote:
>>>> On 6/5/22 2:35 PM, olcott wrote:
>>>>> On 6/5/2022 1:31 PM, Richard Damon wrote:
>>>>>> On 6/5/22 1:02 PM, olcott wrote:
>>>>>>> On 6/5/2022 11:59 AM, Richard Damon wrote:
>>>>>>>> On 6/5/22 12:39 PM, olcott wrote:
>>>>>>>>> On 6/5/2022 11:31 AM, Richard Damon wrote:
>>>>>>>>>> On 6/5/22 11:57 AM, olcott wrote:
>>>>>>>>>>> On 6/5/2022 7:38 AM, Ben wrote:
>>>>>>>>>>>> Alan Mackenzie <acm@muc.de> writes:
>>>>>>>>>>>>
>>>>>>>>>>>>> olcott <NoOne@nowhere.com> wrote:
>>>>>>>>>>>>>> On 6/5/2022 6:12 AM, Alan Mackenzie wrote:
>>>>>>>>>>>>
>>>>>>>>>>>>>>> ...  What would be the turing machine equivalent of an
>>>>>>>>>>>>>>> "abnormal termination" in C?
>>>>>>>>>>>>>
>>>>>>>>>>>>>> An aborted simulation.
>>>>>>>>>>>>>
>>>>>>>>>>>>> There's no such thing on a turing machine.  It either runs
>>>>>>>>>>>>> and halts, or
>>>>>>>>>>>>> it runs forever.
>>>>>>>>>>>>>
>>>>>>>>>>>>> Your aborted simulation is just one final state of a turing
>>>>>>>>>>>>> machine,
>>>>>>>>>>>>> which has thus halted.
>>>>>>>>>>>>
>>>>>>>>>>>> A year ago I tried to get PO to accept a few basic facts
>>>>>>>>>>>> about the
>>>>>>>>>>>> topic.  One of these was
>>>>>>>>>>>>
>>>>>>>>>>>> (B) Every computation that halts, for whatever reason, is a
>>>>>>>>>>>> halting
>>>>>>>>>>>>      computation.
>>>>>>>>>>>>
>>>>>>>>>>>> After much ducking a diving, PO replied "OK".
>>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>> Whatever I said years ago has been superseded by my current
>>>>>>>>>>> understanding:
>>>>>>>>>>>
>>>>>>>>>>> Computation that halts ... the Turing machine will halt
>>>>>>>>>>> whenever it enters a final state. (Linz:1990:234)
>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> And H^ applied to <H^> will Halt if H applied to <H^> <H^>
>>>>>>>>>> rejects its input as non-halting, thus showing that the H was
>>>>>>>>>> wrong.
>>>>>>>>>
>>>>>>>>> You are a brain dead moron on this point.
>>>>>>>>>
>>>>>>>>>
>>>>>>>>
>>>>>>>> Why, for speaking Truth that you can't counter?
>>>>>>>
>>>>>>> The actual truth is beyond your intellectual capacity on this point.
>>>>>>>
>>>>>>
>>>>>> I cam say the same about you, but MY logic is actually correct,
>>>>>> and veriviable by others.
>>>>> Everyone that claims that the correctly simulated input to H(P,P)
>>>>> reaches its final state is woefully incompetent even at basic
>>>>> software engineering.
>>>>>
>>>>>
>>>>
>>>> Why do you say that?
>>>>
>>>> Since the DEFINITION of a correct simulation of something means it
>>>> matchs the behavior of what it is simulating,
>>>
>>> This part is correct.
>>>
>>> _P()
>>> [00001352](01)  55              push ebp
>>> [00001353](02)  8bec            mov ebp,esp
>>> [00001355](03)  8b4508          mov eax,[ebp+08]
>>> [00001358](01)  50              push eax      // push P
>>> [00001359](03)  8b4d08          mov ecx,[ebp+08]
>>> [0000135c](01)  51              push ecx      // push P
>>> [0000135d](05)  e840feffff      call 000011a2 // call H
>>> [00001362](03)  83c408          add esp,+08
>>> [00001365](02)  85c0            test eax,eax
>>> [00001367](02)  7402            jz 0000136b
>>> [00001369](02)  ebfe            jmp 00001369
>>> [0000136b](01)  5d              pop ebp
>>> [0000136c](01)  c3              ret
>>> Size in bytes:(0027) [0000136c]
>>>
>>> It is completely obvious that when H(P,P) correctly emulates its
>>> input that it must emulate the first seven instructions of P. Because
>>> the seventh instruction repeats this process we can know with
>>> complete certainty that the emulated P never reaches its final “ret”
>>> instruction, thus never halts.
>>>
>>
>> So, you can't find an error in the trace you snipped, that implies
>> that you need to accept it as correct.
>>
>> Since you also claim that you can prove the exact opposite, what that
>> proves is that when we add your defintions to the rules of Computation
>> Theory, the field becomes inconsistent.
>>
>> That means that there must be an inconsistency in your rules.
>>
>> YOU FAIL.
>>
>> One of the problems is your statement:
>>
>>> Because the seventh instruction repeats this process
>>
>> Because that isn't what the seventh instruction does (at least not if
>> H isn't just a call to its input).
>>
>> The seventh instruction starts NOT just a "Simulation of the input to
>> H(P,P)" but a DECISION on whether or not this input will Halt. THAT
>> decision is a conditional operation, and thus we now have the
>> conditional in the loop that possibly breaks the infinite loop.
>
> When-so-ever the halt decider must abort its simulation to prevent
> infinite simulation it is always correct to reject its input as non
> halting.

Where do you get that from.

>
> void Infinite_Loop()
> {
>   HERE: goto HERE;
> }
>
> int main()
> {
>   Output("Input_Halts = ", H0(Infinite_Loop));
> }
>
> _Infinite_Loop()
> [00001342](01)  55              push ebp
> [00001343](02)  8bec            mov ebp,esp
> [00001345](02)  ebfe            jmp 00001345
> [00001347](01)  5d              pop ebp
> [00001348](01)  c3              ret
> Size in bytes:(0007) [00001348]
>
>

Fallicy of proof by example.

By your definition, ALL computations must be accepted to be non-halting,
as a halt decider can immediately abort reasoning just as validly that
its input is non-halting.

Remember, your H uses unsound logic, so you can't disallow it. Remember
your logic in H is that If H never aborts its simulation, then the input
is non-halting, but H DOES abort its simulation, so the premise of your
logic isn't true.

On Sun, 5 Jun 2022 15:38:50 -0400
Richard Damon <Richard@Damon-Family.org> wrote:

> On 6/5/22 3:18 PM, Mr Flibble wrote:
> > On Sun, 5 Jun 2022 14:24:17 -0400
> > Richard Damon <Richard@Damon-Family.org> wrote:
> >
> >> On 6/5/22 1:17 PM, Mr Flibble wrote:
> >>> On Sun, 5 Jun 2022 12:57:56 -0400
> >>> Richard Damon <Richard@Damon-Family.org> wrote:
> >>>
> >>>> On 6/5/22 12:37 PM, Mr Flibble wrote:
> >>>>> On Sun, 5 Jun 2022 12:17:48 -0400
> >>>>> Richard Damon <Richard@Damon-Family.org> wrote:
> >>>>>
> >>>>>> On 6/5/22 11:34 AM, Mr Flibble wrote:
> >>>>>>> On Sun, 5 Jun 2022 16:28:05 +0100
> >>>>>>> Andy Walker <anw@cuboid.co.uk> wrote:
> >>>>>>>
> >>>>>>>> On 05/06/2022 14:47, Mr Flibble wrote:
> >>>>>>>>> On Sun, 5 Jun 2022 07:58:42 -0400
> >>>>>>>>> Richard Damon <Richard@Damon-Family.org> wrote:
> >>>>>>>>>> [...] Sort of like how the number Pi has an
> >>>>>>>>>> exact value, but you can never actually express it (because
> >>>>>>>>>> it takes an infinite number of digits).
> >>>>>>>>> PI does not have an exact value; no irrational number has an
> >>>>>>>>> exact value.
> >>>>>>>>
> >>>>>>>> Of course "pi" has an exact value; as do [eg]
> >>>>>>>> "sqrt(2)", "e", and all the other computable real [and
> >>>>>>>> complex] numbers. Whether that value can be expressed in
> >>>>>>>> finite terms in some particular representation is quite
> >>>>>>>> another matter. That in turn depends on the representation;
> >>>>>>>> standard decimals is merely one [common] choice. Note that
> >>>>>>>> in symbolic computer systems, those computable reals are
> >>>>>>>> typically written "pi" [or whatever], and the computer works
> >>>>>>>> with that exactly, so that [eg] "sin^2 (pi/3) == 3/4", not
> >>>>>>>> 0.7499...; and also that in decimal-type notations most
> >>>>>>>> rationals equally have no terminating expansion. Numbers
> >>>>>>>> such as "pi" and "sqrt(2)" are not defined as decimal
> >>>>>>>> expansions but via their properties [eg that "sqrt(2)" is
> >>>>>>>> the unique positive real whose square is 2, or equivalently
> >>>>>>>> that it is the ratio of the diagonal of a square to its
> >>>>>>>> side, and "pi" is the least positive real whose sine is
> >>>>>>>> zero]. Those properties are exact, and tell you all you
> >>>>>>>> ever need to know about those numbers.
> >>>>>>>>
> >>>>>>>> [I have removed my name from the "Subject:"; I don't
> >>>>>>>> know why anyone saw fit to attach it to this debate, such as
> >>>>>>>> it is, on the HP.]
> >>>>>>>
> >>>>>>> What has decimal (base 10) expansion got to do with anything?
> >>>>>>> An irrational number has a non-terminating sequence in ANY
> >>>>>>> base. I am sorry but you are simply mistaken: irrational
> >>>>>>> numbers do NOT have an exact value; this is obvious to anyone
> >>>>>>> who understands logic and uses a sane definition for infinity.
> >>>>>>>
> >>>>>>> /Flibble
> >>>>>>>
> >>>>>>
> >>>>>> How about in base pi? then it is the number 10
> >>>>>
> >>>>> how about base banana? then it is the number 10.
> >>>>>
> >>>>> PI, like banana, is just a symbol representing an irrational
> >>>>> number that has no exact value. To use it here is circular and
> >>>>> therefor erroneous.
> >>>>>
> >>>>>>
> >>>>>> Base pi is an interesting base for some problems.
> >>>>>>
> >>>>>> What is your definition of "an exact value"?
> >>>>>>
> >>>>>> Maybe the problem is you don't quite understand the meaning of
> >>>>>> that term.
> >>>>>
> >>>>> Of course I understand the fucking term. For the purposes of
> >>>>> this discussion an exact value is a real number (non-integer)
> >>>>> that terminates in a base that is not a multiple of itself.
> >>>>>
> >>>>> /Flibble
> >>>>>
> >>>>
> >>>> Where do you get that definition from?
> >>>>
> >>>> So 1/3 isn't an exact value?
> >>>
> >>> 1/3 is 0.1 in base 3 so does have an exact value.
> >>
> >> And base 3 is a multiple of 1/3, which you said wasn't allowed.
> >>
> >>>
> >>> Let me rephrase: for the purposes of this discussion an exact
> >>> value is a real number that either terminates in some base or has
> >>> a repetend in other (non-irrational) bases.
> >>>
> >>> /Flibble
> >>>
> >>
> >> No, that is NOT a correct definition. That isn't a bad definition
> >> of a RATIONAL number, as any number that can be written as a finite
> >> string, or a string with a repetend can be also expressed as a
> >> ratio of two numbers.
> >>
> >> There is nothing in the actual meaning of "exact value" that needs
> >> the value to be expressible as a finite string of digits.
> >
> > You are wrong, and fractally so which is ironic given the topic
> > under discussion.
> >
> > /Flibble
> >
>
> By what reference?
>
> Exact means without approximation
>
> Value, in this context, means the numerical amount.
>
> The number pi, and sqrt(2) met that definition.
>
> There is NO approximation in the definition of either value, pi is
> exact the ratio of the circumferance and diameter of a circle on a
> plane (the circumferance / diameter). This is an exact number.
>
> They represent a numerical amount.
>
> Thus, they meet the definition of an exact value.
>
> You seem to be stuck in logic milleniums old where things that
> couldn't be converted into counting numbers were beyond understanding.

You have yet to prove to me that there is an exact ratio between the
circumference and diameter of a circle: all you have done is a lot of
hand waving.

/Flibble

On 6/5/22 3:33 PM, Mr Flibble wrote:
> On Sun, 5 Jun 2022 15:30:45 -0400
> Richard Damon <Richard@Damon-Family.org> wrote:
>
>> On 6/5/22 3:17 PM, Mr Flibble wrote:
>>> On Sun, 5 Jun 2022 14:17:29 -0400
>>> Richard Damon <Richard@Damon-Family.org> wrote:
>>>
>>>> On 6/5/22 1:22 PM, Mr Flibble wrote:
>>>>> On Sun, 5 Jun 2022 13:05:09 -0400
>>>>> Richard Damon <Richard@Damon-Family.org> wrote:
>>>>>
>>>>>> On 6/5/22 12:56 PM, Mr Flibble wrote:
>>>>>>> On Sun, 5 Jun 2022 12:50:13 -0400
>>>>>>> Richard Damon <Richard@Damon-Family.org> wrote:
>>>>>>>
>>>>>>>> On 6/5/22 12:28 PM, Mr Flibble wrote:
>>>>>>>>> On Sun, 5 Jun 2022 12:22:45 -0400
>>>>>>>>> Richard Damon <Richard@Damon-Family.org> wrote:
>>>>>>>>>
>>>>>>>>>> On 6/5/22 11:49 AM, Mr Flibble wrote:
>>>>>>>>>>> On Sun, 5 Jun 2022 15:44:32 -0000 (UTC)
>>>>>>>>>>> Alan Mackenzie <acm@muc.de> wrote:
>>>>>>>>>>>
>>>>>>>>>>>> Mr Flibble <flibble@reddwarf.jmc> wrote:
>>>>>>>>>>>>> On Sun, 5 Jun 2022 16:28:05 +0100
>>>>>>>>>>>>> Andy Walker <anw@cuboid.co.uk> wrote:
>>>>>>>>>>>>
>>>>>>>>>>>>>> On 05/06/2022 14:47, Mr Flibble wrote:
>>>>>>>>>>>>>>> On Sun, 5 Jun 2022 07:58:42 -0400
>>>>>>>>>>>>>>> Richard Damon <Richard@Damon-Family.org> wrote:
>>>>>>>>>>>>>>>> [...] Sort of like how the number Pi has an
>>>>>>>>>>>>>>>> exact value, but you can never actually express it
>>>>>>>>>>>>>>>> (because it takes an infinite number of digits).
>>>>>>>>>>>>>>> PI does not have an exact value; no irrational number
>>>>>>>>>>>>>>> has an exact value.
>>>>>>>>>>>>
>>>>>>>>>>>>>> Of course "pi" has an exact value; as do [eg]
>>>>>>>>>>>>>> "sqrt(2)", "e", and all the other computable real [and
>>>>>>>>>>>>>> complex] numbers. Whether that value can be expressed in
>>>>>>>>>>>>>> finite terms in some particular representation is quite
>>>>>>>>>>>>>> another matter. That in turn depends on the
>>>>>>>>>>>>>> representation; standard decimals is merely one [common]
>>>>>>>>>>>>>> choice. Note that in symbolic computer systems, those
>>>>>>>>>>>>>> computable reals are typically written "pi" [or
>>>>>>>>>>>>>> whatever], and the computer works with that exactly, so
>>>>>>>>>>>>>> that [eg] "sin^2 (pi/3) == 3/4", not 0.7499...; and also
>>>>>>>>>>>>>> that in decimal-type notations most rationals equally
>>>>>>>>>>>>>> have no terminating expansion. Numbers such as "pi" and
>>>>>>>>>>>>>> "sqrt(2)" are not defined as decimal expansions but via
>>>>>>>>>>>>>> their properties [eg that "sqrt(2)" is the unique
>>>>>>>>>>>>>> positive real whose square is 2, or equivalently that it
>>>>>>>>>>>>>> is the ratio of the diagonal of a square to its side,
>>>>>>>>>>>>>> and "pi" is the least positive real whose sine is zero].
>>>>>>>>>>>>>> Those properties are exact, and tell you all you ever
>>>>>>>>>>>>>> need to know about those numbers.
>>>>>>>>>>>>
>>>>>>>>>>>> [ .... ]
>>>>>>>>>>>>
>>>>>>>>>>>>> What has decimal (base 10) expansion got to do with
>>>>>>>>>>>>> anything? An irrational number has a non-terminating
>>>>>>>>>>>>> sequence in ANY base. I am sorry but you are simply
>>>>>>>>>>>>> mistaken: irrational numbers do NOT have an exact value;
>>>>>>>>>>>>> this is obvious to anyone who understands logic and uses a
>>>>>>>>>>>>> sane definition for infinity.
>>>>>>>>>>>>
>>>>>>>>>>>> That irrational numbers are exact values is clear to
>>>>>>>>>>>> anybody with a degree in maths. Definitions of "infinity"
>>>>>>>>>>>> (of which there are many) have nothing to do with this.
>>>>>>>>>>>
>>>>>>>>>>> You are wrong and fractally so so your degree in maths
>>>>>>>>>>> appears to be worthless. An irrational number's sequence is
>>>>>>>>>>> statistically random, has no fixed point on the number line
>>>>>>>>>>> ergo has no exact representation. Any number with no exact
>>>>>>>>>>> representation has, by definition, no exact value, only an
>>>>>>>>>>> approximation. Infinity has everything to do with this as
>>>>>>>>>>> an irrational's sequence ("digits") never terminates (i.e.
>>>>>>>>>>> it is an INFINITELY long sequence).
>>>>>>>>>>>
>>>>>>>>>>> /Flibble
>>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> Nope. Irrational numbers DO have exact points on the number
>>>>>>>>>> line.
>>>>>>>>>>
>>>>>>>>>> And what does representation have to do with exact value?
>>>>>>>>>>
>>>>>>>>>> Also, irrational numbers sequence of digits are not
>>>>>>>>>> necessarily statistically random, in some representations,
>>>>>>>>>> they can be VERY predictible for some numbers.
>>>>>>>>>>
>>>>>>>>>> One simple construction to show exact position, draw a box
>>>>>>>>>> with sides exactly 1.
>>>>>>>>>>
>>>>>>>>>> Draw a line though opposite corners and make one point the
>>>>>>>>>> value 0.
>>>>>>>>>>
>>>>>>>>>> The other corner will be EXACTLY at the point sqrt(2), so
>>>>>>>>>> that irrational number has an exact point on the number line.
>>>>>>>>>>
>>>>>>>>>> You just don't understand what an exact value means, likely
>>>>>>>>>> because you can't understand things that are somewhat
>>>>>>>>>> abstract.
>>>>>>>>>
>>>>>>>>> An irrational number does not have an exact point on the
>>>>>>>>> number line as it will move about as you "zoom in", you can
>>>>>>>>> keep "zooming in" forever (i.e. infinitely) and it will keep
>>>>>>>>> moving about because the number never terminates.
>>>>>>>>>
>>>>>>>>> If I couldn't understand things that are somewhat abstract
>>>>>>>>> then I wouldn't have a computer science degree (BSc Hons) and
>>>>>>>>> 30 years of industry experience.
>>>>>>>>>
>>>>>>>>> /Flibble
>>>>>>>>>
>>>>>>>>
>>>>>>>> Then why do you think irrational numbers don't have an exact
>>>>>>>> location?
>>>>>>>>
>>>>>>>> I know people with degrees (even with honors) and industry
>>>>>>>> experiance that still show that they don't really understand
>>>>>>>> what they are talking about.
>>>>>>>>
>>>>>>>> The "width" of the point representing the location of an
>>>>>>>> irrational number is just as much "0" as that of a rational
>>>>>>>> number, so specifies just as exact of a location.
>>>>>>>>
>>>>>>>> The fact that we can't write it in a rational base with a
>>>>>>>> finite number of digits doesn't actally mean anything.
>>>>>>>
>>>>>>> 3.1415xxxxxxxxx (a)
>>>>>>> 3.14159xxxxxxxx (b)
>>>>>>>
>>>>>>> (b) is nearer to 3.14160 than 3.14150 that (a) indicates hence
>>>>>>> its value "moves about" as accuracy increases.
>>>>>>>
>>>>>>> /Flibble
>>>>>>>
>>>>>>
>>>>>> But (a) and (b) aren't "pi"
>>>>>
>>>>> No, they are approximations of pi. The value of the approximation
>>>>> changes up to a factor of 1/base as you evaluate it at increasing
>>>>> accuracy.
>>>>>
>>>>>>
>>>>>> All you are showing is that approximations to numbers get better
>>>>>> as they get better, which is just a strange tautology.
>>>>>
>>>>> Yes and therefor they jump about on the number line as accuracy
>>>>> increases.
>>>>>
>>>>>>
>>>>>> The number PI, has only one precise value, the exact ratio of the
>>>>>> circumference of a circle to its diameter on a flat plane (which
>>>>>> will always be the same).
>>>>>
>>>>> pi cannot have a precise value as it neither terminates nor has a
>>>>> repetend in any base.
>>>>>
>>>>>>
>>>>>> The fact that it can't be expressed, isn't an issue on exactness,
>>>>>> but of finite representation.
>>>>>
>>>>> We can only ever have a finite representation.
>>>>>
>>>>>>
>>>>>> Note, that the set of numbers with finite representation is a
>>>>>> countable set, so it isn't surprising that the uncountable
>>>>>> infinity of the reals (that includes the irrationals) is not all
>>>>>> finitely expressible.
>>>>>
>>>>> Stating the obvious.
>>>>>
>>>>> /Flibble
>>>>>
>>>>
>>>> So you DO have problems with abstractions.
>>>>
>>>> The problem is, we don't actually need the finite numerical
>>>> representation of a number if we have the definition of what the
>>>> number is (which is another sort of abstract finite
>>>> representation).
>>>>
>>>> The "finite representation" of pi is the ratio of the circumference
>>>> to the diameter of a circle on a plane.
>>>
>>> No, instead if you use logic you must come to the conclusion that
>>> there is no ratio of the circumference to the diameter of a circle:
>>> rational numbers describe ratios, irrational numbers do not. QED.
>>>
>>> /Flibble
>>>
>>
>> Then you don't understand what is a ratio. Rational numbers are the
>> ratio of INTEGERS, not all ratios. Ratios define the relative
>> magnatude of one number to another, ANY number.
>>
>> Since it is clear you don't understand the meaning of the basic
>> terms, your OPINION about the weightier things becomes suspect.
>
> Prove that there is a ratio between the circumference and diameter of a
> circle. Hint: you can't.
>
> /Flibble
>

On 6/5/22 3:34 PM, Mr Flibble wrote:
> On Sun, 5 Jun 2022 15:32:01 -0400
> Richard Damon <Richard@Damon-Family.org> wrote:
>
>> On 6/5/22 3:19 PM, Mr Flibble wrote:
>>> On Sun, 5 Jun 2022 18:07:46 -0000 (UTC)
>>> Alan Mackenzie <acm@muc.de> wrote:
>>>
>>>> Mr Flibble <flibble@reddwarf.jmc> wrote:
>>>>> On Sun, 5 Jun 2022 12:57:56 -0400
>>>>> Richard Damon <Richard@Damon-Family.org> wrote:
>>>>
>>>> [ .... ]
>>>>
>>>>>> So 1/3 isn't an exact value?
>>>>
>>>>> 1/3 is 0.1 in base 3 so does have an exact value.
>>>>
>>>>> Let me rephrase: for the purposes of this discussion an exact
>>>>> value is a real number that either terminates in some base or has
>>>>> a repetend in other (non-irrational) bases.
>>>>
>>>> So what you seem to be saying is that an exact value is a rational
>>>> number. That, somehow, irrational numbers are inexact. There is
>>>> no basis in modern maths for that last assertion. But for that
>>>> rider - "for the purposes of this discussion" shows that you wish
>>>> to have a discussion based on falsehood and superstition.
>>>
>>> That would be a fair conclusion: irrational numbers are inexact as
>>> it is impossible to evaluate them to infinite precision as infinite
>>> precision is meaningless.
>>>
>>> /Flibble
>>>
>>
>> So, you AGREE that it shows you wish to have a discusion based on
>> falsehood and superstition?
>
> I agree that irrational numbers are inexact.
>
> /Flibble
>

Which has been proved FALSE.

What is inexact about them.

You are just stuck in a circular set of incorrect definitions.

Probably because you can't understand the abstractions needed to move
from the rationals to the reals.

On 6/5/22 3:44 PM, Mr Flibble wrote:
> On Sun, 5 Jun 2022 15:38:50 -0400
> Richard Damon <Richard@Damon-Family.org> wrote:
>
>> On 6/5/22 3:18 PM, Mr Flibble wrote:
>>> On Sun, 5 Jun 2022 14:24:17 -0400
>>> Richard Damon <Richard@Damon-Family.org> wrote:
>>>
>>>> On 6/5/22 1:17 PM, Mr Flibble wrote:
>>>>> On Sun, 5 Jun 2022 12:57:56 -0400
>>>>> Richard Damon <Richard@Damon-Family.org> wrote:
>>>>>
>>>>>> On 6/5/22 12:37 PM, Mr Flibble wrote:
>>>>>>> On Sun, 5 Jun 2022 12:17:48 -0400
>>>>>>> Richard Damon <Richard@Damon-Family.org> wrote:
>>>>>>>
>>>>>>>> On 6/5/22 11:34 AM, Mr Flibble wrote:
>>>>>>>>> On Sun, 5 Jun 2022 16:28:05 +0100
>>>>>>>>> Andy Walker <anw@cuboid.co.uk> wrote:
>>>>>>>>>
>>>>>>>>>> On 05/06/2022 14:47, Mr Flibble wrote:
>>>>>>>>>>> On Sun, 5 Jun 2022 07:58:42 -0400
>>>>>>>>>>> Richard Damon <Richard@Damon-Family.org> wrote:
>>>>>>>>>>>> [...] Sort of like how the number Pi has an
>>>>>>>>>>>> exact value, but you can never actually express it (because
>>>>>>>>>>>> it takes an infinite number of digits).
>>>>>>>>>>> PI does not have an exact value; no irrational number has an
>>>>>>>>>>> exact value.
>>>>>>>>>>
>>>>>>>>>> Of course "pi" has an exact value; as do [eg]
>>>>>>>>>> "sqrt(2)", "e", and all the other computable real [and
>>>>>>>>>> complex] numbers. Whether that value can be expressed in
>>>>>>>>>> finite terms in some particular representation is quite
>>>>>>>>>> another matter. That in turn depends on the representation;
>>>>>>>>>> standard decimals is merely one [common] choice. Note that
>>>>>>>>>> in symbolic computer systems, those computable reals are
>>>>>>>>>> typically written "pi" [or whatever], and the computer works
>>>>>>>>>> with that exactly, so that [eg] "sin^2 (pi/3) == 3/4", not
>>>>>>>>>> 0.7499...; and also that in decimal-type notations most
>>>>>>>>>> rationals equally have no terminating expansion. Numbers
>>>>>>>>>> such as "pi" and "sqrt(2)" are not defined as decimal
>>>>>>>>>> expansions but via their properties [eg that "sqrt(2)" is
>>>>>>>>>> the unique positive real whose square is 2, or equivalently
>>>>>>>>>> that it is the ratio of the diagonal of a square to its
>>>>>>>>>> side, and "pi" is the least positive real whose sine is
>>>>>>>>>> zero]. Those properties are exact, and tell you all you
>>>>>>>>>> ever need to know about those numbers.
>>>>>>>>>>
>>>>>>>>>> [I have removed my name from the "Subject:"; I don't
>>>>>>>>>> know why anyone saw fit to attach it to this debate, such as
>>>>>>>>>> it is, on the HP.]
>>>>>>>>>
>>>>>>>>> What has decimal (base 10) expansion got to do with anything?
>>>>>>>>> An irrational number has a non-terminating sequence in ANY
>>>>>>>>> base. I am sorry but you are simply mistaken: irrational
>>>>>>>>> numbers do NOT have an exact value; this is obvious to anyone
>>>>>>>>> who understands logic and uses a sane definition for infinity.
>>>>>>>>>
>>>>>>>>> /Flibble
>>>>>>>>>
>>>>>>>>
>>>>>>>> How about in base pi? then it is the number 10
>>>>>>>
>>>>>>> how about base banana? then it is the number 10.
>>>>>>>
>>>>>>> PI, like banana, is just a symbol representing an irrational
>>>>>>> number that has no exact value. To use it here is circular and
>>>>>>> therefor erroneous.
>>>>>>>
>>>>>>>>
>>>>>>>> Base pi is an interesting base for some problems.
>>>>>>>>
>>>>>>>> What is your definition of "an exact value"?
>>>>>>>>
>>>>>>>> Maybe the problem is you don't quite understand the meaning of
>>>>>>>> that term.
>>>>>>>
>>>>>>> Of course I understand the fucking term. For the purposes of
>>>>>>> this discussion an exact value is a real number (non-integer)
>>>>>>> that terminates in a base that is not a multiple of itself.
>>>>>>>
>>>>>>> /Flibble
>>>>>>>
>>>>>>
>>>>>> Where do you get that definition from?
>>>>>>
>>>>>> So 1/3 isn't an exact value?
>>>>>
>>>>> 1/3 is 0.1 in base 3 so does have an exact value.
>>>>
>>>> And base 3 is a multiple of 1/3, which you said wasn't allowed.
>>>>
>>>>>
>>>>> Let me rephrase: for the purposes of this discussion an exact
>>>>> value is a real number that either terminates in some base or has
>>>>> a repetend in other (non-irrational) bases.
>>>>>
>>>>> /Flibble
>>>>>
>>>>
>>>> No, that is NOT a correct definition. That isn't a bad definition
>>>> of a RATIONAL number, as any number that can be written as a finite
>>>> string, or a string with a repetend can be also expressed as a
>>>> ratio of two numbers.
>>>>
>>>> There is nothing in the actual meaning of "exact value" that needs
>>>> the value to be expressible as a finite string of digits.
>>>
>>> You are wrong, and fractally so which is ironic given the topic
>>> under discussion.
>>>
>>> /Flibble
>>>
>>
>> By what reference?
>>
>> Exact means without approximation
>>
>> Value, in this context, means the numerical amount.
>>
>> The number pi, and sqrt(2) met that definition.
>>
>> There is NO approximation in the definition of either value, pi is
>> exact the ratio of the circumferance and diameter of a circle on a
>> plane (the circumferance / diameter). This is an exact number.
>>
>> They represent a numerical amount.
>>
>> Thus, they meet the definition of an exact value.
>>
>> You seem to be stuck in logic milleniums old where things that
>> couldn't be converted into counting numbers were beyond understanding.
>
> You have yet to prove to me that there is an exact ratio between the
> circumference and diameter of a circle: all you have done is a lot of
> hand waving.
>
> /Flibble
>

Is your problem that you don't beleive that all circles as similar (and
thus to the same ratio), or that this result is a number?

Both of these are ancient proofs.

On Sun, 5 Jun 2022 15:47:30 -0400
Richard Damon <Richard@Damon-Family.org> wrote:

> On 6/5/22 3:33 PM, Mr Flibble wrote:
> > On Sun, 5 Jun 2022 15:30:45 -0400
> > Richard Damon <Richard@Damon-Family.org> wrote:
> >
> >> On 6/5/22 3:17 PM, Mr Flibble wrote:
> >>> On Sun, 5 Jun 2022 14:17:29 -0400
> >>> Richard Damon <Richard@Damon-Family.org> wrote:
> >>>
> >>>> On 6/5/22 1:22 PM, Mr Flibble wrote:
> >>>>> On Sun, 5 Jun 2022 13:05:09 -0400
> >>>>> Richard Damon <Richard@Damon-Family.org> wrote:
> >>>>>
> >>>>>> On 6/5/22 12:56 PM, Mr Flibble wrote:
> >>>>>>> On Sun, 5 Jun 2022 12:50:13 -0400
> >>>>>>> Richard Damon <Richard@Damon-Family.org> wrote:
> >>>>>>>
> >>>>>>>> On 6/5/22 12:28 PM, Mr Flibble wrote:
> >>>>>>>>> On Sun, 5 Jun 2022 12:22:45 -0400
> >>>>>>>>> Richard Damon <Richard@Damon-Family.org> wrote:
> >>>>>>>>>
> >>>>>>>>>> On 6/5/22 11:49 AM, Mr Flibble wrote:
> >>>>>>>>>>> On Sun, 5 Jun 2022 15:44:32 -0000 (UTC)
> >>>>>>>>>>> Alan Mackenzie <acm@muc.de> wrote:
> >>>>>>>>>>>
> >>>>>>>>>>>> Mr Flibble <flibble@reddwarf.jmc> wrote:
> >>>>>>>>>>>>> On Sun, 5 Jun 2022 16:28:05 +0100
> >>>>>>>>>>>>> Andy Walker <anw@cuboid.co.uk> wrote:
> >>>>>>>>>>>>
> >>>>>>>>>>>>>> On 05/06/2022 14:47, Mr Flibble wrote:
> >>>>>>>>>>>>>>> On Sun, 5 Jun 2022 07:58:42 -0400
> >>>>>>>>>>>>>>> Richard Damon <Richard@Damon-Family.org> wrote:
> >>>>>>>>>>>>>>>> [...] Sort of like how the number Pi has an
> >>>>>>>>>>>>>>>> exact value, but you can never actually express it
> >>>>>>>>>>>>>>>> (because it takes an infinite number of digits).
> >>>>>>>>>>>>>>> PI does not have an exact value; no irrational number
> >>>>>>>>>>>>>>> has an exact value.
> >>>>>>>>>>>>
> >>>>>>>>>>>>>> Of course "pi" has an exact value; as do
> >>>>>>>>>>>>>> [eg] "sqrt(2)", "e", and all the other computable real
> >>>>>>>>>>>>>> [and complex] numbers. Whether that value can be
> >>>>>>>>>>>>>> expressed in finite terms in some particular
> >>>>>>>>>>>>>> representation is quite another matter. That in turn
> >>>>>>>>>>>>>> depends on the representation; standard decimals is
> >>>>>>>>>>>>>> merely one [common] choice. Note that in symbolic
> >>>>>>>>>>>>>> computer systems, those computable reals are typically
> >>>>>>>>>>>>>> written "pi" [or whatever], and the computer works
> >>>>>>>>>>>>>> with that exactly, so that [eg] "sin^2 (pi/3) == 3/4",
> >>>>>>>>>>>>>> not 0.7499...; and also that in decimal-type notations
> >>>>>>>>>>>>>> most rationals equally have no terminating expansion.
> >>>>>>>>>>>>>> Numbers such as "pi" and "sqrt(2)" are not defined as
> >>>>>>>>>>>>>> decimal expansions but via their properties [eg that
> >>>>>>>>>>>>>> "sqrt(2)" is the unique positive real whose square is
> >>>>>>>>>>>>>> 2, or equivalently that it is the ratio of the
> >>>>>>>>>>>>>> diagonal of a square to its side, and "pi" is the
> >>>>>>>>>>>>>> least positive real whose sine is zero]. Those
> >>>>>>>>>>>>>> properties are exact, and tell you all you ever need
> >>>>>>>>>>>>>> to know about those numbers.
> >>>>>>>>>>>>
> >>>>>>>>>>>> [ .... ]
> >>>>>>>>>>>>
> >>>>>>>>>>>>> What has decimal (base 10) expansion got to do with
> >>>>>>>>>>>>> anything? An irrational number has a non-terminating
> >>>>>>>>>>>>> sequence in ANY base. I am sorry but you are simply
> >>>>>>>>>>>>> mistaken: irrational numbers do NOT have an exact value;
> >>>>>>>>>>>>> this is obvious to anyone who understands logic and
> >>>>>>>>>>>>> uses a sane definition for infinity.
> >>>>>>>>>>>>
> >>>>>>>>>>>> That irrational numbers are exact values is clear to
> >>>>>>>>>>>> anybody with a degree in maths. Definitions of
> >>>>>>>>>>>> "infinity" (of which there are many) have nothing to do
> >>>>>>>>>>>> with this.
> >>>>>>>>>>>
> >>>>>>>>>>> You are wrong and fractally so so your degree in maths
> >>>>>>>>>>> appears to be worthless. An irrational number's sequence
> >>>>>>>>>>> is statistically random, has no fixed point on the number
> >>>>>>>>>>> line ergo has no exact representation. Any number with no
> >>>>>>>>>>> exact representation has, by definition, no exact value,
> >>>>>>>>>>> only an approximation. Infinity has everything to do
> >>>>>>>>>>> with this as an irrational's sequence ("digits") never
> >>>>>>>>>>> terminates (i.e. it is an INFINITELY long sequence).
> >>>>>>>>>>>
> >>>>>>>>>>> /Flibble
> >>>>>>>>>>>
> >>>>>>>>>>
> >>>>>>>>>> Nope. Irrational numbers DO have exact points on the number
> >>>>>>>>>> line.
> >>>>>>>>>>
> >>>>>>>>>> And what does representation have to do with exact value?
> >>>>>>>>>>
> >>>>>>>>>> Also, irrational numbers sequence of digits are not
> >>>>>>>>>> necessarily statistically random, in some representations,
> >>>>>>>>>> they can be VERY predictible for some numbers.
> >>>>>>>>>>
> >>>>>>>>>> One simple construction to show exact position, draw a box
> >>>>>>>>>> with sides exactly 1.
> >>>>>>>>>>
> >>>>>>>>>> Draw a line though opposite corners and make one point the
> >>>>>>>>>> value 0.
> >>>>>>>>>>
> >>>>>>>>>> The other corner will be EXACTLY at the point sqrt(2), so
> >>>>>>>>>> that irrational number has an exact point on the number
> >>>>>>>>>> line.
> >>>>>>>>>>
> >>>>>>>>>> You just don't understand what an exact value means, likely
> >>>>>>>>>> because you can't understand things that are somewhat
> >>>>>>>>>> abstract.
> >>>>>>>>>
> >>>>>>>>> An irrational number does not have an exact point on the
> >>>>>>>>> number line as it will move about as you "zoom in", you can
> >>>>>>>>> keep "zooming in" forever (i.e. infinitely) and it will keep
> >>>>>>>>> moving about because the number never terminates.
> >>>>>>>>>
> >>>>>>>>> If I couldn't understand things that are somewhat abstract
> >>>>>>>>> then I wouldn't have a computer science degree (BSc Hons)
> >>>>>>>>> and 30 years of industry experience.
> >>>>>>>>>
> >>>>>>>>> /Flibble
> >>>>>>>>>
> >>>>>>>>
> >>>>>>>> Then why do you think irrational numbers don't have an exact
> >>>>>>>> location?
> >>>>>>>>
> >>>>>>>> I know people with degrees (even with honors) and industry
> >>>>>>>> experiance that still show that they don't really understand
> >>>>>>>> what they are talking about.
> >>>>>>>>
> >>>>>>>> The "width" of the point representing the location of an
> >>>>>>>> irrational number is just as much "0" as that of a rational
> >>>>>>>> number, so specifies just as exact of a location.
> >>>>>>>>
> >>>>>>>> The fact that we can't write it in a rational base with a
> >>>>>>>> finite number of digits doesn't actally mean anything.
> >>>>>>>
> >>>>>>> 3.1415xxxxxxxxx (a)
> >>>>>>> 3.14159xxxxxxxx (b)
> >>>>>>>
> >>>>>>> (b) is nearer to 3.14160 than 3.14150 that (a) indicates hence
> >>>>>>> its value "moves about" as accuracy increases.
> >>>>>>>
> >>>>>>> /Flibble
> >>>>>>>
> >>>>>>
> >>>>>> But (a) and (b) aren't "pi"
> >>>>>
> >>>>> No, they are approximations of pi. The value of the
> >>>>> approximation changes up to a factor of 1/base as you evaluate
> >>>>> it at increasing accuracy.
> >>>>>
> >>>>>>
> >>>>>> All you are showing is that approximations to numbers get
> >>>>>> better as they get better, which is just a strange tautology.
> >>>>>
> >>>>> Yes and therefor they jump about on the number line as accuracy
> >>>>> increases.
> >>>>>
> >>>>>>
> >>>>>> The number PI, has only one precise value, the exact ratio of
> >>>>>> the circumference of a circle to its diameter on a flat plane
> >>>>>> (which will always be the same).
> >>>>>
> >>>>> pi cannot have a precise value as it neither terminates nor has
> >>>>> a repetend in any base.
> >>>>>
> >>>>>>
> >>>>>> The fact that it can't be expressed, isn't an issue on
> >>>>>> exactness, but of finite representation.
> >>>>>
> >>>>> We can only ever have a finite representation.
> >>>>>
> >>>>>>
> >>>>>> Note, that the set of numbers with finite representation is a
> >>>>>> countable set, so it isn't surprising that the uncountable
> >>>>>> infinity of the reals (that includes the irrationals) is not
> >>>>>> all finitely expressible.
> >>>>>
> >>>>> Stating the obvious.
> >>>>>
> >>>>> /Flibble
> >>>>>
> >>>>
> >>>> So you DO have problems with abstractions.
> >>>>
> >>>> The problem is, we don't actually need the finite numerical
> >>>> representation of a number if we have the definition of what the
> >>>> number is (which is another sort of abstract finite
> >>>> representation).
> >>>>
> >>>> The "finite representation" of pi is the ratio of the
> >>>> circumference to the diameter of a circle on a plane.
> >>>
> >>> No, instead if you use logic you must come to the conclusion that
> >>> there is no ratio of the circumference to the diameter of a
> >>> circle: rational numbers describe ratios, irrational numbers do
> >>> not. QED.
> >>>
> >>> /Flibble
> >>>
> >>
> >> Then you don't understand what is a ratio. Rational numbers are the
> >> ratio of INTEGERS, not all ratios. Ratios define the relative
> >> magnatude of one number to another, ANY number.
> >>
> >> Since it is clear you don't understand the meaning of the basic
> >> terms, your OPINION about the weightier things becomes suspect.
> >
> > Prove that there is a ratio between the circumference and diameter
> > of a circle. Hint: you can't.
> >
> > /Flibble
> >
>
> Ratio of numbers A and B is defined as A / B (assuming B is not 0)
>
> The circumference of the circle is a length of the arc all the way
> around the circle, so is a real number.
>
> The diameter of the circle is the length of the line from one side of
> the circle to the other through the center, so is a real number.
> Assuming the circle is not degenerate to a point, that number is not
> zero.
>
> Thus we have two real numbers, the second not zero, so by closure,
> the first divided by the second is a real number.
>
> Thus those numbers have a ratio.

On 05/06/2022 16:16, Alan Mackenzie wrote:
> Mike Terry <news.dead.person.stones@darjeeling.plus.com> wrote:
>> On 05/06/2022 13:14, Alan Mackenzie wrote:
>>> olcott <NoOne@nowhere.com> wrote:
>>>> On 6/5/2022 6:12 AM, Alan Mackenzie wrote:
>>>>> olcott <NoOne@nowhere.com> wrote:
>>>>>> On 6/5/2022 5:14 AM, Mikko wrote:
>>>>>>> On 2022-06-04 19:28:19 +0000, olcott said:
>
>>>>>>>> A Turing machine is said to halt whenever it reaches a
>>>>>>>> configuration for which δ is not defined; this is possible because
>>>>>>>> δ is a partial function. In fact, we will assume that no
>>>>>>>> transitions are defined for any final state so the Turing machine
>>>>>>>> will halt whenever it enters a final state.  (Linz:1990:234)
>
>>>>>>>> Linz, Peter 1990. An Introduction to Formal Languages and Automata.
>>>>>>>> Lexington/Toronto: D. C. Heath and Company.
>
>>>>>>>> When translated into ordinary software engineering terms this means
>>>>>>>> terminated normally. In a C function this means reaching the "ret"
>>>>>>>> instruction.
>
>>>>>>> The best equivalent to "not defined" is not "ret". Instead, "not
>>>>>>> defined" should include at least:
>>>>>>> - HLT or any other instruction that means 'halt'
>>>>>>> - any undefined op code
>>>>>>> - any return or pop instruction if the stack is empty
>>>>>>> - an instruction fetch from a location that is not specifiec by the
>>>>>>> program
>>>>>>> That way the analogy to Linz' definition is much better.
>
>>>>>>> Mikko
>
>>>>>> Reaching a final state is merely the Turing machine way of saying
>>>>>> terminated normally. "ret" is the C way of saying the same thing.
>
>>>>> Sophistry. What would be the turing machine equivalent of an
>>>>> "abnormal termination" in C?
>
>>>> An aborted simulation.
>
>>> There's no such thing on a turing machine. It either runs and halts,
>>> or it runs forever.
>
>>> Your aborted simulation is just one final state of a turing machine,
>>> which has thus halted.
>
>> A TM "aborting" a simulation is just the TM ceasing to calculate
>> computation steps for some computation, and going on to calculate
>> something else instead. It does not mean:
>> a) that the TM (doing the simulation) has halted
>> b) that the simulated computation halts
>> c) that the simulated computation never halts
>
> OK. I've a feeling we're talking more about nice shades of words than
> computer science here, but ....
>
> If the simulation is the entire turing machine, aborting it will bring
> the TM to a halt state. If that simulation is merely part of the TM,
> then the word "halt" has a different meaning when applied to that
> simulation part from when applied to the entire TM. I'm not even sure
> what you mean when you say a part of a TM has halted or not halted.

We are clearly talking at cross purposes - I never talked about /part/ of a TM halting, and like
you, I can't work out what that would mean! I used "halt" only with respect to a computation,
meaning that the computation halts [there is an n such that computation step n is a TM final state].

Reading what you say very carefully, I think that by your definition of simulation, the simulating
TM must be a "pure" simulator that does nothing but simulate computation steps until the simulation
halts, at which point the simulating TM halts (like a UTM). I get that with that interpretation
what you said:

<copied from above>
>>> Your aborted simulation is just one final state of a turing machine,
>>> which has thus halted.

makes sense and is correct. I'd just say I don't think that usage of "simulation" is very useful,
and is DEFINITELY not what PO is talking about (so it would be wrong if applied PO's posts...)

My use of "simulation" is broader: it's simply the activity performed by a TM which consists of
calculating computation steps of some given computation. As such it's just a part of the TM logic.
A TM's typical use of simulation might be something like "..the TM simulates the computation for n
steps, and if the simulation halts during those n steps, the TM [blah blah], /otherwise/ the TM
[blah blah blah]...". Just about every reference in the literature I can recall is something like that.

So... to be 100% clear on what I said:

<copied from above>
>> A TM "aborting" a simulation is just the TM ceasing to calculate
>> computation steps for some computation, and going on to calculate
>> something else instead.

E.g. in PO's P, after P aborts its simulation of P(P), the TM either halts or enters an infinite
loop. (That logic is not part of the simulation, IMO.)

>> It does *NOT* mean:
>> a) that the TM (doing the simulation) has halted

obviously, because now P has gone on to something else...

>> b) that the simulated computation halts
>> c) that the simulated computation never halts

obviously - in general different exacmples of a simulated computation P(I) might halt or never halt,
and this is unaffected by a simulator's decision to simulate no further computation steps. [The TM
may have spotted some pattern in the simulated computation which implies P(I) never halts - that is
a separate matter, but for sure the mere act of "aborting" the simulation doesn't imply P(I) never
halts, or imply that it halts...

Put yet another way, when a TM stops calculating TM steps (aka aborts its simulation), NOTHING
HALTS: not the simulating TM, not the simulated computation, and NOT ANY PART OF EITHER OF THOSE.
(Like you say, what would part of a TM halting mean?)

Mike.

On 6/5/22 3:56 PM, Mr Flibble wrote:
> On Sun, 5 Jun 2022 15:47:30 -0400
> Richard Damon <Richard@Damon-Family.org> wrote:
>
>> On 6/5/22 3:33 PM, Mr Flibble wrote:
>>> On Sun, 5 Jun 2022 15:30:45 -0400
>>> Richard Damon <Richard@Damon-Family.org> wrote:
>>>
>>>> On 6/5/22 3:17 PM, Mr Flibble wrote:
>>>>> On Sun, 5 Jun 2022 14:17:29 -0400
>>>>> Richard Damon <Richard@Damon-Family.org> wrote:
>>>>>
>>>>>> On 6/5/22 1:22 PM, Mr Flibble wrote:
>>>>>>> On Sun, 5 Jun 2022 13:05:09 -0400
>>>>>>> Richard Damon <Richard@Damon-Family.org> wrote:
>>>>>>>
>>>>>>>> On 6/5/22 12:56 PM, Mr Flibble wrote:
>>>>>>>>> On Sun, 5 Jun 2022 12:50:13 -0400
>>>>>>>>> Richard Damon <Richard@Damon-Family.org> wrote:
>>>>>>>>>
>>>>>>>>>> On 6/5/22 12:28 PM, Mr Flibble wrote:
>>>>>>>>>>> On Sun, 5 Jun 2022 12:22:45 -0400
>>>>>>>>>>> Richard Damon <Richard@Damon-Family.org> wrote:
>>>>>>>>>>>
>>>>>>>>>>>> On 6/5/22 11:49 AM, Mr Flibble wrote:
>>>>>>>>>>>>> On Sun, 5 Jun 2022 15:44:32 -0000 (UTC)
>>>>>>>>>>>>> Alan Mackenzie <acm@muc.de> wrote:
>>>>>>>>>>>>>
>>>>>>>>>>>>>> Mr Flibble <flibble@reddwarf.jmc> wrote:
>>>>>>>>>>>>>>> On Sun, 5 Jun 2022 16:28:05 +0100
>>>>>>>>>>>>>>> Andy Walker <anw@cuboid.co.uk> wrote:
>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> On 05/06/2022 14:47, Mr Flibble wrote:
>>>>>>>>>>>>>>>>> On Sun, 5 Jun 2022 07:58:42 -0400
>>>>>>>>>>>>>>>>> Richard Damon <Richard@Damon-Family.org> wrote:
>>>>>>>>>>>>>>>>>> [...] Sort of like how the number Pi has an
>>>>>>>>>>>>>>>>>> exact value, but you can never actually express it
>>>>>>>>>>>>>>>>>> (because it takes an infinite number of digits).
>>>>>>>>>>>>>>>>> PI does not have an exact value; no irrational number
>>>>>>>>>>>>>>>>> has an exact value.
>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> Of course "pi" has an exact value; as do
>>>>>>>>>>>>>>>> [eg] "sqrt(2)", "e", and all the other computable real
>>>>>>>>>>>>>>>> [and complex] numbers. Whether that value can be
>>>>>>>>>>>>>>>> expressed in finite terms in some particular
>>>>>>>>>>>>>>>> representation is quite another matter. That in turn
>>>>>>>>>>>>>>>> depends on the representation; standard decimals is
>>>>>>>>>>>>>>>> merely one [common] choice. Note that in symbolic
>>>>>>>>>>>>>>>> computer systems, those computable reals are typically
>>>>>>>>>>>>>>>> written "pi" [or whatever], and the computer works
>>>>>>>>>>>>>>>> with that exactly, so that [eg] "sin^2 (pi/3) == 3/4",
>>>>>>>>>>>>>>>> not 0.7499...; and also that in decimal-type notations
>>>>>>>>>>>>>>>> most rationals equally have no terminating expansion.
>>>>>>>>>>>>>>>> Numbers such as "pi" and "sqrt(2)" are not defined as
>>>>>>>>>>>>>>>> decimal expansions but via their properties [eg that
>>>>>>>>>>>>>>>> "sqrt(2)" is the unique positive real whose square is
>>>>>>>>>>>>>>>> 2, or equivalently that it is the ratio of the
>>>>>>>>>>>>>>>> diagonal of a square to its side, and "pi" is the
>>>>>>>>>>>>>>>> least positive real whose sine is zero]. Those
>>>>>>>>>>>>>>>> properties are exact, and tell you all you ever need
>>>>>>>>>>>>>>>> to know about those numbers.
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> [ .... ]
>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> What has decimal (base 10) expansion got to do with
>>>>>>>>>>>>>>> anything? An irrational number has a non-terminating
>>>>>>>>>>>>>>> sequence in ANY base. I am sorry but you are simply
>>>>>>>>>>>>>>> mistaken: irrational numbers do NOT have an exact value;
>>>>>>>>>>>>>>> this is obvious to anyone who understands logic and
>>>>>>>>>>>>>>> uses a sane definition for infinity.
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> That irrational numbers are exact values is clear to
>>>>>>>>>>>>>> anybody with a degree in maths. Definitions of
>>>>>>>>>>>>>> "infinity" (of which there are many) have nothing to do
>>>>>>>>>>>>>> with this.
>>>>>>>>>>>>>
>>>>>>>>>>>>> You are wrong and fractally so so your degree in maths
>>>>>>>>>>>>> appears to be worthless. An irrational number's sequence
>>>>>>>>>>>>> is statistically random, has no fixed point on the number
>>>>>>>>>>>>> line ergo has no exact representation. Any number with no
>>>>>>>>>>>>> exact representation has, by definition, no exact value,
>>>>>>>>>>>>> only an approximation. Infinity has everything to do
>>>>>>>>>>>>> with this as an irrational's sequence ("digits") never
>>>>>>>>>>>>> terminates (i.e. it is an INFINITELY long sequence).
>>>>>>>>>>>>>
>>>>>>>>>>>>> /Flibble
>>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>> Nope. Irrational numbers DO have exact points on the number
>>>>>>>>>>>> line.
>>>>>>>>>>>>
>>>>>>>>>>>> And what does representation have to do with exact value?
>>>>>>>>>>>>
>>>>>>>>>>>> Also, irrational numbers sequence of digits are not
>>>>>>>>>>>> necessarily statistically random, in some representations,
>>>>>>>>>>>> they can be VERY predictible for some numbers.
>>>>>>>>>>>>
>>>>>>>>>>>> One simple construction to show exact position, draw a box
>>>>>>>>>>>> with sides exactly 1.
>>>>>>>>>>>>
>>>>>>>>>>>> Draw a line though opposite corners and make one point the
>>>>>>>>>>>> value 0.
>>>>>>>>>>>>
>>>>>>>>>>>> The other corner will be EXACTLY at the point sqrt(2), so
>>>>>>>>>>>> that irrational number has an exact point on the number
>>>>>>>>>>>> line.
>>>>>>>>>>>>
>>>>>>>>>>>> You just don't understand what an exact value means, likely
>>>>>>>>>>>> because you can't understand things that are somewhat
>>>>>>>>>>>> abstract.
>>>>>>>>>>>
>>>>>>>>>>> An irrational number does not have an exact point on the
>>>>>>>>>>> number line as it will move about as you "zoom in", you can
>>>>>>>>>>> keep "zooming in" forever (i.e. infinitely) and it will keep
>>>>>>>>>>> moving about because the number never terminates.
>>>>>>>>>>>
>>>>>>>>>>> If I couldn't understand things that are somewhat abstract
>>>>>>>>>>> then I wouldn't have a computer science degree (BSc Hons)
>>>>>>>>>>> and 30 years of industry experience.
>>>>>>>>>>>
>>>>>>>>>>> /Flibble
>>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> Then why do you think irrational numbers don't have an exact
>>>>>>>>>> location?
>>>>>>>>>>
>>>>>>>>>> I know people with degrees (even with honors) and industry
>>>>>>>>>> experiance that still show that they don't really understand
>>>>>>>>>> what they are talking about.
>>>>>>>>>>
>>>>>>>>>> The "width" of the point representing the location of an
>>>>>>>>>> irrational number is just as much "0" as that of a rational
>>>>>>>>>> number, so specifies just as exact of a location.
>>>>>>>>>>
>>>>>>>>>> The fact that we can't write it in a rational base with a
>>>>>>>>>> finite number of digits doesn't actally mean anything.
>>>>>>>>>
>>>>>>>>> 3.1415xxxxxxxxx (a)
>>>>>>>>> 3.14159xxxxxxxx (b)
>>>>>>>>>
>>>>>>>>> (b) is nearer to 3.14160 than 3.14150 that (a) indicates hence
>>>>>>>>> its value "moves about" as accuracy increases.
>>>>>>>>>
>>>>>>>>> /Flibble
>>>>>>>>>
>>>>>>>>
>>>>>>>> But (a) and (b) aren't "pi"
>>>>>>>
>>>>>>> No, they are approximations of pi. The value of the
>>>>>>> approximation changes up to a factor of 1/base as you evaluate
>>>>>>> it at increasing accuracy.
>>>>>>>
>>>>>>>>
>>>>>>>> All you are showing is that approximations to numbers get
>>>>>>>> better as they get better, which is just a strange tautology.
>>>>>>>
>>>>>>> Yes and therefor they jump about on the number line as accuracy
>>>>>>> increases.
>>>>>>>
>>>>>>>>
>>>>>>>> The number PI, has only one precise value, the exact ratio of
>>>>>>>> the circumference of a circle to its diameter on a flat plane
>>>>>>>> (which will always be the same).
>>>>>>>
>>>>>>> pi cannot have a precise value as it neither terminates nor has
>>>>>>> a repetend in any base.
>>>>>>>
>>>>>>>>
>>>>>>>> The fact that it can't be expressed, isn't an issue on
>>>>>>>> exactness, but of finite representation.
>>>>>>>
>>>>>>> We can only ever have a finite representation.
>>>>>>>
>>>>>>>>
>>>>>>>> Note, that the set of numbers with finite representation is a
>>>>>>>> countable set, so it isn't surprising that the uncountable
>>>>>>>> infinity of the reals (that includes the irrationals) is not
>>>>>>>> all finitely expressible.
>>>>>>>
>>>>>>> Stating the obvious.
>>>>>>>
>>>>>>> /Flibble
>>>>>>>
>>>>>>
>>>>>> So you DO have problems with abstractions.
>>>>>>
>>>>>> The problem is, we don't actually need the finite numerical
>>>>>> representation of a number if we have the definition of what the
>>>>>> number is (which is another sort of abstract finite
>>>>>> representation).
>>>>>>
>>>>>> The "finite representation" of pi is the ratio of the
>>>>>> circumference to the diameter of a circle on a plane.
>>>>>
>>>>> No, instead if you use logic you must come to the conclusion that
>>>>> there is no ratio of the circumference to the diameter of a
>>>>> circle: rational numbers describe ratios, irrational numbers do
>>>>> not. QED.
>>>>>
>>>>> /Flibble
>>>>>
>>>>
>>>> Then you don't understand what is a ratio. Rational numbers are the
>>>> ratio of INTEGERS, not all ratios. Ratios define the relative
>>>> magnatude of one number to another, ANY number.
>>>>
>>>> Since it is clear you don't understand the meaning of the basic
>>>> terms, your OPINION about the weightier things becomes suspect.
>>>
>>> Prove that there is a ratio between the circumference and diameter
>>> of a circle. Hint: you can't.
>>>
>>> /Flibble
>>>
>>
>> Ratio of numbers A and B is defined as A / B (assuming B is not 0)
>>
>> The circumference of the circle is a length of the arc all the way
>> around the circle, so is a real number.
>>
>> The diameter of the circle is the length of the line from one side of
>> the circle to the other through the center, so is a real number.
>> Assuming the circle is not degenerate to a point, that number is not
>> zero.
>>
>> Thus we have two real numbers, the second not zero, so by closure,
>> the first divided by the second is a real number.
>>
>> Thus those numbers have a ratio.
>
> But at least one of those numbers must be irrational otherwise the
> ratio would be a rational number ergo at least one of those numbers is
> inexact ergo the resulting ratio between the circumference
> and diameter of a circle must also be inexact which as an irrational
> number it is. QED.
>
> /Flibble
>

On 6/5/2022 3:07 PM, Mike Terry wrote:
> On 05/06/2022 16:16, Alan Mackenzie wrote:
>> Mike Terry <news.dead.person.stones@darjeeling.plus.com> wrote:
>>> On 05/06/2022 13:14, Alan Mackenzie wrote:
>>>> olcott <NoOne@nowhere.com> wrote:
>>>>> On 6/5/2022 6:12 AM, Alan Mackenzie wrote:
>>>>>> olcott <NoOne@nowhere.com> wrote:
>>>>>>> On 6/5/2022 5:14 AM, Mikko wrote:
>>>>>>>> On 2022-06-04 19:28:19 +0000, olcott said:
>>
>>>>>>>>> A Turing machine is said to halt whenever it reaches a
>>>>>>>>> configuration for which δ is not defined; this is possible because
>>>>>>>>> δ is a partial function. In fact, we will assume that no
>>>>>>>>> transitions are defined for any final state so the Turing machine
>>>>>>>>> will halt whenever it enters a final state.  (Linz:1990:234)
>>
>>>>>>>>> Linz, Peter 1990. An Introduction to Formal Languages and
>>>>>>>>> Automata.
>>>>>>>>> Lexington/Toronto: D. C. Heath and Company.
>>
>>>>>>>>> When translated into ordinary software engineering terms this
>>>>>>>>> means
>>>>>>>>> terminated normally. In a C function this means reaching the "ret"
>>>>>>>>> instruction.
>>
>>>>>>>> The best equivalent to "not defined" is not "ret". Instead, "not
>>>>>>>> defined" should include at least:
>>>>>>>> - HLT or any other instruction that means 'halt'
>>>>>>>> - any undefined op code
>>>>>>>> - any return or pop instruction if the stack is empty
>>>>>>>> - an instruction fetch from a location that is not specifiec by the
>>>>>>>>      program
>>>>>>>> That way the analogy to Linz' definition is much better.
>>
>>>>>>>> Mikko
>>
>>>>>>> Reaching a final state is merely the Turing machine way of saying
>>>>>>> terminated normally. "ret" is the C way of saying the same thing.
>>
>>>>>> Sophistry.  What would be the turing machine equivalent of an
>>>>>> "abnormal termination" in C?
>>
>>>>> An aborted simulation.
>>
>>>> There's no such thing on a turing machine.  It either runs and halts,
>>>> or it runs forever.
>>
>>>> Your aborted simulation is just one final state of a turing machine,
>>>> which has thus halted.
>>
>>> A TM "aborting" a simulation is just the TM ceasing to calculate
>>> computation steps for some computation, and going on to calculate
>>> something else instead.  It does not mean:
>>> a)  that the TM (doing the simulation) has halted
>>> b)  that the simulated computation halts
>>> c)  that the simulated computation never halts
>>
>> OK.  I've a feeling we're talking more about nice shades of words than
>> computer science here, but ....
>>
>> If the simulation is the entire turing machine, aborting it will bring
>> the TM to a halt state.  If that simulation is merely part of the TM,
>> then the word "halt" has a different meaning when applied to that
>> simulation part from when applied to the entire TM.  I'm not even sure
>> what you mean when you say a part of a TM has halted or not halted.
>
> We are clearly talking at cross purposes - I never talked about /part/
> of a TM halting, and like you, I can't work out what that would mean!  I
> used "halt" only with respect to a computation, meaning that the
> computation halts [there is an n such that computation step n is a TM
> final state].
>
> Reading what you say very carefully, I think that by your definition of
> simulation, the simulating TM must be a "pure" simulator that does
> nothing but simulate computation steps until the simulation halts, at
> which point the simulating TM halts (like a UTM).  I get that with that
> interpretation what you said:
>
> <copied from above>
> >>> Your aborted simulation is just one final state of a turing machine,
> >>> which has thus halted.
>
>  makes sense and is correct.  I'd just say I don't think that usage of
> "simulation" is very useful, and is DEFINITELY not what PO is talking
> about (so it would be wrong if applied PO's posts...)
>
> My use of "simulation" is broader: it's simply the activity performed by
> a TM which consists of calculating computation steps of some given
> computation.  As such it's just a part of the TM logic. A TM's typical
> use of simulation might be something like "..the TM simulates the
> computation for n steps, and if the simulation halts during those n
> steps, the TM [blah blah], /otherwise/ the TM [blah blah blah]...".
> Just about every reference in the literature I can recall is something
> like that.
>
> So... to be 100% clear on what I said:
>
> <copied from above>
> >> A TM "aborting" a simulation is just the TM ceasing to calculate
> >> computation steps for some computation, and going on to calculate
> >> something else instead.
>
> E.g. in PO's P, after P aborts its simulation of P(P), the TM either
> halts or enters an infinite loop.  (That logic is not part of the
> simulation, IMO.)
>
> >> It does *NOT* mean:
> >> a)  that the TM (doing the simulation) has halted
>
> obviously, because now P has gone on to something else...
>
> >> b)  that the simulated computation halts
> >> c)  that the simulated computation never halts
>
> obviously - in general different exacmples of a simulated computation
> P(I) might halt or never halt, and this is unaffected by a simulator's
> decision to simulate no further computation steps.  [The TM may have
> spotted some pattern in the simulated computation which implies P(I)
> never halts - that is a separate matter, but for sure the mere act of
> "aborting" the simulation doesn't imply P(I) never halts, or imply that
> it halts...
>
> Put yet another way, when a TM stops calculating TM steps (aka aborts
> its simulation), NOTHING HALTS: not the simulating TM, not the simulated
> computation, and NOT ANY PART OF EITHER OF THOSE. (Like you say, what
> would part of a TM halting mean?)
>
>
> Mike.

THIS IS EXACTLY WHAT I MEAN.
H0(Infinite_Loop) correctly determines that its input never halts.

If we really want to get totally tediously precise (tedium utterly
freaks me out) we can say that the finite string machine code specified
as an input parameter to the C function H0 specifies a sequence of
instructions that would never reach their "ret" instruction.

void Infinite_Loop()
{ HERE: goto HERE;
}

int main()
{ Output("Input_Halts = ", H0(Infinite_Loop));
}

_Infinite_Loop()
[00001342](01) 55 push ebp
[00001343](02) 8bec mov ebp,esp
[00001345](02) ebfe jmp 00001345
[00001347](01) 5d pop ebp
[00001348](01) c3 ret
Size in bytes:(0007) [00001348]

_main()
[00001372](01) 55 push ebp
[00001373](02) 8bec mov ebp,esp
[00001375](05) 6842130000 push 00001342
[0000137a](05) e833fdffff call 000010b2
[0000137f](03) 83c404 add esp,+04
[00001382](01) 50 push eax
[00001383](05) 6823040000 push 00000423
[00001388](05) e8e5f0ffff call 00000472
[0000138d](03) 83c408 add esp,+08
[00001390](02) 33c0 xor eax,eax
[00001392](01) 5d pop ebp
[00001393](01) c3 ret
Size in bytes:(0034) [00001393]

machine stack stack machine assembly
address address data code language
======== ======== ======== ========= =============
[00001372][0010228f][00000000] 55 push ebp
[00001373][0010228f][00000000] 8bec mov ebp,esp
[00001375][0010228b][00001342] 6842130000 push 00001342 // push
_Infinite_Loop
[0000137a][00102287][0000137f] e833fdffff call 000010b2 // call H0

Begin Local Halt Decider Simulation Execution Trace Stored at:212343
[00001342][00212333][00212337] 55 push ebp
[00001343][00212333][00212337] 8bec mov ebp,esp
[00001345][00212333][00212337] ebfe jmp 00001345
[00001345][00212333][00212337] ebfe jmp 00001345
Local Halt Decider: Infinite Loop Detected Simulation Stopped

On Sun, 5 Jun 2022 16:09:13 -0400
Richard Damon <Richard@Damon-Family.org> wrote:

> On 6/5/22 3:56 PM, Mr Flibble wrote:
> > On Sun, 5 Jun 2022 15:47:30 -0400
> > Richard Damon <Richard@Damon-Family.org> wrote:
> >
> >> On 6/5/22 3:33 PM, Mr Flibble wrote:
> >>> On Sun, 5 Jun 2022 15:30:45 -0400
> >>> Richard Damon <Richard@Damon-Family.org> wrote:
> >>>
> >>>> On 6/5/22 3:17 PM, Mr Flibble wrote:
> >>>>> On Sun, 5 Jun 2022 14:17:29 -0400
> >>>>> Richard Damon <Richard@Damon-Family.org> wrote:
> >>>>>
> >>>>>> On 6/5/22 1:22 PM, Mr Flibble wrote:
> >>>>>>> On Sun, 5 Jun 2022 13:05:09 -0400
> >>>>>>> Richard Damon <Richard@Damon-Family.org> wrote:
> >>>>>>>
> >>>>>>>> On 6/5/22 12:56 PM, Mr Flibble wrote:
> >>>>>>>>> On Sun, 5 Jun 2022 12:50:13 -0400
> >>>>>>>>> Richard Damon <Richard@Damon-Family.org> wrote:
> >>>>>>>>>
> >>>>>>>>>> On 6/5/22 12:28 PM, Mr Flibble wrote:
> >>>>>>>>>>> On Sun, 5 Jun 2022 12:22:45 -0400
> >>>>>>>>>>> Richard Damon <Richard@Damon-Family.org> wrote:
> >>>>>>>>>>>
> >>>>>>>>>>>> On 6/5/22 11:49 AM, Mr Flibble wrote:
> >>>>>>>>>>>>> On Sun, 5 Jun 2022 15:44:32 -0000 (UTC)
> >>>>>>>>>>>>> Alan Mackenzie <acm@muc.de> wrote:
> >>>>>>>>>>>>>
> >>>>>>>>>>>>>> Mr Flibble <flibble@reddwarf.jmc> wrote:
> >>>>>>>>>>>>>>> On Sun, 5 Jun 2022 16:28:05 +0100
> >>>>>>>>>>>>>>> Andy Walker <anw@cuboid.co.uk> wrote:
> >>>>>>>>>>>>>>
> >>>>>>>>>>>>>>>> On 05/06/2022 14:47, Mr Flibble wrote:
> >>>>>>>>>>>>>>>>> On Sun, 5 Jun 2022 07:58:42 -0400
> >>>>>>>>>>>>>>>>> Richard Damon <Richard@Damon-Family.org> wrote:
> >>>>>>>>>>>>>>>>>> [...] Sort of like how the number Pi has an
> >>>>>>>>>>>>>>>>>> exact value, but you can never actually express it
> >>>>>>>>>>>>>>>>>> (because it takes an infinite number of digits).
> >>>>>>>>>>>>>>>>> PI does not have an exact value; no irrational
> >>>>>>>>>>>>>>>>> number has an exact value.
> >>>>>>>>>>>>>>
> >>>>>>>>>>>>>>>> Of course "pi" has an exact value; as
> >>>>>>>>>>>>>>>> do [eg] "sqrt(2)", "e", and all the other computable
> >>>>>>>>>>>>>>>> real [and complex] numbers. Whether that value can be
> >>>>>>>>>>>>>>>> expressed in finite terms in some particular
> >>>>>>>>>>>>>>>> representation is quite another matter. That in turn
> >>>>>>>>>>>>>>>> depends on the representation; standard decimals is
> >>>>>>>>>>>>>>>> merely one [common] choice. Note that in symbolic
> >>>>>>>>>>>>>>>> computer systems, those computable reals are
> >>>>>>>>>>>>>>>> typically written "pi" [or whatever], and the
> >>>>>>>>>>>>>>>> computer works with that exactly, so that [eg]
> >>>>>>>>>>>>>>>> "sin^2 (pi/3) == 3/4", not 0.7499...; and also that
> >>>>>>>>>>>>>>>> in decimal-type notations most rationals equally
> >>>>>>>>>>>>>>>> have no terminating expansion. Numbers such as "pi"
> >>>>>>>>>>>>>>>> and "sqrt(2)" are not defined as decimal expansions
> >>>>>>>>>>>>>>>> but via their properties [eg that "sqrt(2)" is the
> >>>>>>>>>>>>>>>> unique positive real whose square is 2, or
> >>>>>>>>>>>>>>>> equivalently that it is the ratio of the diagonal of
> >>>>>>>>>>>>>>>> a square to its side, and "pi" is the least positive
> >>>>>>>>>>>>>>>> real whose sine is zero]. Those properties are
> >>>>>>>>>>>>>>>> exact, and tell you all you ever need to know about
> >>>>>>>>>>>>>>>> those numbers.
> >>>>>>>>>>>>>>
> >>>>>>>>>>>>>> [ .... ]
> >>>>>>>>>>>>>>
> >>>>>>>>>>>>>>> What has decimal (base 10) expansion got to do with
> >>>>>>>>>>>>>>> anything? An irrational number has a non-terminating
> >>>>>>>>>>>>>>> sequence in ANY base. I am sorry but you are simply
> >>>>>>>>>>>>>>> mistaken: irrational numbers do NOT have an exact
> >>>>>>>>>>>>>>> value; this is obvious to anyone who understands
> >>>>>>>>>>>>>>> logic and uses a sane definition for infinity.
> >>>>>>>>>>>>>>
> >>>>>>>>>>>>>> That irrational numbers are exact values is clear to
> >>>>>>>>>>>>>> anybody with a degree in maths. Definitions of
> >>>>>>>>>>>>>> "infinity" (of which there are many) have nothing to do
> >>>>>>>>>>>>>> with this.
> >>>>>>>>>>>>>
> >>>>>>>>>>>>> You are wrong and fractally so so your degree in maths
> >>>>>>>>>>>>> appears to be worthless. An irrational number's
> >>>>>>>>>>>>> sequence is statistically random, has no fixed point on
> >>>>>>>>>>>>> the number line ergo has no exact representation. Any
> >>>>>>>>>>>>> number with no exact representation has, by definition,
> >>>>>>>>>>>>> no exact value, only an approximation. Infinity has
> >>>>>>>>>>>>> everything to do with this as an irrational's sequence
> >>>>>>>>>>>>> ("digits") never terminates (i.e. it is an INFINITELY
> >>>>>>>>>>>>> long sequence).
> >>>>>>>>>>>>>
> >>>>>>>>>>>>> /Flibble
> >>>>>>>>>>>>>
> >>>>>>>>>>>>
> >>>>>>>>>>>> Nope. Irrational numbers DO have exact points on the
> >>>>>>>>>>>> number line.
> >>>>>>>>>>>>
> >>>>>>>>>>>> And what does representation have to do with exact value?
> >>>>>>>>>>>>
> >>>>>>>>>>>> Also, irrational numbers sequence of digits are not
> >>>>>>>>>>>> necessarily statistically random, in some
> >>>>>>>>>>>> representations, they can be VERY predictible for some
> >>>>>>>>>>>> numbers.
> >>>>>>>>>>>>
> >>>>>>>>>>>> One simple construction to show exact position, draw a
> >>>>>>>>>>>> box with sides exactly 1.
> >>>>>>>>>>>>
> >>>>>>>>>>>> Draw a line though opposite corners and make one point
> >>>>>>>>>>>> the value 0.
> >>>>>>>>>>>>
> >>>>>>>>>>>> The other corner will be EXACTLY at the point sqrt(2), so
> >>>>>>>>>>>> that irrational number has an exact point on the number
> >>>>>>>>>>>> line.
> >>>>>>>>>>>>
> >>>>>>>>>>>> You just don't understand what an exact value means,
> >>>>>>>>>>>> likely because you can't understand things that are
> >>>>>>>>>>>> somewhat abstract.
> >>>>>>>>>>>
> >>>>>>>>>>> An irrational number does not have an exact point on the
> >>>>>>>>>>> number line as it will move about as you "zoom in", you
> >>>>>>>>>>> can keep "zooming in" forever (i.e. infinitely) and it
> >>>>>>>>>>> will keep moving about because the number never
> >>>>>>>>>>> terminates.
> >>>>>>>>>>>
> >>>>>>>>>>> If I couldn't understand things that are somewhat abstract
> >>>>>>>>>>> then I wouldn't have a computer science degree (BSc Hons)
> >>>>>>>>>>> and 30 years of industry experience.
> >>>>>>>>>>>
> >>>>>>>>>>> /Flibble
> >>>>>>>>>>>
> >>>>>>>>>>
> >>>>>>>>>> Then why do you think irrational numbers don't have an
> >>>>>>>>>> exact location?
> >>>>>>>>>>
> >>>>>>>>>> I know people with degrees (even with honors) and industry
> >>>>>>>>>> experiance that still show that they don't really
> >>>>>>>>>> understand what they are talking about.
> >>>>>>>>>>
> >>>>>>>>>> The "width" of the point representing the location of an
> >>>>>>>>>> irrational number is just as much "0" as that of a rational
> >>>>>>>>>> number, so specifies just as exact of a location.
> >>>>>>>>>>
> >>>>>>>>>> The fact that we can't write it in a rational base with a
> >>>>>>>>>> finite number of digits doesn't actally mean anything.
> >>>>>>>>>
> >>>>>>>>> 3.1415xxxxxxxxx (a)
> >>>>>>>>> 3.14159xxxxxxxx (b)
> >>>>>>>>>
> >>>>>>>>> (b) is nearer to 3.14160 than 3.14150 that (a) indicates
> >>>>>>>>> hence its value "moves about" as accuracy increases.
> >>>>>>>>>
> >>>>>>>>> /Flibble
> >>>>>>>>>
> >>>>>>>>
> >>>>>>>> But (a) and (b) aren't "pi"
> >>>>>>>
> >>>>>>> No, they are approximations of pi. The value of the
> >>>>>>> approximation changes up to a factor of 1/base as you evaluate
> >>>>>>> it at increasing accuracy.
> >>>>>>>
> >>>>>>>>
> >>>>>>>> All you are showing is that approximations to numbers get
> >>>>>>>> better as they get better, which is just a strange
> >>>>>>>> tautology.
> >>>>>>>
> >>>>>>> Yes and therefor they jump about on the number line as
> >>>>>>> accuracy increases.
> >>>>>>>
> >>>>>>>>
> >>>>>>>> The number PI, has only one precise value, the exact ratio of
> >>>>>>>> the circumference of a circle to its diameter on a flat plane
> >>>>>>>> (which will always be the same).
> >>>>>>>
> >>>>>>> pi cannot have a precise value as it neither terminates nor
> >>>>>>> has a repetend in any base.
> >>>>>>>
> >>>>>>>>
> >>>>>>>> The fact that it can't be expressed, isn't an issue on
> >>>>>>>> exactness, but of finite representation.
> >>>>>>>
> >>>>>>> We can only ever have a finite representation.
> >>>>>>>
> >>>>>>>>
> >>>>>>>> Note, that the set of numbers with finite representation is a
> >>>>>>>> countable set, so it isn't surprising that the uncountable
> >>>>>>>> infinity of the reals (that includes the irrationals) is not
> >>>>>>>> all finitely expressible.
> >>>>>>>
> >>>>>>> Stating the obvious.
> >>>>>>>
> >>>>>>> /Flibble
> >>>>>>>
> >>>>>>
> >>>>>> So you DO have problems with abstractions.
> >>>>>>
> >>>>>> The problem is, we don't actually need the finite numerical
> >>>>>> representation of a number if we have the definition of what
> >>>>>> the number is (which is another sort of abstract finite
> >>>>>> representation).
> >>>>>>
> >>>>>> The "finite representation" of pi is the ratio of the
> >>>>>> circumference to the diameter of a circle on a plane.
> >>>>>
> >>>>> No, instead if you use logic you must come to the conclusion
> >>>>> that there is no ratio of the circumference to the diameter of a
> >>>>> circle: rational numbers describe ratios, irrational numbers do
> >>>>> not. QED.
> >>>>>
> >>>>> /Flibble
> >>>>>
> >>>>
> >>>> Then you don't understand what is a ratio. Rational numbers are
> >>>> the ratio of INTEGERS, not all ratios. Ratios define the relative
> >>>> magnatude of one number to another, ANY number.
> >>>>
> >>>> Since it is clear you don't understand the meaning of the basic
> >>>> terms, your OPINION about the weightier things becomes suspect.
> >>>
> >>> Prove that there is a ratio between the circumference and diameter
> >>> of a circle. Hint: you can't.
> >>>
> >>> /Flibble
> >>>
> >>
> >> Ratio of numbers A and B is defined as A / B (assuming B is not 0)
> >>
> >> The circumference of the circle is a length of the arc all the way
> >> around the circle, so is a real number.
> >>
> >> The diameter of the circle is the length of the line from one side
> >> of the circle to the other through the center, so is a real number.
> >> Assuming the circle is not degenerate to a point, that number is
> >> not zero.
> >>
> >> Thus we have two real numbers, the second not zero, so by closure,
> >> the first divided by the second is a real number.
> >>
> >> Thus those numbers have a ratio.
> >
> > But at least one of those numbers must be irrational otherwise the
> > ratio would be a rational number ergo at least one of those numbers
> > is inexact ergo the resulting ratio between the circumference
> > and diameter of a circle must also be inexact which as an irrational
> > number it is. QED.
> >
> > /Flibble
> >
>
> You are ASSUMING that irrational numbers are inexact, and that is a
> false statement so you logic is unsound.
>
> Note, the square root of 2 can be proved exact by simple geometric
> construction, thus disproving your claim that irrational numbers are
> inexact.
>