On Thursday, 18 August 2022 at 02:54:17 UTC+2, richar...@gmail.com wrote:
> There IS a decidable < operation on elements of R.
What do you mean by "decidable"!?! You must have somen notion of "decidability" far removed from the usual notion pertaining to Turing Machine-decidable.

> The fact that "closer to infinity", since infinity isn't a normal value
> of the field is defined differently than "closer to x" where x IS a
> member of the field isn't a problem.
>
> That is just operator overloading.
I understand generics/overloading just fine, you sophist.

You overloaded the notion of "decidability" in your previous comment to mean something other than Turing Decidable.
> x is closer to infinity than y if x > y, which holds for all x and y
> that are actually members of the Real Number system.
But x > y doesn't tell you ANYTHING about "closeness to infinity" ?!?!?!?
x > y is about DISTANCE FROM ZERO!

x > y is just a proposition.

If x is further away from 0 than y then (x>y) is a theorem!
If x is NOT further away from 0 than y then not(x>y) is a theorem!

On Thursday, 18 August 2022 at 14:02:14 UTC+2, richar...@gmail.com wrote:
> No, I think that is likely, as you prove you are dumber that others.
Dude! I am infinitely dumb.

And you are still dumber.

On Thursday, 18 August 2022 at 14:01:23 UTC+2, richar...@gmail.com wrote:
> Nothing.
> lim, as an operator, needs a function to apply to.
No shit, sherlock! That's how all higher order functions work.

To avoid confusing yourself any further apply it to the identity function: f(x) = x

lim(x -> ∞) f(x)

> What is "1 + " mean?
Do you mean "+ 1" or "1 +"? Those are different things!

"1 +" means "1 applied to +".
"+ 1" means "+ applied to 1"

"lim(x -> ∞) f(x)" is not the same thing as "f(x)lim(x -> ∞)

On Thursday, 18 August 2022 at 14:21:00 UTC+2, Alan Mackenzie wrote:
> > You fucking idiot.
> You've lost the argument, so you descend to abuse.
This is worth repeating.... YOU FUCKING IDIOT.

This game is not about "winning" or "losing" arguments.
This game is about understanding.

If you don't understand then it doesn't matter whether you win or lose the argument. You lose either way.

Base(0) and Base(1) are perfectly valid bases for counting!
Base(0) -> [ 0, 00, 000, 0000, ....]
Base(1) -> [ 1, 11, 111, 1111, ...]
And BECAUSE there is a bijection between the two sets the "=" operator is a map!

0 = 1
00 = 11
000 = 000
....

But your over-educated, over-abstracted idiot ass thinks that all number systems are "fundamentally the same".
I mean - they are.

Once you abstract away the fact on whether they index on SOMETHING(1) or NOTHING(0).

On 18/08/2022 12:29, Malcolm McLean wrote:
> Base zero [...]

If you only have one symbol to use, then maths gets a little
boring. Balanced binary is more interesting. Basic idea: start from
O [ == zero, or the origin]. If some point X on the number line is to
the right of where you are, go right, indicated by R; if to the left,
go left, indicated by L. For suitable choices of how far you go, every
point on the line can be reached by a sequence such as RRRLLRLLRRR.
"Suitable" has certain desirable properties:

-- having gone R and then L, no further sequence should take you
to or past where the initial R took you to, else you can reach
points in several ways; so the steps should get shorter.
Similarly for L and then R.
-- OTOH, the steps should not get shorter too quickly, else points
become unreachable.
-- [So the obvious choice is that steps should halve, then every
point is accessible in exactly one way, closely related to the
binary expansion of its value. But other rules are possible.]
-- [But this should not apply to initial sequences of RRR... or
LLL..., else you can't reach points past twice the initial
step length.]
-- For mathematical simplicity, it turns out [but this is not
"obvious"] that the initial steps should all be the same length,
call it 1. So, eg, the above sequence RRRLLRLLRRR corresponds
to 1 + 1 + 1 - 1/2 - 1/4 + 1/8 - 1/16 - 1/32 + 1/64 + 1/128 +
1/256 == 2 79/256 == 10.01001111 in binary.
-- With the above choice, ordinary arithmetic is v easy [much the
same as the usual binary arithmetic, except that there is no
"-" sign, negation comes simply from swapping L and R.
-- Eg, 1/3 == RLLRLRLR... == RL(LR) where (...) indicates recurring
patterns.
-- Note that R(L) is less than any normal positive fraction; and
(R) is greater than any finite integer; cf infinitesimal and
infinity. Further developments left as an exercise.

--
Andy Walker, Nottingham.
Andy's music pages: www.cuboid.me.uk/andy/Music
Composer of the day: www.cuboid.me.uk/andy/Music/Composers/Grieg

On Thursday, 18 August 2022 at 18:51:38 UTC+2, Andy Walker wrote:
> If you only have one symbol to use, then maths gets a little
> boring. Balanced binary is more interesting. Basic idea: start from
> O [ == zero, or the origin].
Hahahahahaha. You cultural ignoramus.

For thousands of years 0/zero was NOT the origin.
For thousands of years there was noway to represent NOTHING in your number system!
Because. numbers stood for SOMETHING, not NOTHING.

For thousands of years there were no negative numbers either!

>If some point X on the number line is to
> the right of where you are, go right, indicated by R; if to the left,
> go left, indicated by L. For suitable choices of how far you go, every
> point on the line can be reached by a sequence such as RRRLLRLLRRR.
> "Suitable" has certain desirable properties: -- having gone R and then L, no further sequence should take you
> to or past where the initial R took you to, else you can reach
> points in several ways; so the steps should get shorter.
> Similarly for L and then R.
You over-educated ignoramus!

Until 0 was invented there was no Left(1)!
Until negative numbers were invented there was no Left(0) either!!!

On Thursday, 18 August 2022 at 18:51:38 UTC+2, Andy Walker wrote:
> On 18/08/2022 12:29, Malcolm McLean wrote:
> > Base zero [...]
>
> If you only have one symbol to use, then maths gets a little
> boring. Balanced binary is more interesting. Basic idea: start from
> O [ == zero, or the origin]. If some point X on the number line is to
> the right of where you are, go right, indicated by R; if to the left,
> go left, indicated by L. For suitable choices of how far you go, every
> point on the line can be reached by a sequence such as RRRLLRLLRRR.
> "Suitable" has certain desirable properties:
>
> -- having gone R and then L, no further sequence should take you
> to or past where the initial R took you to, else you can reach
> points in several ways; so the steps should get shorter.
> Similarly for L and then R.
> -- OTOH, the steps should not get shorter too quickly, else points
> become unreachable.
> -- [So the obvious choice is that steps should halve, then every
> point is accessible in exactly one way, closely related to the
> binary expansion of its value. But other rules are possible.]
> -- [But this should not apply to initial sequences of RRR... or
> LLL..., else you can't reach points past twice the initial
> step length.]
> -- For mathematical simplicity, it turns out [but this is not
> "obvious"] that the initial steps should all be the same length,
> call it 1. So, eg, the above sequence RRRLLRLLRRR corresponds
> to 1 + 1 + 1 - 1/2 - 1/4 + 1/8 - 1/16 - 1/32 + 1/64 + 1/128 +
> 1/256 == 2 79/256 == 10.01001111 in binary.
> -- With the above choice, ordinary arithmetic is v easy [much the
> same as the usual binary arithmetic, except that there is no
> "-" sign, negation comes simply from swapping L and R.
> -- Eg, 1/3 == RLLRLRLR... == RL(LR) where (...) indicates recurring
> patterns.
> -- Note that R(L) is less than any normal positive fraction; and
> (R) is greater than any finite integer; cf infinitesimal and
> infinity. Further developments left as an exercise.
>
> --
> Andy Walker, Nottingham.
> Andy's music pages: www.cuboid.me.uk/andy/Music
> Composer of the day: www.cuboid.me.uk/andy/Music/Composers/Grieg
Also... do you even understand the notion of an unordered n-tuple ?!?
Imagine a set with no structure. A set in which the < operator doesn't work for any pair of elements.

A set in which "left of" and "right of" are meaningless notions.

Skep Dick <skepdick22@gmail.com> wrote:
> On Thursday, 18 August 2022 at 18:51:38 UTC+2, Andy Walker wrote:
>> On 18/08/2022 12:29, Malcolm McLean wrote:
>> > Base zero [...]

>> If you only have one symbol to use, then maths gets a little
>> boring. Balanced binary is more interesting. Basic idea: start from
>> O [ == zero, or the origin]. If some point X on the number line is to
>> the right of where you are, go right, indicated by R; if to the left,
>> go left, indicated by L. For suitable choices of how far you go, every
>> point on the line can be reached by a sequence such as RRRLLRLLRRR.
>> "Suitable" has certain desirable properties:

>> -- having gone R and then L, no further sequence should take you
>> to or past where the initial R took you to, else you can reach
>> points in several ways; so the steps should get shorter.
>> Similarly for L and then R.
>> -- OTOH, the steps should not get shorter too quickly, else points
>> become unreachable.
>> -- [So the obvious choice is that steps should halve, then every
>> point is accessible in exactly one way, closely related to the
>> binary expansion of its value. But other rules are possible.]
>> -- [But this should not apply to initial sequences of RRR... or
>> LLL..., else you can't reach points past twice the initial
>> step length.]
>> -- For mathematical simplicity, it turns out [but this is not
>> "obvious"] that the initial steps should all be the same length,
>> call it 1. So, eg, the above sequence RRRLLRLLRRR corresponds
>> to 1 + 1 + 1 - 1/2 - 1/4 + 1/8 - 1/16 - 1/32 + 1/64 + 1/128 +
>> 1/256 == 2 79/256 == 10.01001111 in binary.
>> -- With the above choice, ordinary arithmetic is v easy [much the
>> same as the usual binary arithmetic, except that there is no
>> "-" sign, negation comes simply from swapping L and R.
>> -- Eg, 1/3 == RLLRLRLR... == RL(LR) where (...) indicates recurring
>> patterns.
>> -- Note that R(L) is less than any normal positive fraction; and
>> (R) is greater than any finite integer; cf infinitesimal and
>> infinity. Further developments left as an exercise.

>> --
>> Andy Walker, Nottingham.
>> Andy's music pages: www.cuboid.me.uk/andy/Music
>> Composer of the day: www.cuboid.me.uk/andy/Music/Composers/Grieg

> Also... do you even understand the notion of an unordered n-tuple ?!?
> Imagine a set with no structure. A set in which the < operator doesn't
> work for any pair of elements.

Yes, in mathematics we call such a thing a set. More precisely, a finite
set. If you fail to define a < relationship, then the < operator "won't
work".....

> A set in which "left of" and "right of" are meaningless notions.

..... if you fail to define "left of" and "right of" too, then they will
be meaningless. Andy certainly understands these things.

Were you trying to make a point? If so, what was it?

--
Alan Mackenzie (Nuremberg, Germany).

On Thursday, August 18, 2022 at 12:52:36 AM UTC-7, Skep Dick wrote:
> On Wednesday, 17 August 2022 at 23:09:09 UTC+2, dklei...@gmail.com wrote:
> > On Wednesday, August 17, 2022 at 12:08:11 AM UTC-7, Skep Dick wrote:
> > > On Wednesday, 17 August 2022 at 07:15:58 UTC+2, dklei...@gmail.com wrote:
> > > > Once chosen 0 does not change and the isomorphism
> > > > stops being of interest.
> > > It stops being of interest ??!?!?!
> > >
> > > -∞<----------------------0-> +∞
> > > is isomorphic to
> > > -∞<-0----------------------> +∞
> > >
> > > It gang-rapes my intuition that the isomorphism is being preserved despite my moving the "0" left and right!
> > >
> > There is a complete continuum of either side of 0. Seems obvious from the meaning of infinity.
> Are you actually capable of reading complete English sentences?!? Maybe your cultural/Mathematical biases are getting in your way of understanding me. Lets replace the symbol "0" with the symbol "X". Let x represent a free variable on the continuum.
>
> -∞<-X----------------------> +∞
>
> What does it mean for x to approach -∞; or +-∞ if there is ALWAYS a complete continuum on either side of ANY x ?!?

It means just what it says. X is a countably
infinite sequence of numbers such each
member is larger than the previous one.

Infinity is a weird beast.

On Thursday, August 18, 2022 at 12:56:39 AM UTC-7, Skep Dick wrote:
> On Wednesday, 17 August 2022 at 23:14:49 UTC+2, dklei...@gmail.com wrote:
> > On Wednesday, August 17, 2022 at 5:04:40 AM UTC-7, Skep Dick wrote:
> > > On Wednesday, 17 August 2022 at 13:54:24 UTC+2, richar...@gmail.com wrote:
> > > > Why is that a problem, many things that are True are unintuitive.
> > > Because Truth in Mathematics is ALWAYS relative to a model!
> > >
> > > And the model is semantic (intuition!) not syntactic (proofs).
> > > > Especially when you throw things like infinity into the mix, as we have
> > > > trouble really understanding it.
> > > I mean... your first sign of trouble is the ∃ operator. Which one is true?
> > >
> > > ∃∞ or ¬∃∞ ?
> > >
> > > And the answer is... whichever model you CHOOSE.
> > > > Maybe the biggest part of the issue is you are thinking of infinity
> > > > being AT the > symbol, when actually that is telling you it isn't there
> > > > but off a REAL FAR DISTANCE in that direction.
> > > Maybe your problem is that you think there is a DISTANCE metric to infinity?
> > Metrics are a poverty-stricken way to talk about limits. You need a topology.
> You are not hearing or understanding a word of English!
>
> I have given you a topology! The continuum itself.
>
> -∞<-------------------------------> +∞
>
> And I have given you a free variable - x.
>
> -∞<-------x----------------------> +∞
>
> Now thell me what lim(x-> +∞); or lim(x -> -∞) means!

That's not what topology means.

lim(x-> +∞) by itself is meaningless. To give it meaning
we have to have some function f(x) and ask what
lim(x-> +∞) f(x) means.

lim(x-> +∞) f(x) = g means that for any p there is a q
such that for all x such that q < x
- (f(x) - g) < p < f(x) - g)

If this is hard to understand this is where most people
who are not mathematicians stop. You have lots of
company.

On Thursday, 18 August 2022 at 20:16:07 UTC+2, Alan Mackenzie wrote:
> Yes, in mathematics we call such a thing a set.
> Yes, in mathematics we call such a thing a set. More precisely, a finite
> set. If you fail to define a < relationship, then the < operator "won't
> work".....
> > A set in which "left of" and "right of" are meaningless notions.
> .... if you fail to define "left of" and "right of" too, then they will
> be meaningless. Andy certainly understands these things.
>
> Were you trying to make a point? If so, what was it?
Blah blah blah!

This is an n-tuple: {x,x} that is not a set.

Andy Walker <anw@cuboid.co.uk> writes:

> On 18/08/2022 12:29, Malcolm McLean wrote:
>> Base zero [...]
>
> If you only have one symbol to use, then maths gets a little
> boring. Balanced binary is more interesting. Basic idea: start from
> O [ == zero, or the origin]. If some point X on the number line is to
> the right of where you are, go right, indicated by R; if to the left,
> go left, indicated by L. For suitable choices of how far you go, every
> point on the line can be reached by a sequence such as RRRLLRLLRRR.
> "Suitable" has certain desirable properties:
>
> -- having gone R and then L, no further sequence should take you
> to or past where the initial R took you to, else you can reach
> points in several ways; so the steps should get shorter.
> Similarly for L and then R.
> -- OTOH, the steps should not get shorter too quickly, else points
> become unreachable.
> -- [So the obvious choice is that steps should halve, then every
> point is accessible in exactly one way, closely related to the
> binary expansion of its value. But other rules are possible.]
> -- [But this should not apply to initial sequences of RRR... or
> LLL..., else you can't reach points past twice the initial
> step length.]
> -- For mathematical simplicity, it turns out [but this is not
> "obvious"] that the initial steps should all be the same length,
> call it 1. So, eg, the above sequence RRRLLRLLRRR corresponds
> to 1 + 1 + 1 - 1/2 - 1/4 + 1/8 - 1/16 - 1/32 + 1/64 + 1/128 +
> 1/256 == 2 79/256 == 10.01001111 in binary.
> -- With the above choice, ordinary arithmetic is v easy [much the
> same as the usual binary arithmetic, except that there is no
> "-" sign, negation comes simply from swapping L and R.
> -- Eg, 1/3 == RLLRLRLR... == RL(LR) where (...) indicates recurring
> patterns.
> -- Note that R(L) is less than any normal positive fraction; and
> (R) is greater than any finite integer; cf infinitesimal and
> infinity. Further developments left as an exercise.

If RL(LR) = 1/3, why is R(L) not 0? R(L) would indeed be, as you say,
less than any normal positive fraction, but not special in any way.

The simple Hackenbush numbers have a "rounding up" bit in the conversion
rule, so RL(LR) is not exactly 1/3 but 1/3 + 1/𝜔 and R(L) = 1/𝜔.

--
Ben.

On Thursday, 18 August 2022 at 20:16:07 UTC+2, Alan Mackenzie wrote:
> .... if you fail to define "left of" and "right of" too, then they will
> be meaningless. Andy certainly understands these things.
So can we apply this rule to sets also?

If you fail to define"sets" then they will be meaningless also, right?

Or do sets get a free pass?

On Thursday, 18 August 2022 at 20:29:11 UTC+2, dklei...@gmail.com wrote:
> It means just what it says. X is a countably
> infinite sequence of numbers such each
> member is larger than the previous one.
>
> Infinity is a weird beast.
I don't understand what you mean by "countably infinite sequence". Are you confusing the discreteness of the symbols in the sequence (their topology) with the ability to finish counting the elements in the sequence?

On 8/18/2022 12:48 PM, dklei...@gmail.com wrote:
> On Thursday, August 18, 2022 at 12:56:39 AM UTC-7, Skep Dick wrote:
>> On Wednesday, 17 August 2022 at 23:14:49 UTC+2, dklei...@gmail.com wrote:
>>> On Wednesday, August 17, 2022 at 5:04:40 AM UTC-7, Skep Dick wrote:
>>>> On Wednesday, 17 August 2022 at 13:54:24 UTC+2, richar...@gmail.com wrote:
>>>>> Why is that a problem, many things that are True are unintuitive.
>>>> Because Truth in Mathematics is ALWAYS relative to a model!
>>>>
>>>> And the model is semantic (intuition!) not syntactic (proofs).
>>>>> Especially when you throw things like infinity into the mix, as we have
>>>>> trouble really understanding it.
>>>> I mean... your first sign of trouble is the ∃ operator. Which one is true?
>>>>
>>>> ∃∞ or ¬∃∞ ?
>>>>
>>>> And the answer is... whichever model you CHOOSE.
>>>>> Maybe the biggest part of the issue is you are thinking of infinity
>>>>> being AT the > symbol, when actually that is telling you it isn't there
>>>>> but off a REAL FAR DISTANCE in that direction.
>>>> Maybe your problem is that you think there is a DISTANCE metric to infinity?
>>> Metrics are a poverty-stricken way to talk about limits. You need a topology.
>> You are not hearing or understanding a word of English!
>>
>> I have given you a topology! The continuum itself.
>>
>> -∞<-------------------------------> +∞
>>
>> And I have given you a free variable - x.
>>
>> -∞<-------x----------------------> +∞
>>
>> Now thell me what lim(x-> +∞); or lim(x -> -∞) means!
>
> That's not what topology means.
>
> lim(x-> +∞) by itself is meaningless. To give it meaning
> we have to have some function f(x) and ask what
> lim(x-> +∞) f(x) means.
>
> lim(x-> +∞) f(x) = g means that for any p there is a q
> such that for all x such that q < x
> - (f(x) - g) < p < f(x) - g)
I think you meant to say |f(x) - g| < p, where |e| is the absolute value
of e.
> If this is hard to understand this is where most people
> who are not mathematicians stop. You have lots of
> company.
--
Jeff Barnett

On Thursday, August 18, 2022 at 11:48:47 AM UTC-7, dklei...@gmail.com wrote:
> On Thursday, August 18, 2022 at 12:56:39 AM UTC-7, Skep Dick wrote:
> > On Wednesday, 17 August 2022 at 23:14:49 UTC+2, dklei...@gmail.com wrote:
> > > On Wednesday, August 17, 2022 at 5:04:40 AM UTC-7, Skep Dick wrote:
> > > > On Wednesday, 17 August 2022 at 13:54:24 UTC+2, richar...@gmail.com wrote:
> > > > > Why is that a problem, many things that are True are unintuitive.
> > > > Because Truth in Mathematics is ALWAYS relative to a model!
> > > >
> > > > And the model is semantic (intuition!) not syntactic (proofs).
> > > > > Especially when you throw things like infinity into the mix, as we have
> > > > > trouble really understanding it.
> > > > I mean... your first sign of trouble is the ∃ operator. Which one is true?
> > > >
> > > > ∃∞ or ¬∃∞ ?
> > > >
> > > > And the answer is... whichever model you CHOOSE.
> > > > > Maybe the biggest part of the issue is you are thinking of infinity
> > > > > being AT the > symbol, when actually that is telling you it isn't there
> > > > > but off a REAL FAR DISTANCE in that direction.
> > > > Maybe your problem is that you think there is a DISTANCE metric to infinity?
> > > Metrics are a poverty-stricken way to talk about limits. You need a topology.
> > You are not hearing or understanding a word of English!
> >
> > I have given you a topology! The continuum itself.
> >
> > -∞<-------------------------------> +∞
> >
> > And I have given you a free variable - x.
> >
> > -∞<-------x----------------------> +∞
> >
> > Now thell me what lim(x-> +∞); or lim(x -> -∞) means!
> That's not what topology means.
>
> lim(x-> +∞) by itself is meaningless. To give it meaning
> we have to have some function f(x) and ask what
> lim(x-> +∞) f(x) means.
>
> lim(x-> +∞) f(x) = g means that for any p there is a q
> such that for all x such that q < x
> - (f(x) - g) < p < f(x) - g)
>
> If this is hard to understand this is where most people
> who are not mathematicians stop. You have lots of
> company.
>
OOPS -p < f(q) - g < + p

Skep Dick <skepdick22@gmail.com> wrote:
> On Thursday, 18 August 2022 at 20:16:07 UTC+2, Alan Mackenzie wrote:
>> .... if you fail to define "left of" and "right of" too, then they will
>> be meaningless. Andy certainly understands these things.
> So can we apply this rule to sets also?

Here we go again, .....

> If you fail to define"sets" then they will be meaningless also, right?

Sets are defined, so your "If" isn't interesting.

> Or do sets get a free pass?

I know what your next planned move is. I'll tell you in advance, it
isn't interesting. We've heard it all before, and it's right tiresome.

--
Alan Mackenzie (Nuremberg, Germany).

On Thursday, 18 August 2022 at 21:20:29 UTC+2, Alan Mackenzie wrote:
> Sets are defined, so your "If" isn't interesting.
No they aren't.

There is literally no formal definition of what a set is in Math!

> > Or do sets get a free pass?
> I know what your next planned move is. I'll tell you in advance, it
> isn't interesting. We've heard it all before, and it's right tiresome.
What "move" you moron? This is not a chess game.

Set are NOT defined. We talk ABOUT sets. We try to understand what sets are LIKE given our attempts to characterize them with axioms.

This is a well accepted fact!

On 8/18/2022 11:49 AM, Skep Dick wrote:
> On Thursday, 18 August 2022 at 18:51:38 UTC+2, Andy Walker wrote:
>> On 18/08/2022 12:29, Malcolm McLean wrote:
>>> Base zero [...]
>>
>> If you only have one symbol to use, then maths gets a little
>> boring. Balanced binary is more interesting. Basic idea: start from
>> O [ == zero, or the origin]. If some point X on the number line is to
>> the right of where you are, go right, indicated by R; if to the left,
>> go left, indicated by L. For suitable choices of how far you go, every
>> point on the line can be reached by a sequence such as RRRLLRLLRRR.
>> "Suitable" has certain desirable properties:
>>
>> -- having gone R and then L, no further sequence should take you
>> to or past where the initial R took you to, else you can reach
>> points in several ways; so the steps should get shorter.
>> Similarly for L and then R.
>> -- OTOH, the steps should not get shorter too quickly, else points
>> become unreachable.
>> -- [So the obvious choice is that steps should halve, then every
>> point is accessible in exactly one way, closely related to the
>> binary expansion of its value. But other rules are possible.]
>> -- [But this should not apply to initial sequences of RRR... or
>> LLL..., else you can't reach points past twice the initial
>> step length.]
>> -- For mathematical simplicity, it turns out [but this is not
>> "obvious"] that the initial steps should all be the same length,
>> call it 1. So, eg, the above sequence RRRLLRLLRRR corresponds
>> to 1 + 1 + 1 - 1/2 - 1/4 + 1/8 - 1/16 - 1/32 + 1/64 + 1/128 +
>> 1/256 == 2 79/256 == 10.01001111 in binary.
>> -- With the above choice, ordinary arithmetic is v easy [much the
>> same as the usual binary arithmetic, except that there is no
>> "-" sign, negation comes simply from swapping L and R.
>> -- Eg, 1/3 == RLLRLRLR... == RL(LR) where (...) indicates recurring
>> patterns.
>> -- Note that R(L) is less than any normal positive fraction; and
>> (R) is greater than any finite integer; cf infinitesimal and
>> infinity. Further developments left as an exercise.
>>
>> --
>> Andy Walker, Nottingham.
>> Andy's music pages: www.cuboid.me.uk/andy/Music
>> Composer of the day: www.cuboid.me.uk/andy/Music/Composers/Grieg
> Also... do you even understand the notion of an unordered n-tuple ?!?
> Imagine a set with no structure. A set in which the < operator doesn't work for any pair of elements.
>
> A set in which "left of" and "right of" are meaningless notions.

Mathematicians call it a "set with multiplicities" and computer
scientist call it a "bag". There is some set, say E, that are elements
and a bag is defined as an unordered collection of some, none, or all of
the elements of E and is unordered. (The bag is unordered whether or not
you have an order for E.) Another, almost equivalent definition is that
a bag is a function on E to N, the natural numbers. This definition is
actually different than the first since it limits the amount of each
element that can be in a bag.
--
Jeff Barnett

On Thursday, August 18, 2022 at 12:11:26 PM UTC-7, Skep Dick wrote:
> On Thursday, 18 August 2022 at 20:29:11 UTC+2, dklei...@gmail.com wrote:
> > It means just what it says. X is a countably
> > infinite sequence of numbers such each
> > member is larger than the previous one.
> >
> > Infinity is a weird beast.
> I don't understand what you mean by "countably infinite sequence". Are you confusing the discreteness of the symbols in the sequence (their topology) with the ability to finish counting the elements in the sequence?

Your background with digital computers is
coloring your understand of mathematics.
Mathematics has no concept of process. A
reference to X in lim (X -> Y) implies some
single unchanging entity X not any kind of
variable. What mathematics does with process
is work with the completed process as a
single whole.

Skep Dick <skepdick22@gmail.com> wrote:
> On Thursday, 18 August 2022 at 21:20:29 UTC+2, Alan Mackenzie wrote:
>> Sets are defined, so your "If" isn't interesting.
> No they aren't.

> There is literally no formal definition of what a set is in Math!

As predicted. Sets are defined axiomatically.

>> > Or do sets get a free pass?
>> I know what your next planned move is. I'll tell you in advance, it
>> isn't interesting. We've heard it all before, and it's right tiresome.
> What "move" you moron? This is not a chess game.

The one you made above.

> Set are NOT defined. We talk ABOUT sets. We try to understand what
> sets are LIKE given our attempts to characterize them with axioms.

You don't have the intellectual capacity of a mathematician, all of whom
understand axiomatic definition. You don't. You just don't get it.
Your level of understanding is not up to that needed to do mathematics.

> This is a well accepted fact!

Only by cranks.

--
Alan Mackenzie (Nuremberg, Germany).

On Thursday, 18 August 2022 at 21:28:40 UTC+2, dklei...@gmail.com wrote:
> On Thursday, August 18, 2022 at 12:11:26 PM UTC-7, Skep Dick wrote:
> > On Thursday, 18 August 2022 at 20:29:11 UTC+2, dklei...@gmail.com wrote:
> > > It means just what it says. X is a countably
> > > infinite sequence of numbers such each
> > > member is larger than the previous one.
> > >
> > > Infinity is a weird beast.
> > I don't understand what you mean by "countably infinite sequence". Are you confusing the discreteness of the symbols in the sequence (their topology) with the ability to finish counting the elements in the sequence?
> Your background with digital computers is
> coloring your understand of mathematics.
> Mathematics has no concept of process.
Buuuuulshit!

https://en.wikipedia.org/wiki/%CE%A0-calculus

>A reference to X in lim (X -> Y) implies some
> single unchanging entity X not any kind of
> variable. What mathematics does with process
> is work with the completed process as a
> single whole.
Please explain to me what you mean by "completed". when dealing with an infinite stream of symbols?

On 8/18/2022 1:24 PM, Jeff Barnett wrote:
> On 8/18/2022 11:49 AM, Skep Dick wrote:
>> On Thursday, 18 August 2022 at 18:51:38 UTC+2, Andy Walker wrote:
>>> On 18/08/2022 12:29, Malcolm McLean wrote:
>>>> Base zero [...]
>>>
>>> If you only have one symbol to use, then maths gets a little
>>> boring. Balanced binary is more interesting. Basic idea: start from
>>> O [ == zero, or the origin]. If some point X on the number line is to
>>> the right of where you are, go right, indicated by R; if to the left,
>>> go left, indicated by L. For suitable choices of how far you go, every
>>> point on the line can be reached by a sequence such as RRRLLRLLRRR.
>>> "Suitable" has certain desirable properties:
>>>
>>> -- having gone R and then L, no further sequence should take you
>>> to or past where the initial R took you to, else you can reach
>>> points in several ways; so the steps should get shorter.
>>> Similarly for L and then R.
>>> -- OTOH, the steps should not get shorter too quickly, else points
>>> become unreachable.
>>> -- [So the obvious choice is that steps should halve, then every
>>> point is accessible in exactly one way, closely related to the
>>> binary expansion of its value. But other rules are possible.]
>>> -- [But this should not apply to initial sequences of RRR... or
>>> LLL..., else you can't reach points past twice the initial
>>> step length.]
>>> -- For mathematical simplicity, it turns out [but this is not
>>> "obvious"] that the initial steps should all be the same length,
>>> call it 1. So, eg, the above sequence RRRLLRLLRRR corresponds
>>> to 1 + 1 + 1 - 1/2 - 1/4 + 1/8 - 1/16 - 1/32 + 1/64 + 1/128 +
>>> 1/256 == 2 79/256 == 10.01001111 in binary.
>>> -- With the above choice, ordinary arithmetic is v easy [much the
>>> same as the usual binary arithmetic, except that there is no
>>> "-" sign, negation comes simply from swapping L and R.
>>> -- Eg, 1/3 == RLLRLRLR... == RL(LR) where (...) indicates recurring
>>> patterns.
>>> -- Note that R(L) is less than any normal positive fraction; and
>>> (R) is greater than any finite integer; cf infinitesimal and
>>> infinity. Further developments left as an exercise.
>>>
>>> --
>>> Andy Walker, Nottingham.
>>> Andy's music pages: www.cuboid.me.uk/andy/Music
>>> Composer of the day: www.cuboid.me.uk/andy/Music/Composers/Grieg
>> Also...  do you even understand the notion of an unordered n-tuple ?!?
>> Imagine a set with no structure. A set in which the < operator doesn't
>> work for any pair of elements.
>>
>> A set in which "left of" and "right of" are meaningless notions.
>
> Mathematicians call it a "set with multiplicities" and computer
> scientist call it a "bag". There is some set, say E, that are elements
> and a bag is defined as an unordered collection of some, none, or all of
> the elements of E and is unordered. (The bag is unordered whether or not
> you have an order for E.) Another, almost equivalent definition is that
> a bag is a function on E to N, the natural numbers. This definition is
> actually different than the first since it limits the amount of each
> element that can be in a bag.
I wrote BS above: "There is some set, say E, that are elements and a bag
is defined as an unordered collection of some, none, or all of the
elements of E and is unordered." should say
There is some set, say E, that are elements and a bag is defined as an
unordered collection of some number, possibly zero, of EACH element of E.
Sorry for the goof.
--
Jeff Barnett

On Thursday, 18 August 2022 at 21:33:40 UTC+2, Alan Mackenzie wrote:
> Skep Dick <skepd...@gmail.com> wrote:
> > On Thursday, 18 August 2022 at 21:20:29 UTC+2, Alan Mackenzie wrote:
> >> Sets are defined, so your "If" isn't interesting.
> > No they aren't.
>
> > There is literally no formal definition of what a set is in Math!
> As predicted. Sets are defined axiomatically.
No! They are not.

> > What "move" you moron? This is not a chess game.
> The one you made above.
Stating facts is "making moves"?!?!?

> You don't have the intellectual capacity of a mathematician, all of whom
> understand axiomatic definition. You don't. You just don't get it.
Evientially I have the intelelctual capacity to understand facts.

Sets are NOT defined in Mathematics.

> Only by cranks.
Ooooh. Is that how it works?

Everybody who understands the fact that sets are not formally defined is a crank.

That's one way to defed your religion from critique.

Skep Dick <skepdick22@gmail.com> wrote:
> On Thursday, 18 August 2022 at 21:33:40 UTC+2, Alan Mackenzie wrote:
>> Skep Dick <skepd...@gmail.com> wrote:
>> > On Thursday, 18 August 2022 at 21:20:29 UTC+2, Alan Mackenzie wrote:
>> >> Sets are defined, so your "If" isn't interesting.
>> > No they aren't.

>> > There is literally no formal definition of what a set is in Math!
>> As predicted. Sets are defined axiomatically.
> No! They are not.

As I said in my last post, such is beyond your understanding. Look at it
this way: The foundatations of mathematics are based on set theory. If
sets were not defined, the whole theory would collapse like a house of
cards. This is seen not to have happened.

>> > What "move" you moron? This is not a chess game.
>> The one you made above.
> Stating facts is "making moves"?!?!?

You lack the understanding to know what a mathematical fact is.

>> You don't have the intellectual capacity of a mathematician, all of whom
>> understand axiomatic definition. You don't. You just don't get it.
> Evidentially I have the intellectual capacity to understand facts.

You haven't produced any such evidence.

> Sets are NOT defined in Mathematics.

You're wrong, here. Mathematicians know, accept, and use the definition
of sets.

>> Only by cranks.
> Ooooh. Is that how it works?

I'm afraid it is. Only cranks and mathematicians care about it. The
former get it wrong.

> Everybody who understands the fact that sets are not formally defined
> is a crank.

Not quite. Many people pass through that state of misunderstanding.
Some get stuck on it. They tend to become cranks. A defining
characteristic of a crank is to disparage what is beyond his
understanding. This is what you are doing.

> That's one way to defend your religion from critique.

I'd hardly characterise the content of your last few posts as critique.

--
Alan Mackenzie (Nuremberg, Germany).