On 9/28/2022 10:28 PM, Ross A. Finlayson wrote:
> On Wednesday, September 28, 2022 at 10:23:37 PM UTC-7, Chris M. Thomasson wrote:
>> On 9/28/2022 10:13 PM, Chris M. Thomasson wrote:
>>> On 9/28/2022 9:32 PM, Ross A. Finlayson wrote:
>>>> On Wednesday, September 28, 2022 at 9:28:37 PM UTC-7, Chris M.
>>>> Thomasson wrote:
>>>>> On 9/24/2022 9:57 AM, Sergi o wrote:
>>>>>> On 9/21/2022 2:17 PM, mitchr...@gmail.com wrote:
>> [...]
>>> Humm... Interesting. Wrt 13/3 - 1 is = 3.333..., so we can scoot over
>>> the endless three's, so to speak... Just like my direct solution does
>>> for 333...34. Add one, to the the ending 4. Fair enough?
>> Ouch. I mean add one to the ending three to make the four, in 333...34.
>>
>> The floor operations basically strips off the infinite threes as in
>> floor(1/3) = 0
>> i[n] = floor((1/3) * 10^(n+2) + 1)
>
> Oh you mean storage?
>

No. I mean getting rid of the string of infinite 3's (think base 10
representation), 1/3 = .3333333....

Fine.

floor(1/3) = 0
floor(.3333333...) = 0

However, floor(123456789.1011121314) = 123456789

To get 333...34 in a direct function, well, exploiting 1/3 sure seems to
work! ;^)

i[n] = floor((1/3) * 10^(n+2) + 1)

On Wednesday, September 28, 2022 at 10:51:05 PM UTC-7, Chris M. Thomasson wrote:
> On 9/28/2022 10:28 PM, Ross A. Finlayson wrote:
> > On Wednesday, September 28, 2022 at 10:23:37 PM UTC-7, Chris M. Thomasson wrote:
> >> On 9/28/2022 10:13 PM, Chris M. Thomasson wrote:
> >>> On 9/28/2022 9:32 PM, Ross A. Finlayson wrote:
> >>>> On Wednesday, September 28, 2022 at 9:28:37 PM UTC-7, Chris M.
> >>>> Thomasson wrote:
> >>>>> On 9/24/2022 9:57 AM, Sergi o wrote:
> >>>>>> On 9/21/2022 2:17 PM, mitchr...@gmail.com wrote:
> >> [...]
> >>> Humm... Interesting. Wrt 13/3 - 1 is = 3.333..., so we can scoot over
> >>> the endless three's, so to speak... Just like my direct solution does
> >>> for 333...34. Add one, to the the ending 4. Fair enough?
> >> Ouch. I mean add one to the ending three to make the four, in 333...34.
> >>
> >> The floor operations basically strips off the infinite threes as in
> >> floor(1/3) = 0
> >> i[n] = floor((1/3) * 10^(n+2) + 1)
> >
> > Oh you mean storage?
> >
> No. I mean getting rid of the string of infinite 3's (think base 10
> representation), 1/3 = .3333333....
>
> Fine.
>
> floor(1/3) = 0
> floor(.3333333...) = 0
>
> However, floor(123456789.1011121314) = 123456789
>
> To get 333...34 in a direct function, well, exploiting 1/3 sure seems to
> work! ;^)
> i[n] = floor((1/3) * 10^(n+2) + 1)

Subtracting 1/3 twice?

Figuring out how many bits in the floating point,
here the idea is to, ..., "subtract 1/3 twice, get
..333...333... somewhere 3444...", that the "somewhere"
was where the floating point unit, would extend.
I.e., it might be fair, "the subtracted 1/3 twice
each extended all the way down .333..., leaving .3444...,
only wherever the floating point unit is beyond precision".

I figure it would be close or same to any other usual calculations,
also "for all the bits in the FPU value what result its bits is
its representation".

I.e. the Floating Point Unit is much bigger than Fixed Point,
the register width usually enough 32/64 doubles and 80 bit intel floats.
Then though keeping the guarantees of fixed point, for floating point,
I mostly look at floating point for how to implement fixed point arithmetic,
and mostly I use the vector registers and about having that the MMX is
used for shifting and so on instead of multimedia along ... multiplying transforms.

Then, though again the floating point unit is bigger, it doesn't have to
do integer arithmetic, I'd like to think that the floating point units, are
given to statistics, for where words are small in small alphabets, in words,
while floating point values are "80 bits".

Which is good to have defined as "10 8 bits".

Also I forget that it's a real number in [0, infinity) not just [0,1],
where "infinity" excuse me is, "Not A Number", "NaN", Not-a-number.

I guess the 754 and 854 kind of define this, IEEE floats,
with platforms "do or don't", IEEE float.

Now I've also written a scritto and if you'll excuse me I'll write it here.

You know, with a C runtime, and socket, and, ..., a C runtime, all sorts C then C++ code compile, ....

Then, some, event machine, is that let's say that there is JavaScript runtime, and, there is
"static JavaScript runtime", in terms of, say, "C runtime, ...".

This is enough library most tools make a POSIX.

"Why old POSIX?" Yeah, "why old POSIX", ....

Tooling down POSIX, basically is for reflecting the derivation of POSIX,
in the line terminal and the setting. Then, there is for making in tooling,
what are the environment and configuration, what makes terminal,
results a terminal.

The console and the terminal, here for kernel mode user land,
reflects a usual enough runtime quite unremarkable.

Then though it results for the framework and platform, for basically giving off
the hardware access, what results that resources are actually well-defined again,
putting nice conventions all over what results "cooperative multithreading under
constant bounds with immediate suspend is nice", what would result under routine
that for example it's usually organized in for example "an overall adaptive resource",
all sorts what running emulation makes run inspecting code under running it, in
terms of binary objects then running them.

So, a terminal, then for a shell, would be usually enough for the filesystem,
what results "as a computer, it looks like a shell, ...". The framebuffer is usually
considered a single area, in what is for example full frame console emulation,
that must provide a shell console and terminal, then with a usual user's minimal
semantics, simplest tools laying around, and homes.

Here the console emulation itself is often where the framebuffer, under console-only,
writes console to the framebuffer, for example where otherwise "what is the framebuffer"
is a "virtual framebuffer" if it's made virtual, there's no patent reasons not to drive output
directly to RAM besides the view adapter.

So, this operating system should be along the lines of "the system has 4GB of RAM and
1GB Quad CPU and an entire Terabyte of disk", then to run "the system has 32 bit address
virtual RAM according to the runtime", 1 Megabyte of Disk, and Zero RAM.

Then for driving full HD and especially for split-screen HD, or 30/60 refresh, it's usually
to be expected that the drivers are in a DMA convention must be. There's also the disk
controller, far be it for me to queue the spool to the disk controller anyways, "putting
all the I/O through the disk controller".

(What results a disk controller.)

Here that's the idea, the I/O, to work the I/O, in terms of: basically cooperating with
the disk controller network I/O, and, the fact that disk "read-often" or "read-random",
makes "memory-mapped files" for the C runtime, disk controller cache and virtual memory.

So, the I/O, is broken down into protocols, the Internet protocols these days about
result to "compression", and, "encryption", and catalog, what result block coders
that all computer nework I/O code executes in their block coding, with "TLS session"
and "Deflate/Gzip un-encumbered compression", or data format.

Then protocols vary for example their routine all transfer, for making:

JPEG protocols <- there are them
"Internet protocols", ..., data transfer in data formats

This way there is for making organization, "here are your videos", "here is your code",
"this is your shell", ....

I'd love to think I could make a computer, that I could program by tapping at the shell.

Probably best start with "disk controller and DMA".

Ross A. Finlayson has brought this to us :
> On Wednesday, September 28, 2022 at 2:17:06 PM UTC-7, Jim Burns wrote:
>> On 9/28/2022 11:27 AM, Ross A. Finlayson wrote:
>>
>>> The split or cut is where the point's line
>>> can bend, ....
>> This is not in the same spacetime continuum
>> as true.
>>> The point of gaplessness of points is
>>> then it's a line, or curve.
>> No,
>> the point of gaplessness is that
>> a function continuous _at every point_
>> will not have jumps.
>>
>> Gaplessness makes sure that "every point"
>> includes enough points in order for
>> it to be true that there are no jumps.
>
> Yeah, that was unusually loose, without connecting it directly
> to the definition of continuous functions for any curves but
> straight lines.
>
> Then it looks just like smooth analysis, ....

It still allows for cusps and corners.

On Wednesday, September 28, 2022 at 5:04:13 PM UTC-4, Jim Burns wrote:
> On 9/28/2022 8:57 AM, Timothy Golden wrote:
> > On Tuesday, September 27, 2022
> > at 2:51:30 PM UTC-4, Timothy Golden wrote:
> >> On Tuesday, September 27, 2022
> >> at 1:56:45 PM UTC-4, Jim Burns wrote:
>
> >>> 333...34 fails that test of being a natural number
> >>> because,
> >>> whatever Sea Monsters be after the line,
> >>> no step crosses the line.
> >>>
> >>> No step crosses the line
> >>> because
> >>> each step which starts on the near side
> >>> ends on the near side.
> >>>
> >>> The presence or absence of Sea Monsters
> >>> doesn't enter the discussion.
> >>> It follows from the nature of 4, 34, 334, ...
> >>>
> >>> That is why I say 333...34 is not a natural number.
> >
> > This is a lie in these last two lines.
> >
> > It doesn't follow at all.
> > every instance in that sequence
> > is natural valued.
> What does it look like to be an instance in
> that sequence?
>
> i[0] = 4
> i[i+1] = 10*i[i] - 6
> AND
> you _can_ get there from here.
>
> What does it looks like to be able to
> get there (i[n]) from here {4)?
>
> For each _split_
> there is a way across that split
> a last-before j and a first-after k
> for which we require k = 10*j - 6
>
> ( A _split_ is two subsets BEFORE and AFTER,
> ( both non-empty, together containing
> ( each instance from 4 to i[n],
> ( and each of BEFORE ≺ each of AFTER
>
> That's what it looks like to be able to
> get there (i[n]) from here (4).
> > By induction we get that
> > the ultimate value is natural.
> No,
> by induction we get that
> each instance is a natural.
>
> Being a natural follows from
> being able to get there (AKA induction).
>
> | Assume otherwise.
> | Assume i[n] is NOT a natural.
> | i[n] ∉ ℕ
> |
> | This split BEFOREₙ AFTERₙ exists
> | BEFOREₙ = { j | ∀j' =< j : j' ∈ ℕ }
> | AFTERₙ = { k | ∃k' =< k : k' ∉ ℕ }
> |
> | By assumption, we _can_ get there from here.
> | Last-before jₙ in BEFOREₙ
> | and first-after kₙ in AFTERₙ exist.
> |
> | From the definitions of BEFOREₙ and AFTERₙ
> | ∀j ∈ BEFOREₙ : j ∈ ℕ
> |
> | kₙ is first in AFTERₙ
> | ∀j < kₙ : j ∈ BEFOREₙ
> | ∀j < kₙ : j ∈ ℕ
> | ~∃j < kₙ : j ∉ ℕ
> |
> | kₙ ∈ AFTERₙ
> | kₙ ∉ ℕ or ∃j < kₙ : j ∉ ℕ
> | But ~∃j < kₙ : j ∉ ℕ
> | kₙ ∉ ℕ
> |
> | jₙ ∈ BEFOREₙ
> | jₙ ∈ ℕ
> |
> | jₙ ∈ ℕ kₙ ∉ ℕ and
> | kₙ immediately follows jₙ in the sequence.
> |
> | However,
> | kₙ immediately follows jₙ in the sequence.
> | kₙ = 10*jₙ - 6
> | if jₙ ∈ ℕ then kₙ ∈ ℕ
> | Contradiction.
>
> Therefore
> i[n] IS a natural.
> > By induction we get that
> > the ultimate value is natural.
> No,
> we get that
> each instance i[n] is followed by i[n+1]
> Each instance i[n] is not the ultimate value.
>
> We get that
> no ultimate value exists.
> > Let's actually implement it
> > within the sequence now:
> > 4, 34, 334, ..., 333...34
> If
> you can't get there (333...34) from here (4)
> then
> there (333...34) is not within the sequence.
>
> There (333...34) is after all of 3,34,334,3334,...
> Getting there requires a step j to 10*j-6
> from not after all 3,34,334,3334,...
> to all after all 3,34,334,3334,...

It is correct that there is no step to infinity.
However this sequence was selected by you, and agreeable to me.
It was selected because the pattern, repetitious as it is, does lead up to the ultimate value that is under scrutiny.
In this way the sequence is not critical to the value, but it helps cover the ground.
Allow me to state again:
4 is natural. 34 is natural. 334 is natural. 3334 is natural.
Therefor 333...34 is natural
You can stop whenever you like and the value will be natural.
Therefor there is no logic present which prohibits 333...34 from being natural, which you were claiming is a logical conclusion.
I can understand that you don't like 333...34, but here we are trying to analyze it cleanly.
Your construction in i and i[i] is flawed so I won't touch it for the sake of simplicity.

Once again you've managed to dodge the usage of the ellipsis and its own conflicted meaning. There to me lays the crux for rejection, but it is a crux that you have crossed already into I think. It is this fact that you willingly go there on other familiar occasions, but this subtle variation has not been tried yet. Clearly it is outside of the stuff in the books. Well it could easily be in somebody's writings but they are obscure.

>
> There is no such step.
> You can't get there from here.
> 333...34 is not in the sequence.
> > That we can terminate this sequence
> You are mistaken about what is needed
> to be in the sequence:
> You _can_ get there from here.
>
> Something not-in the sequence can be
> after all of the sequence, but it is
> still not-in the sequence.
> > That we can terminate this sequence
> > is only possibly due to the embedded ellipsis,
> > which you seem fond of ignoring.
> I don't always ignore your '...'
> Repeating myself is boring.
>
> Again, you are mistaken about what '...' means.
> > In that the ellipsis itself is inductive in nature:
> The ellipsis replaces detailed description
> where that description is already obvious
> and would only clutter the page, chalkboard,
> or window.
No Jim. You have completely dodged the halting problem of the ellipses.
The fact that we could go on forever could have been left out of all of analysis, since it should be clear in the analyst's mind that there is no need to stop the analysis at three instances. Therefor no usage of the ellipsis would be necessary under your own rendition. The truth is that the ellipsis implies something that the human cannot achieve. Neither can the computer.. That you are struggling with this issue on the flattest instance of such usage, yet one which grows so much more quickly than your usual usages, exposes a weakness in your frame of thought. I know better than to punch your buttons, but as we carry on here your own refusal to address the iterative nature of the ellipsis as if it is not even present cannot go well.

I think in such a flat case as this the amount of analysis required to grant induction is only slight. There really is nothing clever to do here. In this regard we have not only simplified induction, but we've gotten away with a constant instance of infinity capable of computation as well. We owe it all to the modulo form or our number system; something refused by traditional natural analysis which can only creep along. The fact that you cannot disclose any analysis on this side of the fence is troubling. I have come over to your side in agreement with some of your own points. Indeed, I am not taking a charlatan approach here. The sequestering of these forms to a branch of mathematics is entirely appropriate. As to what else sits on that branch... here there may be quite some discussion. In this case the ellipsis discloses that there is more information and that the details are not covered yet.

>
> It is the description which the ellipsis
> replaces which is or is not inductive.
> All the ellipsis says is "You know what I mean".
> > this point is not getting any attention in
> > the discussion.
> Once you've been told you're reading '...'
> incorrectly, there's not much left to say
> -- at least, until you read it correctly.
> > Clearly the value is inductive.
> You can't get there from here.
> It is not inductive.

Well, now, this is becoming infantile.
I will say again that if I am misusing the terminology 'induction' then whatever careful definition denies me likely will expose the correct term, and when it does under its dissection we will still be staring down the ellipsis and grinding the juncture of the two for a neat whirring sound. A child could say that we will never get there, and they'd be right. That is the nature of infinity. Back on Peano the same mixing effect can be seen, as if the successor function is something new and different. I honestly never meant to delve into natural analysis at all. I pushed on the magnitude of the polysign rudiment sx and all this other stuff falls out. I suppose I am nearing a curriculum now, yet it seems awfully thin. Ahh, but shouldn't it be that way when decent mathematics is found? The quantity of correctives that I can lay out now is substantial. That ambiguities do exist in mathematics places this subject in a very different realm than I was told about it. How fascinating that philosophy could find space here. Physics too. I feel quite secure that this is valid math mainly because of its computability. At first it is unclear, but by practicing with these values their inductive form proves itself to be consistent.

On Thursday, September 29, 2022 at 6:48:06 AM UTC-7, timba...@gmail.com wrote:
> On Wednesday, September 28, 2022 at 5:04:13 PM UTC-4, Jim Burns wrote:
> > On 9/28/2022 8:57 AM, Timothy Golden wrote:
> > > On Tuesday, September 27, 2022
> > > at 2:51:30 PM UTC-4, Timothy Golden wrote:
> > >> On Tuesday, September 27, 2022
> > >> at 1:56:45 PM UTC-4, Jim Burns wrote:
> >
> > >>> 333...34 fails that test of being a natural number
> > >>> because,
> > >>> whatever Sea Monsters be after the line,
> > >>> no step crosses the line.
> > >>>
> > >>> No step crosses the line
> > >>> because
> > >>> each step which starts on the near side
> > >>> ends on the near side.
> > >>>
> > >>> The presence or absence of Sea Monsters
> > >>> doesn't enter the discussion.
> > >>> It follows from the nature of 4, 34, 334, ...
> > >>>
> > >>> That is why I say 333...34 is not a natural number.
> > >
> > > This is a lie in these last two lines.
> > >
> > > It doesn't follow at all.
> > > every instance in that sequence
> > > is natural valued.
> > What does it look like to be an instance in
> > that sequence?
> >
> > i[0] = 4
> > i[i+1] = 10*i[i] - 6
> > AND
> > you _can_ get there from here.
> >
> > What does it looks like to be able to
> > get there (i[n]) from here {4)?
> >
> > For each _split_
> > there is a way across that split
> > a last-before j and a first-after k
> > for which we require k = 10*j - 6
> >
> > ( A _split_ is two subsets BEFORE and AFTER,
> > ( both non-empty, together containing
> > ( each instance from 4 to i[n],
> > ( and each of BEFORE ≺ each of AFTER
> >
> > That's what it looks like to be able to
> > get there (i[n]) from here (4).
> > > By induction we get that
> > > the ultimate value is natural.
> > No,
> > by induction we get that
> > each instance is a natural.
> >
> > Being a natural follows from
> > being able to get there (AKA induction).
> >
> > | Assume otherwise.
> > | Assume i[n] is NOT a natural.
> > | i[n] ∉ ℕ
> > |
> > | This split BEFOREₙ AFTERₙ exists
> > | BEFOREₙ = { j | ∀j' =< j : j' ∈ ℕ }
> > | AFTERₙ = { k | ∃k' =< k : k' ∉ ℕ }
> > |
> > | By assumption, we _can_ get there from here.
> > | Last-before jₙ in BEFOREₙ
> > | and first-after kₙ in AFTERₙ exist.
> > |
> > | From the definitions of BEFOREₙ and AFTERₙ
> > | ∀j ∈ BEFOREₙ : j ∈ ℕ
> > |
> > | kₙ is first in AFTERₙ
> > | ∀j < kₙ : j ∈ BEFOREₙ
> > | ∀j < kₙ : j ∈ ℕ
> > | ~∃j < kₙ : j ∉ ℕ
> > |
> > | kₙ ∈ AFTERₙ
> > | kₙ ∉ ℕ or ∃j < kₙ : j ∉ ℕ
> > | But ~∃j < kₙ : j ∉ ℕ
> > | kₙ ∉ ℕ
> > |
> > | jₙ ∈ BEFOREₙ
> > | jₙ ∈ ℕ
> > |
> > | jₙ ∈ ℕ kₙ ∉ ℕ and
> > | kₙ immediately follows jₙ in the sequence.
> > |
> > | However,
> > | kₙ immediately follows jₙ in the sequence.
> > | kₙ = 10*jₙ - 6
> > | if jₙ ∈ ℕ then kₙ ∈ ℕ
> > | Contradiction.
> >
> > Therefore
> > i[n] IS a natural.
> > > By induction we get that
> > > the ultimate value is natural.
> > No,
> > we get that
> > each instance i[n] is followed by i[n+1]
> > Each instance i[n] is not the ultimate value.
> >
> > We get that
> > no ultimate value exists.
> > > Let's actually implement it
> > > within the sequence now:
> > > 4, 34, 334, ..., 333...34
> > If
> > you can't get there (333...34) from here (4)
> > then
> > there (333...34) is not within the sequence.
> >
> > There (333...34) is after all of 3,34,334,3334,...
> > Getting there requires a step j to 10*j-6
> > from not after all 3,34,334,3334,...
> > to all after all 3,34,334,3334,...
> It is correct that there is no step to infinity.
> However this sequence was selected by you, and agreeable to me.
> It was selected because the pattern, repetitious as it is, does lead up to the ultimate value that is under scrutiny.
> In this way the sequence is not critical to the value, but it helps cover the ground.
> Allow me to state again:
> 4 is natural. 34 is natural. 334 is natural. 3334 is natural.
> Therefor 333...34 is natural
> You can stop whenever you like and the value will be natural.
> Therefor there is no logic present which prohibits 333...34 from being natural, which you were claiming is a logical conclusion.
> I can understand that you don't like 333...34, but here we are trying to analyze it cleanly.
> Your construction in i and i[i] is flawed so I won't touch it for the sake of simplicity.
>
> Once again you've managed to dodge the usage of the ellipsis and its own conflicted meaning. There to me lays the crux for rejection, but it is a crux that you have crossed already into I think. It is this fact that you willingly go there on other familiar occasions, but this subtle variation has not been tried yet. Clearly it is outside of the stuff in the books. Well it could easily be in somebody's writings but they are obscure.
> >
> > There is no such step.
> > You can't get there from here.
> > 333...34 is not in the sequence.
> > > That we can terminate this sequence
> > You are mistaken about what is needed
> > to be in the sequence:
> > You _can_ get there from here.
> >
> > Something not-in the sequence can be
> > after all of the sequence, but it is
> > still not-in the sequence.
> > > That we can terminate this sequence
> > > is only possibly due to the embedded ellipsis,
> > > which you seem fond of ignoring.
> > I don't always ignore your '...'
> > Repeating myself is boring.
> >
> > Again, you are mistaken about what '...' means.
> > > In that the ellipsis itself is inductive in nature:
> > The ellipsis replaces detailed description
> > where that description is already obvious
> > and would only clutter the page, chalkboard,
> > or window.
> No Jim. You have completely dodged the halting problem of the ellipses.
> The fact that we could go on forever could have been left out of all of analysis, since it should be clear in the analyst's mind that there is no need to stop the analysis at three instances. Therefor no usage of the ellipsis would be necessary under your own rendition. The truth is that the ellipsis implies something that the human cannot achieve. Neither can the computer. That you are struggling with this issue on the flattest instance of such usage, yet one which grows so much more quickly than your usual usages, exposes a weakness in your frame of thought. I know better than to punch your buttons, but as we carry on here your own refusal to address the iterative nature of the ellipsis as if it is not even present cannot go well.
>
> I think in such a flat case as this the amount of analysis required to grant induction is only slight. There really is nothing clever to do here. In this regard we have not only simplified induction, but we've gotten away with a constant instance of infinity capable of computation as well. We owe it all to the modulo form or our number system; something refused by traditional natural analysis which can only creep along. The fact that you cannot disclose any analysis on this side of the fence is troubling. I have come over to your side in agreement with some of your own points. Indeed, I am not taking a charlatan approach here. The sequestering of these forms to a branch of mathematics is entirely appropriate. As to what else sits on that branch... here there may be quite some discussion. In this case the ellipsis discloses that there is more information and that the details are not covered yet.
> >
> > It is the description which the ellipsis
> > replaces which is or is not inductive.
> > All the ellipsis says is "You know what I mean".
> > > this point is not getting any attention in
> > > the discussion.
> > Once you've been told you're reading '...'
> > incorrectly, there's not much left to say
> > -- at least, until you read it correctly.
> > > Clearly the value is inductive.
> > You can't get there from here.
> > It is not inductive.
> Well, now, this is becoming infantile.
> I will say again that if I am misusing the terminology 'induction' then whatever careful definition denies me likely will expose the correct term, and when it does under its dissection we will still be staring down the ellipsis and grinding the juncture of the two for a neat whirring sound. A child could say that we will never get there, and they'd be right. That is the nature of infinity. Back on Peano the same mixing effect can be seen, as if the successor function is something new and different. I honestly never meant to delve into natural analysis at all. I pushed on the magnitude of the polysign rudiment sx and all this other stuff falls out. I suppose I am nearing a curriculum now, yet it seems awfully thin. Ahh, but shouldn't it be that way when decent mathematics is found? The quantity of correctives that I can lay out now is substantial. That ambiguities do exist in mathematics places this subject in a very different realm than I was told about it. How fascinating that philosophy could find space here. Physics too. I feel quite secure that this is valid math mainly because of its computability. At first it is unclear, but by practicing with these values their inductive form proves itself to be consistent.
>
> The best features of the new analysis is back on the decimal value and its own natural valued forbearance. Back here the juncture of the reals and the naturals was felt just by implementing a secondary form of unity. Whether you count large values in tens or twelves, as was done for some things back in the day, and maybe donuts still come out this way, structural analysis suggests that the implementation of such units is all that rational analysis actually needs, and this then takes us to an even more general application of unity beyond the decimal point. Yet as radix concerns become a part of the awareness of these digital forms then it simply these radix forms with the decimal point in tow which allow for such representations as:
> 1/3 = 0.1
> and the digital power can commence here too so that ninths are not such a foreign thing to the thirds. Yet for the rational analysis they lay off by six steps, whereas here they are but one step away. When the natural analyst threw away his digits he gave up quite a lot. Taking them back what we see is that where he landed us by the time he gets to the continuum is contaminated. Not only are the real values lacking in correspondence to reality; they make claims that are not possible to implement. The form that comes from the natural valued interpretation augmented by the decimal position (unital position) yields values that every engineer and physicist is comfortable with already. For this reason this attack on mathematics will stand the test of time. How long will mathematics withstand the attack? Given what politician's are willing to do these days it could be quite some time.

On Wednesday, September 28, 2022 at 11:13:40 PM UTC-7, Ross A. Finlayson wrote:
> On Wednesday, September 28, 2022 at 10:51:05 PM UTC-7, Chris M. Thomasson wrote:
> > On 9/28/2022 10:28 PM, Ross A. Finlayson wrote:
> > > On Wednesday, September 28, 2022 at 10:23:37 PM UTC-7, Chris M. Thomasson wrote:
> > >> On 9/28/2022 10:13 PM, Chris M. Thomasson wrote:
> > >>> On 9/28/2022 9:32 PM, Ross A. Finlayson wrote:
> > >>>> On Wednesday, September 28, 2022 at 9:28:37 PM UTC-7, Chris M.
> > >>>> Thomasson wrote:
> > >>>>> On 9/24/2022 9:57 AM, Sergi o wrote:
> > >>>>>> On 9/21/2022 2:17 PM, mitchr...@gmail.com wrote:
> > >> [...]
> > >>> Humm... Interesting. Wrt 13/3 - 1 is = 3.333..., so we can scoot over
> > >>> the endless three's, so to speak... Just like my direct solution does
> > >>> for 333...34. Add one, to the the ending 4. Fair enough?
> > >> Ouch. I mean add one to the ending three to make the four, in 333...34.
> > >>
> > >> The floor operations basically strips off the infinite threes as in
> > >> floor(1/3) = 0
> > >> i[n] = floor((1/3) * 10^(n+2) + 1)
> > >
> > > Oh you mean storage?
> > >
> > No. I mean getting rid of the string of infinite 3's (think base 10
> > representation), 1/3 = .3333333....
> >
> > Fine.
> >
> > floor(1/3) = 0
> > floor(.3333333...) = 0
> >
> > However, floor(123456789.1011121314) = 123456789
> >
> > To get 333...34 in a direct function, well, exploiting 1/3 sure seems to
> > work! ;^)
> > i[n] = floor((1/3) * 10^(n+2) + 1)
> Subtracting 1/3 twice?
>
> Figuring out how many bits in the floating point,
> here the idea is to, ..., "subtract 1/3 twice, get
> .333...333... somewhere 3444...", that the "somewhere"
> was where the floating point unit, would extend.
> I.e., it might be fair, "the subtracted 1/3 twice
> each extended all the way down .333..., leaving .3444...,
> only wherever the floating point unit is beyond precision".
>
> I figure it would be close or same to any other usual calculations,
> also "for all the bits in the FPU value what result its bits is
> its representation".
>
> I.e. the Floating Point Unit is much bigger than Fixed Point,
> the register width usually enough 32/64 doubles and 80 bit intel floats.
> Then though keeping the guarantees of fixed point, for floating point,
> I mostly look at floating point for how to implement fixed point arithmetic,
> and mostly I use the vector registers and about having that the MMX is
> used for shifting and so on instead of multimedia along ... multiplying transforms.
>
> Then, though again the floating point unit is bigger, it doesn't have to
> do integer arithmetic, I'd like to think that the floating point units, are
> given to statistics, for where words are small in small alphabets, in words,
> while floating point values are "80 bits".
>
> Which is good to have defined as "10 8 bits".
>
> Also I forget that it's a real number in [0, infinity) not just [0,1],
> where "infinity" excuse me is, "Not A Number", "NaN", Not-a-number.
>
> I guess the 754 and 854 kind of define this, IEEE floats,
> with platforms "do or don't", IEEE float.
>
>
>
>
> Now I've also written a scritto and if you'll excuse me I'll write it here.
>
> You know, with a C runtime, and socket, and, ..., a C runtime, all sorts C then C++ code compile, ....
>
> Then, some, event machine, is that let's say that there is JavaScript runtime, and, there is
> "static JavaScript runtime", in terms of, say, "C runtime, ...".
>
> This is enough library most tools make a POSIX.
>
> "Why old POSIX?" Yeah, "why old POSIX", ....
>
> Tooling down POSIX, basically is for reflecting the derivation of POSIX,
> in the line terminal and the setting. Then, there is for making in tooling,
> what are the environment and configuration, what makes terminal,
> results a terminal.
>
> The console and the terminal, here for kernel mode user land,
> reflects a usual enough runtime quite unremarkable.
>
> Then though it results for the framework and platform, for basically giving off
> the hardware access, what results that resources are actually well-defined again,
> putting nice conventions all over what results "cooperative multithreading under
> constant bounds with immediate suspend is nice", what would result under routine
> that for example it's usually organized in for example "an overall adaptive resource",
> all sorts what running emulation makes run inspecting code under running it, in
> terms of binary objects then running them.
>
> So, a terminal, then for a shell, would be usually enough for the filesystem,
> what results "as a computer, it looks like a shell, ...". The framebuffer is usually
> considered a single area, in what is for example full frame console emulation,
> that must provide a shell console and terminal, then with a usual user's minimal
> semantics, simplest tools laying around, and homes.
>
> Here the console emulation itself is often where the framebuffer, under console-only,
> writes console to the framebuffer, for example where otherwise "what is the framebuffer"
> is a "virtual framebuffer" if it's made virtual, there's no patent reasons not to drive output
> directly to RAM besides the view adapter.
>
> So, this operating system should be along the lines of "the system has 4GB of RAM and
> 1GB Quad CPU and an entire Terabyte of disk", then to run "the system has 32 bit address
> virtual RAM according to the runtime", 1 Megabyte of Disk, and Zero RAM.
>
>
> Then for driving full HD and especially for split-screen HD, or 30/60 refresh, it's usually
> to be expected that the drivers are in a DMA convention must be. There's also the disk
> controller, far be it for me to queue the spool to the disk controller anyways, "putting
> all the I/O through the disk controller".
>
> (What results a disk controller.)
>
> Here that's the idea, the I/O, to work the I/O, in terms of: basically cooperating with
> the disk controller network I/O, and, the fact that disk "read-often" or "read-random",
> makes "memory-mapped files" for the C runtime, disk controller cache and virtual memory.
>
> So, the I/O, is broken down into protocols, the Internet protocols these days about
> result to "compression", and, "encryption", and catalog, what result block coders
> that all computer nework I/O code executes in their block coding, with "TLS session"
> and "Deflate/Gzip un-encumbered compression", or data format.
>
> Then protocols vary for example their routine all transfer, for making:
>
> JPEG protocols <- there are them
> "Internet protocols", ..., data transfer in data formats
>
> This way there is for making organization, "here are your videos", "here is your code",
> "this is your shell", ....
>
> I'd love to think I could make a computer, that I could program by tapping at the shell.
>
> Probably best start with "disk controller and DMA".

https://blogs.synopsys.com/expressyourself/2013/07/29/what-is-dma-and-what-good-is-it-to-me/

Hmm..., it says "PCI Express is the new DMA, also it's DMA".

"A PCI Bus has no DMA controller", ....

"Most PCIe devices are DMA masters", .... "the main memory is always a slave".

Ohh...,

So, the PCIe devices are fed data in memory, off the PCIe message bus would-be.

"A basic PCIe link includes two, low voltage, differentially driven signal pairs,
a transmit pair and a receive pair, ...".

"... Mapped legacy signals into Message Bus ...".

"PCI, DP, USB, and SATA, ..."

Hmm, there is IDE and there is SCSI, there is Serial ATA, and, there is for figuring
out the steady-stage memory logic and "classical disk controller", in terms of
what a "classical disk-controller for example with access control" vis-a-vis
"a very large bank of flash RAM called the disk", here is in terms of whatever
it actually is, however its communicated and under whatever conditions it does.

"Down to the TLP: How PCI express devices talk"

On 9/28/2022 11:13 PM, Ross A. Finlayson wrote:
> On Wednesday, September 28, 2022 at 10:51:05 PM UTC-7, Chris M. Thomasson wrote:
>> On 9/28/2022 10:28 PM, Ross A. Finlayson wrote:
>>> On Wednesday, September 28, 2022 at 10:23:37 PM UTC-7, Chris M. Thomasson wrote:
>>>> On 9/28/2022 10:13 PM, Chris M. Thomasson wrote:
>>>>> On 9/28/2022 9:32 PM, Ross A. Finlayson wrote:
>>>>>> On Wednesday, September 28, 2022 at 9:28:37 PM UTC-7, Chris M.
>>>>>> Thomasson wrote:
>>>>>>> On 9/24/2022 9:57 AM, Sergi o wrote:
>>>>>>>> On 9/21/2022 2:17 PM, mitchr...@gmail.com wrote:
>>>> [...]
>>>>> Humm... Interesting. Wrt 13/3 - 1 is = 3.333..., so we can scoot over
>>>>> the endless three's, so to speak... Just like my direct solution does
>>>>> for 333...34. Add one, to the the ending 4. Fair enough?
>>>> Ouch. I mean add one to the ending three to make the four, in 333...34.
>>>>
>>>> The floor operations basically strips off the infinite threes as in
>>>> floor(1/3) = 0
>>>> i[n] = floor((1/3) * 10^(n+2) + 1)
>>>
>>> Oh you mean storage?
>>>
>> No. I mean getting rid of the string of infinite 3's (think base 10
>> representation), 1/3 = .3333333....
>>
>> Fine.
>>
>> floor(1/3) = 0
>> floor(.3333333...) = 0
>>
>> However, floor(123456789.1011121314) = 123456789
>>
>> To get 333...34 in a direct function, well, exploiting 1/3 sure seems to
>> work! ;^)
>> i[n] = floor((1/3) * 10^(n+2) + 1)
>
> Subtracting 1/3 twice?
[...]

:^)

I am just using the floor function to zap all the .333's into oblivion.
Quick and easy. Another way, without floor could be, say to get 333...34
directly, using subtraction like you suggested:

(1/3 * 100) - (1/3) + 1 = 34
(1/3 * 1000) - (1/3) + 1 = 334
(1/3 * 10000) - (1/3) + 1 = 3334

on and on. There is no need to depend on a predecessor and/or successor.
It's direct. Therefore:

i[n] = (1/3 * 10^(n + 2)) - (1/3) + 1
_________________________

i[5] = (1/3 * 10^(5 + 2)) - (1/3) + 1 = 3333334

i[6] = (1/3 * 10^(6 + 2)) - (1/3) + 1 = 33333334

i[7] = (1/3 * 10^(7 + 2)) - (1/3) + 1 = 333333334
_________________________

I like the floor version, but subtracting works as well. Thanks.

i[0] = (1/3 * 10^(0 + 2)) - (1/3) + 1 = 34

;^D

On 9/29/2022 11:37 AM, Chris M. Thomasson wrote:
> On 9/28/2022 11:13 PM, Ross A. Finlayson wrote:
>> On Wednesday, September 28, 2022 at 10:51:05 PM UTC-7, Chris M.
>> Thomasson wrote:
>>> On 9/28/2022 10:28 PM, Ross A. Finlayson wrote:
>>>> On Wednesday, September 28, 2022 at 10:23:37 PM UTC-7, Chris M.
>>>> Thomasson wrote:
>>>>> On 9/28/2022 10:13 PM, Chris M. Thomasson wrote:
>>>>>> On 9/28/2022 9:32 PM, Ross A. Finlayson wrote:
>>>>>>> On Wednesday, September 28, 2022 at 9:28:37 PM UTC-7, Chris M.
>>>>>>> Thomasson wrote:
>>>>>>>> On 9/24/2022 9:57 AM, Sergi o wrote:
>>>>>>>>> On 9/21/2022 2:17 PM, mitchr...@gmail.com wrote:
>>>>> [...]
>>>>>> Humm... Interesting. Wrt 13/3 - 1 is = 3.333..., so we can scoot over
>>>>>> the endless three's, so to speak... Just like my direct solution does
>>>>>> for 333...34. Add one, to the the ending 4. Fair enough?
>>>>> Ouch. I mean add one to the ending three to make the four, in
>>>>> 333...34.
>>>>>
>>>>> The floor operations basically strips off the infinite threes as in
>>>>> floor(1/3) = 0
>>>>> i[n] = floor((1/3) * 10^(n+2) + 1)
>>>>
>>>> Oh you mean storage?
>>>>
>>> No. I mean getting rid of the string of infinite 3's (think base 10
>>> representation), 1/3 = .3333333....
>>>
>>> Fine.
>>>
>>> floor(1/3) = 0
>>> floor(.3333333...) = 0
>>>
>>> However, floor(123456789.1011121314) = 123456789
>>>
>>> To get 333...34 in a direct function, well, exploiting 1/3 sure seems to
>>> work! ;^)
>>> i[n] = floor((1/3) * 10^(n+2) + 1)
>>
>> Subtracting 1/3 twice?
> [...]
>
> :^)
>
> I am just using the floor function to zap all the .333's into oblivion.
> Quick and easy. Another way, without floor could be, say to get 333...34
> directly, using subtraction like you suggested:
>
> (1/3 * 100) - (1/3) + 1 = 34
> (1/3 * 1000) - (1/3) + 1 = 334
> (1/3 * 10000) - (1/3) + 1 = 3334
>
> on and on. There is no need to depend on a predecessor and/or successor.
> It's direct. Therefore:
>
> i[n] = (1/3 * 10^(n + 2)) - (1/3) + 1
> _________________________
>
> i[5] = (1/3 * 10^(5 + 2)) - (1/3) + 1 = 3333334
>
> i[6] = (1/3 * 10^(6 + 2)) - (1/3) + 1 = 33333334
>
> i[7] = (1/3 * 10^(7 + 2)) - (1/3) + 1 = 333333334
> _________________________
>
> I like the floor version, but subtracting works as well. Thanks.
>
> i[0] = (1/3 * 10^(0 + 2)) - (1/3) + 1 = 34
>
> ;^D
>

((1/3 * 10^(0 + 2)) - (1/3) + 1) = (floor((1/3) * 10^(0+2) + 1)) = TRUE

;^)

On Thursday, September 29, 2022 at 3:01:41 AM UTC-7, FromTheRafters wrote:
> Ross A. Finlayson has brought this to us :
> > On Wednesday, September 28, 2022 at 2:17:06 PM UTC-7, Jim Burns wrote:
> >> On 9/28/2022 11:27 AM, Ross A. Finlayson wrote:
> >>
> >>> The split or cut is where the point's line
> >>> can bend, ....
> >> This is not in the same spacetime continuum
> >> as true.
> >>> The point of gaplessness of points is
> >>> then it's a line, or curve.
> >> No,
> >> the point of gaplessness is that
> >> a function continuous _at every point_
> >> will not have jumps.
> >>
> >> Gaplessness makes sure that "every point"
> >> includes enough points in order for
> >> it to be true that there are no jumps.
> >
> > Yeah, that was unusually loose, without connecting it directly
> > to the definition of continuous functions for any curves but
> > straight lines.
> >
> > Then it looks just like smooth analysis, ....
> It still allows for cusps and corners.

Quarter-angles?

It's like there's orthogonality, and there's quarter-angles.

The quarter angle is the quadrant's angle.

(Rotation is the angle.)

On Friday, September 30, 2022 at 7:54:45 AM UTC-7, Ross A. Finlayson wrote:
> On Thursday, September 29, 2022 at 3:01:41 AM UTC-7, FromTheRafters wrote:
> > Ross A. Finlayson has brought this to us :
> > > On Wednesday, September 28, 2022 at 2:17:06 PM UTC-7, Jim Burns wrote:
> > >> On 9/28/2022 11:27 AM, Ross A. Finlayson wrote:
> > >>
> > >>> The split or cut is where the point's line
> > >>> can bend, ....
> > >> This is not in the same spacetime continuum
> > >> as true.
> > >>> The point of gaplessness of points is
> > >>> then it's a line, or curve.
> > >> No,
> > >> the point of gaplessness is that
> > >> a function continuous _at every point_
> > >> will not have jumps.
> > >>
> > >> Gaplessness makes sure that "every point"
> > >> includes enough points in order for
> > >> it to be true that there are no jumps.
> > >
> > > Yeah, that was unusually loose, without connecting it directly
> > > to the definition of continuous functions for any curves but
> > > straight lines.
> > >
> > > Then it looks just like smooth analysis, ....
> > It still allows for cusps and corners.
> Quarter-angles?
>
> It's like there's orthogonality, and there's quarter-angles.
>
> The quarter angle is the quadrant's angle.
>
> (Rotation is the angle.)

"The geometry where all angles are right", ....

There's the isosceles triangle opposite an angle,
"what is the angle". The right angle formed on
the circle, is pi/4 radians times two, intersecting
the circle according to the radius of the circle,
what is the opening of the compass, that it happens
that the inscribed squares' side length, is the
opening of the compass, that the square's root two
is the diameter.

On 9/29/2022 9:48 AM, Timothy Golden wrote:
> On Wednesday, September 28, 2022
> at 5:04:13 PM UTC-4, Jim Burns wrote:

>> There (333...34) is after all of 3,34,334,3334,...
>> Getting there requires a step j to 10*j-6
>> from not after all 3,34,334,3334,...
>> to all after all 3,34,334,3334,...
>
> It is correct that
> there is no step to infinity.

I'm saying something a little different.

There is no step out of finity.

That follows from the description of finity,
not from the description of infinity,
so it is very hard to avoid with some
(hypothetical) clever adjustment of infinity.
_Finity_ is unchanged.

> However this sequence was selected by you,
> and agreeable to me.

I selected it and asked
"Is that what you (TG) mean by 333...34?"
Please don't forget that I'm discussing
4,34,334,3334,... in order to understand
what you're trying to tell us.

> It was selected because the pattern,
> repetitious as it is,
> does lead up to the ultimate value
> that is under scrutiny.

No,
I remember why I (JB) selected it.
I selected it because I thought you (TG)
might be using 333...34 to refer to
the thing after all of 4,34,334,3334,...

Do you see the difference?
The difference is the point I want to make.

There is no ultimate value of 4,34,334,3334,...

The sense in which 4,34,334,... leads to 4,
leads to 34, leads to 334, leads to 3334,
_that_ sense does not apply to anything
after all of 4,34,334,3334,...
_that_ sense does not apply to an ultimate value.

In other words,
you can't get there from here.

> In this way the sequence is not critical
> to the value, but it helps cover the ground.
>
> Allow me to state again:
> 4 is natural.
> 34 is natural.
> 334 is natural.
> 3334 is natural.
> Therefor 333...34 is natural
> You can stop whenever you like
> and the value will be natural.

I see these _different_ possibilities for
what you (TG) mean by 333...34
(1)
333...34 is first after all of 4,34,334,...

(2)
333...34 is the set {4,34,334,3334,...}

(3)
333...34 is any of the set {4,34,334,3334,...}

My best guess is (1).
Please tell me if you mean something else.

> Therefor there is no logic present which
> prohibits 333...34 from being natural,
> which you were claiming is a logical conclusion.

My conclusion applies to (1)
Are we talking about (1)?

This is not how induction works.
| 4 is natural.
| ...
| Therefor 333...34 is natural
| You can stop whenever you like
| and the value will be natural.

Yes,
you can keep going, 4, 34, 333333334,
and each will be a natural number.
But the question is: is 333...34 one of them?
Keeping-going does not address that question.

>>> Clearly the value is inductive.
>>
>> You can't get there from here.
>> It is not inductive.
>
> Well, now, this is becoming infantile.

I don't want to offend.
I'm trying to find common ground upon which
you and I can communicate.
For that purpose, "infantile" is a feature,
not a bug.

It would be nice if we could have used '...' and
"inductive" to communicate. It looks as though
that is not going to happen.

----
Consider a few examples of recursive
-- I'm going to call them "definitions"
even though that's problematic --
a few examples of recursive definitions.

(Si = i+1 the successor of i)

Adding i:
j + 0 = j
j + Si = S(j + i)

Multiplying by i:
j*0 = 0
j*Si = (j*i) + j

Raising to the power of i:
j^0 = 1
j^Si = (j^i)*j

Factorial of i
0! = 1
(Si)! = (i+1)*(i!)

We want a definition to tell us what
a term means which we don't know
using terms which we know.

But see what we have here with these definitions.
'+' is on both sides.
'*' is on both sides.
'^' is on both sides.
'!' is on both sides.

These definitions can't be telling us what
'+' ('*' '^' '!') means which we don't know
using the same '+' ('*' '^' '!') which we know.
We can't be having our cake and eating it too.

However,
we can give better descriptions of '+'
('*' '^' '!') which do not have that problem.

The key assumption is that
you _can_ get there from here.

For a particular sum j+k, we assume that
a finite sequence ⟨j+0 j+1 ⋯ j+k⟩ exists.
Each split has a last-before j+i and
a first-after j+(i+1).
The 0ₜₕ value is j.
The kₜₕ value, whatever that value is,
is the defined sum.

Addition defined, without using '+' or '⋯' :

j+k = m ⟺
∃𝑺,
∀i < k, ∀mᵢ : (⟨mᵢ i⟩ ∈ 𝑺 ⟺ ⟨Smᵢ Si⟩ ∈ 𝑺)
∧ ⟨j 0⟩ ∈ 𝑺 ∧ ⟨m k⟩ ∈ 𝑺

𝑺 is the finite sequence of increments from j
to m, indexed by i.
If 𝑺 exists, j+k = m. If not, then not.

And similarly for defining '*' '^' '!'
without '*' '^' '!' or '⋯'

"You _can_ get there from here" is
an irreplaceable concept for important parts
of computing, even if it spends most of its
time buried. But being buried is the fate of
most foundations.

On Friday, September 30, 2022 at 2:43:32 PM UTC-4, Jim Burns wrote:
> On 9/29/2022 9:48 AM, Timothy Golden wrote:
> > On Wednesday, September 28, 2022
> > at 5:04:13 PM UTC-4, Jim Burns wrote:
>
> >> There (333...34) is after all of 3,34,334,3334,...
> >> Getting there requires a step j to 10*j-6
> >> from not after all 3,34,334,3334,...
> >> to all after all 3,34,334,3334,...
> >
> > It is correct that
> > there is no step to infinity.
> I'm saying something a little different.
>
> There is no step out of finity.
>
> That follows from the description of finity,
> not from the description of infinity,
> so it is very hard to avoid with some
> (hypothetical) clever adjustment of infinity.
> _Finity_ is unchanged.
> > However this sequence was selected by you,
> > and agreeable to me.
> I selected it and asked
> "Is that what you (TG) mean by 333...34?"
> Please don't forget that I'm discussing
> 4,34,334,3334,... in order to understand
> what you're trying to tell us.
> > It was selected because the pattern,
> > repetitious as it is,
> > does lead up to the ultimate value
> > that is under scrutiny.
> No,
> I remember why I (JB) selected it.
> I selected it because I thought you (TG)
> might be using 333...34 to refer to
> the thing after all of 4,34,334,3334,...
>
> Do you see the difference?
> The difference is the point I want to make.
>
> There is no ultimate value of 4,34,334,3334,...
>
> The sense in which 4,34,334,... leads to 4,
> leads to 34, leads to 334, leads to 3334,
> _that_ sense does not apply to anything
> after all of 4,34,334,3334,...
> _that_ sense does not apply to an ultimate value.
>
> In other words,
> you can't get there from here.
> > In this way the sequence is not critical
> > to the value, but it helps cover the ground.
> >
> > Allow me to state again:
> > 4 is natural.
> > 34 is natural.
> > 334 is natural.
> > 3334 is natural.
> > Therefor 333...34 is natural
> > You can stop whenever you like
> > and the value will be natural.
> I see these _different_ possibilities for
> what you (TG) mean by 333...34
> (1)
> 333...34 is first after all of 4,34,334,...
>
> (2)
> 333...34 is the set {4,34,334,3334,...}
>
> (3)
> 333...34 is any of the set {4,34,334,3334,...}
>
> My best guess is (1).
> Please tell me if you mean something else.
> > Therefor there is no logic present which
> > prohibits 333...34 from being natural,
> > which you were claiming is a logical conclusion.
> My conclusion applies to (1)
> Are we talking about (1)?
>
> This is not how induction works.
> | 4 is natural.
> | ...
> | Therefor 333...34 is natural
> | You can stop whenever you like
> | and the value will be natural.
> Yes,
> you can keep going, 4, 34, 333333334,
> and each will be a natural number.
> But the question is: is 333...34 one of them?
> Keeping-going does not address that question.
> >>> Clearly the value is inductive.
> >>
> >> You can't get there from here.
> >> It is not inductive.
> >
> > Well, now, this is becoming infantile.
> I don't want to offend.
> I'm trying to find common ground upon which
> you and I can communicate.
> For that purpose, "infantile" is a feature,
> not a bug.
>
> It would be nice if we could have used '...' and
> "inductive" to communicate. It looks as though
> that is not going to happen.
>
> ----
> Consider a few examples of recursive
> -- I'm going to call them "definitions"
> even though that's problematic --
> a few examples of recursive definitions.
>
> (Si = i+1 the successor of i)
>
> Adding i:
> j + 0 = j
> j + Si = S(j + i)
>
> Multiplying by i:
> j*0 = 0
> j*Si = (j*i) + j
>
> Raising to the power of i:
> j^0 = 1
> j^Si = (j^i)*j
>
> Factorial of i
> 0! = 1
> (Si)! = (i+1)*(i!)
>
> We want a definition to tell us what
> a term means which we don't know
> using terms which we know.
>
> But see what we have here with these definitions.
> '+' is on both sides.
> '*' is on both sides.
> '^' is on both sides.
> '!' is on both sides.
>
> These definitions can't be telling us what
> '+' ('*' '^' '!') means which we don't know
> using the same '+' ('*' '^' '!') which we know.
> We can't be having our cake and eating it too.
>
> However,
> we can give better descriptions of '+'
> ('*' '^' '!') which do not have that problem.
>
> The key assumption is that
> you _can_ get there from here.
> For a particular sum j+k, we assume that
> a finite sequence ⟨j+0 j+1 ⋯ j+k⟩ exists.
> Each split has a last-before j+i and
> a first-after j+(i+1).
> The 0ₜₕ value is j.
> The kₜₕ value, whatever that value is,
> is the defined sum.
>
> Addition defined, without using '+' or '⋯' :
>
> j+k = m ⟺
> ∃𝑺,
> ∀i < k, ∀mᵢ : (⟨mᵢ i⟩ ∈ 𝑺 ⟺ ⟨Smᵢ Si⟩ ∈ 𝑺)
> ∧ ⟨j 0⟩ ∈ 𝑺 ∧ ⟨m k⟩ ∈ 𝑺
>
> 𝑺 is the finite sequence of increments from j
> to m, indexed by i.
> If 𝑺 exists, j+k = m. If not, then not.
>
>
> And similarly for defining '*' '^' '!'
> without '*' '^' '!' or '⋯'
>
> "You _can_ get there from here" is
> an irreplaceable concept for important parts
> of computing, even if it spends most of its
> time buried. But being buried is the fate of
> most foundations.

I understand that these inductive values that I am working with are new. Their product is found by induction. So at least here there is something fancy to do with induction, but it does boil down to pattern matching; again the 'infantile' system, but simplicity has to be a part of this feat.
x = 333...34
xx = 111...1555...56
Clearly I did not compute an infinity of digits here. This is the inductive result of considering the product on those predecessors that you have fixated upon. In the arbitrary precision math application sage I can write:
sage: x = 333333333333333333333333333333334
sage: x*x
111111111111111111111111111111111555555555555555555555555555555556
and further I can alter the quantity of threes in the expression and witness no change in the result other than the length of the string of digits.
Certainly this procedure is inductive. The addition of more three digits will not alter the pattern in the string.

We know that the number of digits in a square can take twice as many digits as its predecessor and that is exactly what we witness here. The form of infinity of the square of the infinite value 333...34 does indeed hold up. Granting a digital aleph one for the initial value we have achieved aleph two here computationally. Of course even my usage of 'aleph' will require some naming or care that will distinguish it. What is nice though is that we need it within this theory.

There is no need to require the predecessors in order to consider the value 333...34. At least it is not at all a detailed choice as you've marked up three options on. That is all a diversion. We happen to be lucky on this value as its square holds a simplistic form. Not so for 222...2:
sage: x=2222222222222222222222222222222222222
sage: x*x
4938271604938271604938271604938271603950617283950617283950617283950617284
Of course the resolution has been found and it will entail implementing some aleph marks in order to disambiguate the quantity of digits versus the length of the repetition. There is even a squirrely bit in the middle that will dance around, but thanks to induction we can break through all of this. After all our pure form is ultimately infinite in nature and so the pattern will come clean. One thing is for certain: each set of ellipses indicates an aleph is present. The quantity of digits in the repetition then has not actually blown up even though the complexity has risen.

We have to admit that it is only thanks to the repetitious qualities of these values that we can even make these claims. No doubt any reader who comes along would like to flip one of those 2's above to another digit. Well, this can technically be done at the head or at the tail of the value, and the pattern detection can grow cumbersome, but the thing that makes these values work is the ellipsis which implies repetition. You could try to work the square on a value such as x=123123123...123 however determining the result is likely best done by the computer. I do not have the theory that states how many digits would be in the repetition. Possibly that has been done in some other number theoretic system. Possibly it will one day be done on this theory. The point that I am making is that we do rely upon repetition in order that we achieve infinity. We have given up the complete freedom of digit n and instead we can state clearly every digit n simply. In effect this is a position of compromise, yet the result is new. Almost. There is one prior usage that is too well known:
1/3 =0.333...
and along with it comes rather a lot of repeating decimals. So you see we are not terribly far from existing numbers. The structured approach allows us to pull the decimal point from this value and regard it as a natural 333....3 and upon doing so we see that the decimal point encoded was taking an infinite position. In this regard the original expression might be said to be ambiguous. So such usages of the ellipsis go, and of course this detail of structural integrity is brought to us only in the last half century. Prior to this compiler integrity was a matter of the nod of the head. Yes, some heads were quite serious, and many were quite particular about their presentations. Here we have a new such presentation that leans more on modern technology and less of the scratch of the head approach.

On 10/1/2022 8:11 AM, Timothy Golden wrote:
> On Friday, September 30, 2022 at 2:43:32 PM UTC-4, Jim Burns wrote:
>> On 9/29/2022 9:48 AM, Timothy Golden wrote:
>>> On Wednesday, September 28, 2022
>>> at 5:04:13 PM UTC-4, Jim Burns wrote:
>>
>>>> There (333...34) is after all of 3,34,334,3334,...
>>>> Getting there requires a step j to 10*j-6
>>>> from not after all 3,34,334,3334,...
>>>> to all after all 3,34,334,3334,...
>>>
>>> It is correct that
>>> there is no step to infinity.
>> I'm saying something a little different.
>>
>> There is no step out of finity.
>>
>> That follows from the description of finity,
>> not from the description of infinity,
>> so it is very hard to avoid with some
>> (hypothetical) clever adjustment of infinity.
>> _Finity_ is unchanged.
>>> However this sequence was selected by you,
>>> and agreeable to me.
>> I selected it and asked
>> "Is that what you (TG) mean by 333...34?"
>> Please don't forget that I'm discussing
>> 4,34,334,3334,... in order to understand
>> what you're trying to tell us.
>>> It was selected because the pattern,
>>> repetitious as it is,
>>> does lead up to the ultimate value
>>> that is under scrutiny.
>> No,
>> I remember why I (JB) selected it.
>> I selected it because I thought you (TG)
>> might be using 333...34 to refer to
>> the thing after all of 4,34,334,3334,...
>>
>> Do you see the difference?
>> The difference is the point I want to make.
>>
>> There is no ultimate value of 4,34,334,3334,...
>>
>> The sense in which 4,34,334,... leads to 4,
>> leads to 34, leads to 334, leads to 3334,
>> _that_ sense does not apply to anything
>> after all of 4,34,334,3334,...
>> _that_ sense does not apply to an ultimate value.
>>
>> In other words,
>> you can't get there from here.
>>> In this way the sequence is not critical
>>> to the value, but it helps cover the ground.
>>>
>>> Allow me to state again:
>>> 4 is natural.
>>> 34 is natural.
>>> 334 is natural.
>>> 3334 is natural.
>>> Therefor 333...34 is natural
>>> You can stop whenever you like
>>> and the value will be natural.
>> I see these _different_ possibilities for
>> what you (TG) mean by 333...34
>> (1)
>> 333...34 is first after all of 4,34,334,...
>>
>> (2)
>> 333...34 is the set {4,34,334,3334,...}
>>
>> (3)
>> 333...34 is any of the set {4,34,334,3334,...}
>>
>> My best guess is (1).
>> Please tell me if you mean something else.
>>> Therefor there is no logic present which
>>> prohibits 333...34 from being natural,
>>> which you were claiming is a logical conclusion.
>> My conclusion applies to (1)
>> Are we talking about (1)?
>>
>> This is not how induction works.
>> | 4 is natural.
>> | ...
>> | Therefor 333...34 is natural
>> | You can stop whenever you like
>> | and the value will be natural.
>> Yes,
>> you can keep going, 4, 34, 333333334,
>> and each will be a natural number.
>> But the question is: is 333...34 one of them?
>> Keeping-going does not address that question.
>>>>> Clearly the value is inductive.
>>>>
>>>> You can't get there from here.
>>>> It is not inductive.
>>>
>>> Well, now, this is becoming infantile.
>> I don't want to offend.
>> I'm trying to find common ground upon which
>> you and I can communicate.
>> For that purpose, "infantile" is a feature,
>> not a bug.
>>
>> It would be nice if we could have used '...' and
>> "inductive" to communicate. It looks as though
>> that is not going to happen.
>>
>> ----
>> Consider a few examples of recursive
>> -- I'm going to call them "definitions"
>> even though that's problematic --
>> a few examples of recursive definitions.
>>
>> (Si = i+1 the successor of i)
>>
>> Adding i:
>> j + 0 = j
>> j + Si = S(j + i)
>>
>> Multiplying by i:
>> j*0 = 0
>> j*Si = (j*i) + j
>>
>> Raising to the power of i:
>> j^0 = 1
>> j^Si = (j^i)*j
>>
>> Factorial of i
>> 0! = 1
>> (Si)! = (i+1)*(i!)
>>
>> We want a definition to tell us what
>> a term means which we don't know
>> using terms which we know.
>>
>> But see what we have here with these definitions.
>> '+' is on both sides.
>> '*' is on both sides.
>> '^' is on both sides.
>> '!' is on both sides.
>>
>> These definitions can't be telling us what
>> '+' ('*' '^' '!') means which we don't know
>> using the same '+' ('*' '^' '!') which we know.
>> We can't be having our cake and eating it too.
>>
>> However,
>> we can give better descriptions of '+'
>> ('*' '^' '!') which do not have that problem.
>>
>> The key assumption is that
>> you _can_ get there from here.
>> For a particular sum j+k, we assume that
>> a finite sequence ⟨j+0 j+1 ⋯ j+k⟩ exists.
>> Each split has a last-before j+i and
>> a first-after j+(i+1).
>> The 0ₜₕ value is j.
>> The kₜₕ value, whatever that value is,
>> is the defined sum.
>>
>> Addition defined, without using '+' or '⋯' :
>>
>> j+k = m ⟺
>> ∃𝑺,
>> ∀i < k, ∀mᵢ : (⟨mᵢ i⟩ ∈ 𝑺 ⟺ ⟨Smᵢ Si⟩ ∈ 𝑺)
>> ∧ ⟨j 0⟩ ∈ 𝑺 ∧ ⟨m k⟩ ∈ 𝑺
>>
>> 𝑺 is the finite sequence of increments from j
>> to m, indexed by i.
>> If 𝑺 exists, j+k = m. If not, then not.
>>
>>
>> And similarly for defining '*' '^' '!'
>> without '*' '^' '!' or '⋯'
>>
>> "You _can_ get there from here" is
>> an irreplaceable concept for important parts
>> of computing, even if it spends most of its
>> time buried. But being buried is the fate of
>> most foundations.
>
> I understand that these inductive values that I am working with are new. Their product is found by induction. So at least here there is something fancy to do with induction, but it does boil down to pattern matching; again the 'infantile' system, but simplicity has to be a part of this feat.
> x = 333...34
> xx = 111...1555...56
> Clearly I did not compute an infinity of digits here. This is the inductive result of considering the product on those predecessors that you have fixated upon. In the arbitrary precision math application sage I can write:
> sage: x = 333333333333333333333333333333334
> sage: x*x
> 111111111111111111111111111111111555555555555555555555555555555556
> and further I can alter the quantity of threes in the expression and witness no change in the result other than the length of the string of digits.
> Certainly this procedure is inductive. The addition of more three digits will not alter the pattern in the string.
>
> We know that the number of digits in a square can take twice as many digits as its predecessor and that is exactly what we witness here. The form of infinity of the square of the infinite value 333...34 does indeed hold up. Granting a digital aleph one for the initial value we have achieved aleph two here computationally. Of course even my usage of 'aleph' will require some naming or care that will distinguish it. What is nice though is that we need it within this theory.
>
> There is no need to require the predecessors in order to consider the value 333...34. At least it is not at all a detailed choice as you've marked up three options on. That is all a diversion. We happen to be lucky on this value as its square holds a simplistic form. Not so for 222...2:
> sage: x=2222222222222222222222222222222222222
> sage: x*x
> 4938271604938271604938271604938271603950617283950617283950617283950617284

Click here to read the complete article

On 10/1/2022 9:11 AM, Timothy Golden wrote:

> I understand that these inductive values that I am
> working with are new. Their product is found
> by induction. So at least here there is something
> fancy to do with induction, but it does boil down
> to pattern matching; again the 'infantile' system,
> but simplicity has to be a part of this feat.
> x = 333...34
> xx = 111...1555...56
>
> Clearly I did not compute an infinity of digits here.
> This is the inductive result of considering the
> product on those predecessors that you have fixated
> upon. In the arbitrary precision math application sage
> I can write:
> sage: x = 333333333333333333333333333333334
> sage: x*x
> 111111111111111111111111111111111555555555555555555555555555555556
> and further I can alter the quantity of threes
> in the expression and witness no change in the result
> other than the length of the string of digits.
>
> Certainly this procedure is inductive.
> The addition of more three digits will not alter
> the pattern in the string.

I see these _different_ possibilities for
what you (TG) mean by 333...34
(1)
333...34 is first after all of 4,34,334,...

(2)
333...34 is the set {4,34,334,3334,...}

(3)
333...34 is any of the set {4,34,334,3334,...}

(4)
333...34 is something that hasn't occurred to me.

For (1), "You _can_ get there from here"
is embedded in the meaning of 333...34
For each of 4,34,334,... you _can_
For anything after each of 4,34,334,...
you _can't_
By "from here", I mean "from 0"

For (2), a set of numbers is not a number.
I consider (2) least likely to be what you mean.

For (3), "any of" treats the number of 3's
as a _variable_ There is at least one other
poster who misunderstands what a _variable_ is.

If I say x is in {1,2,3},
perhaps x=1 or x=2 or x=3
But x _doesn't change_ despite "variable"

"x is in {1,2,3}" _incompletely identifies_ x
With more information, perhaps two of its
currently-possible values could be ruled out.
Perhaps not.

Using a variable x in a claim allows us to
finitely state a claim that is true of
infinitely-many. It is the incomplete
identification which allows us to do that.

If (3) is what you mean, you need better
notation. It looks to me as though,
if we insert '333' in '333...34'
we get '333...34'. It looks impossible to
discuss inserting '333' if we can't
distinguish '333...34'-before from
'333...34'-after.

> There is no need to require the predecessors
> in order to consider the value 333...34.

Then, not option (1)?

The kind of induction I've been discussion
assumes we are discussing things for which
"You _can_ get there from here" is true.
| | If
| P(0) ∧ ∀j, P(j) ⟹ P(j+1)
| then
| ∀k, P(k)

There is also transfinite induction, for _ordinals_
This assumes we are discussing things for which
"If any exist, a first exists" is true.
| | If
| ∀j, (∀i<j, P(i)) ⟹ P(j)
| then
| ∀k, P(k)

Transfinite induction also considers predecessors,
meaning ordinals that come before (<)
Not all ordinals have immediate predecessors.

Transfinite induction might also be something
other than what you (TG) are talking about.
But it deals very well with infinities.

On 9/29/2022 1:13 AM, Chris M. Thomasson wrote:

> Interesting. Mitch said 13/3.
> Well, it most definitely can be exploited
> in multiple ways:
>
> 13/3 = 4.33333333333333...
>
> 4 + 1/3 = 13/3 ?
>
> Except, its backwards wrt 333...34, wrt:
>
> 34
> 334
> 3334
> ...

You remind me of a nice article Fred Jeffries
told sci.logic about, ca. 4 years ago.

Joel David Hamkins gives a non-standard order for
the natural numbers, the _final-digit order_
In that non-standard order, with the open sets
which it implies, addition and multiplication
are continuous.

And that is interesting but
the reason I mention it is that I think
we can fill in the single-point gaps in the
naturals -- when in the final-digit order --
with _non-standard_ naturals which look
something like 333...34 and its ilk.

I'm thinking of something along the lines
of the Dedekind construction of the real
numbers.

I think it's unlikely that this is what
Timothy Golden is talking about, but
it looks like fun (for geeky values
of "fun").

http://jdh.hamkins.org/topological-models-of-arithmetic/
https://arxiv.org/abs/1808.01270

Newsgroups: sci.logic
Subject: Topological models of (nonstandard) arithmetic
Date: Mon, 6 Aug 2018 11:56:58 -0700 (PDT)
From: FredJeffries <fredjeffries@gmail.com>
Message-ID: <36516443-e30c-4fc3-b4df-d6b953156d78@googlegroups.com>

On Saturday, October 1, 2022 at 2:20:57 PM UTC-4, Jim Burns wrote:
> On 10/1/2022 9:11 AM, Timothy Golden wrote:
>
> > I understand that these inductive values that I am
> > working with are new. Their product is found
> > by induction. So at least here there is something
> > fancy to do with induction, but it does boil down
> > to pattern matching; again the 'infantile' system,
> > but simplicity has to be a part of this feat.
> > x = 333...34
> > xx = 111...1555...56
> >
> > Clearly I did not compute an infinity of digits here.
> > This is the inductive result of considering the
> > product on those predecessors that you have fixated
> > upon. In the arbitrary precision math application sage
> > I can write:
> > sage: x = 333333333333333333333333333333334
> > sage: x*x
> > 111111111111111111111111111111111555555555555555555555555555555556
> > and further I can alter the quantity of threes
> > in the expression and witness no change in the result
> > other than the length of the string of digits.
> >
> > Certainly this procedure is inductive.
> > The addition of more three digits will not alter
> > the pattern in the string.
> I see these _different_ possibilities for
> what you (TG) mean by 333...34
> (1)
> 333...34 is first after all of 4,34,334,...
>
> (2)
> 333...34 is the set {4,34,334,3334,...}
>
> (3)
> 333...34 is any of the set {4,34,334,3334,...}
> (4)
> 333...34 is something that hasn't occurred to me.
>
>
> For (1), "You _can_ get there from here"
> is embedded in the meaning of 333...34
> For each of 4,34,334,... you _can_
> For anything after each of 4,34,334,...
> you _can't_
> By "from here", I mean "from 0"
>
> For (2), a set of numbers is not a number.
> I consider (2) least likely to be what you mean.
>
> For (3), "any of" treats the number of 3's
> as a _variable_ There is at least one other
> poster who misunderstands what a _variable_ is.
>
> If I say x is in {1,2,3},
> perhaps x=1 or x=2 or x=3
> But x _doesn't change_ despite "variable"
>
> "x is in {1,2,3}" _incompletely identifies_ x
> With more information, perhaps two of its
> currently-possible values could be ruled out.
> Perhaps not.
>
> Using a variable x in a claim allows us to
> finitely state a claim that is true of
> infinitely-many. It is the incomplete
> identification which allows us to do that.
>
> If (3) is what you mean, you need better
> notation. It looks to me as though,
> if we insert '333' in '333...34'
> we get '333...34'. It looks impossible to
> discuss inserting '333' if we can't
> distinguish '333...34'-before from
> '333...34'-after.
> > There is no need to require the predecessors
> > in order to consider the value 333...34.
> Then, not option (1)?

The predecessor to 333...34 is 333...33.
I've explained what I understand of this construction many times over here.
I don't have to select one of your four options which are not my language at all.
I do think that the sequence you've fixated on is good, but it is not the source of this value.
My thinking is actually quite a bit broader, and to express it once again is not a problem.
The rational number relies upon division. Not only this but it relies upon a type of division that lacks closure.
Beyond embedding an operator and two values and trying to make it seem like a singular value the level of conflict that the rational value has raised could have been alleviated. Witness that the meaning of the decimal point in a value can be taken as marking the unitary position of what is otherwise a natural number. Witness that extending this thinking further, should we consider the value six as unity in a secondary system, then as we count by ones in the primary system it follows that this same count will be as sixths in the secondary system. In this regard all of our numerical forms, and I can happily forgo the irrational numbers as they again do the embedded operator thing, which I just got rid of via this secondary unital concept, are in fact natural valued upon removing the secondary details. Those secondary details are augmentations to the natural value. The decimal point, for instance can be simply pulled from any value. It's position could be recorded so that it can be put back later, let's say. Now this argument conveniently lands us at the predecessor of 333...34, for in the lore of old mathematics still used in this day we have
1/3 = 0.333...
and so my own usage of repeating digits is not so unfamiliar is it? Ahh... but these digits disappear to fleetingly small values that we need not worry about, right? These threes of this one third expression are different from the threes of that extremely large value, right? Under the decimal interpretation just given we can pull the decimal point. We witness that its position relative to unity is infinite, and so under the new system this old value may be argued as poorly constructed, for structurally there is no ability to code that representation. Well, this argument overlooks the ellipses entirely, and so the ambiguities somewhat finger point at each other; often this is the case. Regardless, pulling the decimal point and now regarding
333...
as a natural value what harm is there in admitting that
333... = 333...33 ?
Indeed we can now express the successor; our chosen instance of debate; but how nice it is to actually have such a thing in hand.

Filling out a bit more ground here, to what degree are numbers two-sided? Do they have a head and a tail? Can they go prancing about slinking around looking for a treat to eat? That sounds more like a variable. Here these are statues; fixed in position and without a doubt fully defined. Every digit is so well expressed that you'd have to be a moron not to see it. Yet I suppose there are a few who do witness this very refusal. Well, to them I would say that taking the freedom to construct when subtle changes are made at the base of mathematics that such results are possible.

Clearly the freedom with which mathematicians use the ellipsis in their notation allows for this usage. Clearly the existing work that we will find in many a text exposing the repeating digits of one third will act as precedent. There is a strangeness about it all; yes. As to why it was not done before: I think this is the most apt that can alleviate the strangeness. Obviously these points are uncomfortable to discuss, for entering the ground this way we've just trashed the rational value. Yes, we did recover it along the way without any need for division. Did I mention that division is not even a fundamental operator? It is a reverse operator. Without a product that yields the divisor no such value is even guaranteed to exist. And even then there are exceptions. This is what your real value is built upon? No; it cannot be so. It quite literally cannot be so. Habituation and repetition amongst the human race explains too much, I'm afraid. I suppose it doesn't hurt that these old laws are enforced under threat of failure. Not only that but for the first twenty years or so of your life... some of which you are even expected to pay for... and quite a bit, too.

I do believe that the option to reject the usage of ellipses is a valid choice. As to what is left of mathematics upon doing so becomes a large problem.
The ambiguities of the ellipsis and its usage have been laid out. The halting problem ensues. The exact contradiction is well exposed and so to touch upon such ambiguity will be dodged every step of the way by those wishing to uphold the status quo.

Permitting the usage of the ellipses; legitimating them on this one episode of 333...34 we find some further strange ground. I had gone to the trouble of explaining the product of such values and its inductive nature earlier. The need of an aleph mark sometimes makes itself known. Going back to about here: https://groups.google.com/g/sci.math/c/tHL3zLZ97v0/m/RROBL-skAQAJ I do see those aleph marks coming into being.

What is really strange about this is that we can confuse those aleph marks with the decimal point and land in a sense of continuum that some who are so obsessed with natural valued infinity feel in their gut. Once again this can be taken as a unital interpretation but this time taking the aleph one form of infinity as unital. For this sort the notion that the tail of their values goes on and on to some infinite precision form gives them comfort. I don't feel this comfort myself. I see the form, but claiming that this has the ring of truth about it is too much.

As I recall, Jim, you have not bothered to address the usage of such values as 0.333...
Would you say that the far end of the number is ill-defined?
Or can we admit that the tail of this value goes like
0.333...33 ?

Of course this is just the first step.
Next I'll have you pulling away that decimal point.
Increment the value.
Now we've got our creature feature.

On 10/2/2022 8:47 AM, Timothy Golden wrote:

> The predecessor to 333...34 is 333...33.

Having a predecessor is not enough
to make 333..34 inductive
==meaning==
Even if we know
| P(0) ∧ ∀j, P(j) ⟹ P(j+1)
we don't know
| P(333..34)

When we do know a claim of that form,
for example, the way we would know
P(10^(10^(10^(10^(10))))+1)

how we know is NOT because somebody declared
10^(10^(10^(10^(10))))+1 is inductive.
Our knowledge follows from what
10^(10^(10^(10^(10))))+1 means,
an important part of which being:
you _can_ get there from here (from 0)

> I don't have to select one of your four options
> which are not my language at all.

You're absolutely right.
You don't have to tell anyone
what you're talking about.

So, are you happy with how that's
working out for you?

> As I recall, Jim, you have not bothered to
> address the usage of such values as 0.333...
> Would you say that the far end of the number
> is ill-defined?

The far end of 0.333... does not exist.
It is a square circle,
a description that is not true of anything.

> Or can we admit that the tail of this value goes like
> 0.333...33 ?

Each digit of 0.333... is at a place
for which you can get there from here.
For each split of the decimal-string 0.333...
there is a last-before and a first-after.

Did I remember to mention that it's not clear
what you mean by '333...33'? I think I did.

I'm guessing that the '33' at the end of
0.333...33 is different from the others,
so you _can't_ get there from here.

There are plenty of things like that
but they aren't natural numbers.

> We now have infinities that can increment.

https://en.wikipedia.org/wiki/Ordinal_arithmetic
https://en.wikipedia.org/wiki/Cardinal_number#Cardinal_arithmetic

You're welcome.

On Sunday, October 2, 2022 at 10:15:48 AM UTC-4, Jim Burns wrote:
> On 10/2/2022 8:47 AM, Timothy Golden wrote:
>
> > The predecessor to 333...34 is 333...33.
> Having a predecessor is not enough
> to make 333..34 inductive
> ==meaning==
> Even if we know
> | P(0) ∧ ∀j, P(j) ⟹ P(j+1)
> we don't know
> | P(333..34)
>
> When we do know a claim of that form,
> for example, the way we would know
> P(10^(10^(10^(10^(10))))+1)
>
> how we know is NOT because somebody declared
> 10^(10^(10^(10^(10))))+1 is inductive.
> Our knowledge follows from what
> 10^(10^(10^(10^(10))))+1 means,
> an important part of which being:
> you _can_ get there from here (from 0)
> > I don't have to select one of your four options
> > which are not my language at all.
> You're absolutely right.
> You don't have to tell anyone
> what you're talking about.
>
> So, are you happy with how that's
> working out for you?
> > As I recall, Jim, you have not bothered to
> > address the usage of such values as 0.333...
> > Would you say that the far end of the number
> > is ill-defined?
> The far end of 0.333... does not exist.
> It is a square circle,
> a description that is not true of anything.
> > Or can we admit that the tail of this value goes like
> > 0.333...33 ?
> Each digit of 0.333... is at a place
> for which you can get there from here.
> For each split of the decimal-string 0.333...
> there is a last-before and a first-after.
>
> Did I remember to mention that it's not clear
> what you mean by '333...33'? I think I did.
>
> I'm guessing that the '33' at the end of
> 0.333...33 is different from the others,
> so you _can't_ get there from here.
>
> There are plenty of things like that
> but they aren't natural numbers.
> > We now have infinities that can increment.
> https://en.wikipedia.org/wiki/Ordinal_arithmetic
> https://en.wikipedia.org/wiki/Cardinal_number#Cardinal_arithmetic
>
> You're welcome.

I am sorry that you are so sour to what could be called a new branch of mathematics. That this interpretation rings clean and through is a part of the largess of the theory. To fully appreciate this claim we have to admit that the natural value as continuous would require an infinite absurdum such as this one. To cast the real value as gray I believe is the mature way; it is to remain finite as you preach to do. You are on that branch of denial of the ellipses. As to what else you must cast aside as a purist in this form: I think this has yet to be filled out, yet as we see you dodge the halting problem of the ellipsis, and by this I do think I mean every ellipsis: here I cannot comment so much as you can.

I can't help that some of your own bitterness is due to the long slow crawl that the natural value as preached by Peano and his sort; a method which ignores the modulo quality of number completely: Then to see how quickly computations can refine such values as gears whirring along; the clickety-clack almost gone, but for a strange harmonic as the digits drive downward in precision... or was it upward? There is no question that the natural interpretation thanks especially to the new decimal point interpretation is upward unconditionally and naturally, too. The idea that you might be pleased by this realization awaits you.

I really truly do appreciate the energy that you expend here. Exasperated as you may feel. Perhaps I grant myself too much room, but if I don't who else will? So many ideas get pinched for they are unconsequential. That these interpreti (plural, actually) are consequential and form a simplification of standing mathematics is a thing of beauty; again awaiting reception.

On 10/2/2022 7:08 PM, Timothy Golden wrote:

> I can't help that some of your own bitterness
> is due to

> yet as we see you dodge the halting problem of
> the ellipsis,

The ellipsis does not have a halting problem.
The ellipsis replaces description that is
too well-known to need repeating.

For example,
| | ℕ = {0,1,2,3,...}

stands for
| | Inductive(ℕ) ∧
| ~∃𝐵 ⫋ ℕ : Inductive(𝐵)
| | Inductive(𝐵) <->
| 0 ∈ 𝐵 ∧ ∀k ∈ 𝐵 : k+1 ∈ 𝐵

The set ℕ of natural numbers does not halt.

I suppose that is considered a problem by
people who do not want the natural numbers
to be the natural numbers.

But now, we've left mathematics and
entered psychology. I am less interested in
why you do not want the natural numbers
to be the natural numbers.

On Monday, October 3, 2022 at 1:55:43 AM UTC+2, Jim Burns wrote:
> On 10/2/2022 7:08 PM, Timothy Golden wrote: <nonsense>
>
> The ellipsis does not have a halting problem.
> The ellipsis replaces description that is
> too well-known to need repeating.

Indeed!

> For example,
>
> | ℕ = {0, 1, 2, 3, ...}
>
> stands for
>
> | Inductive(ℕ) ∧
> | ~∃𝐵 ⫋ ℕ : Inductive(𝐵)

[ with

> | Inductive(𝐵) :<->
> | 0 ∈ 𝐵 ∧ ∀k ∈ 𝐵 : k+1 ∈ 𝐵 ]

and

∀k ∈ ℕ : k+1 =/= 0 ∧
∀k,k' ∈ ℕ : k+1 = k'+1 -> k = k'

[ as well as

1 = 0+1
2 = 1+1
3 = 2+1 ].

(After all, IN = {0, 1} with 0 =/= 1, 0+1 = 1 and 1+1 = 0, 0 = 2 and 1 = 3 would also allow to write ℕ = {0, 1, 2, 3, ...}. Moreover in this case Inductive(ℕ) ∧ ~∃𝐵 ⫋ ℕ : Inductive(𝐵) would hold too.)

From this "description" of IN (not taking into account 1 = 0+1, 2 = 1+1, 3 = 2+1) we can prove that there's a bijection from IN onto IN \ {0} (and hence from IN onto a proper subset of IN), which allows to conclude that IN is an "infinite" set (due to Dedekind).

On 10/1/2022 3:03 PM, Jim Burns wrote:
> On 9/29/2022 1:13 AM, Chris M. Thomasson wrote:
>
>> Interesting. Mitch said 13/3.
>> Well, it most definitely can be exploited
>> in multiple ways:
>>
>> 13/3 = 4.33333333333333...
>>
>> 4 + 1/3 = 13/3 ?
>>
>> Except, its backwards wrt 333...34, wrt:
>>
>> 34
>> 334
>> 3334
>> ...
>
> You remind me of a nice article Fred Jeffries
> told sci.logic about, ca. 4 years ago.
>
> Joel David Hamkins gives a non-standard order for
> the natural numbers, the _final-digit order_
> In that non-standard order, with the open sets
> which it implies, addition and multiplication
> are continuous.
>
> And that is interesting but
> the reason I mention it is that I think
> we can fill in the single-point gaps in the
> naturals -- when in the final-digit order --
> with _non-standard_ naturals which look
> something like 333...34 and its ilk.
>
> I'm thinking of something along the lines
> of the Dedekind construction of the real
> numbers.
>
> I think it's unlikely that this is what
> Timothy Golden is talking about, but
> it looks like fun (for geeky values
> of "fun").
>
>
> http://jdh.hamkins.org/topological-models-of-arithmetic/
> https://arxiv.org/abs/1808.01270
>
> Newsgroups: sci.logic
> Subject: Topological models of (nonstandard) arithmetic
> Date: Mon, 6 Aug 2018 11:56:58 -0700 (PDT)
> From: FredJeffries <fredjeffries@gmail.com>
> Message-ID: <36516443-e30c-4fc3-b4df-d6b953156d78@googlegroups.com>

Love the fractal tree:

http://jdh.hamkins.org/wp-content/uploads/Final-digits-order-2.jpg

;^)

Thanks for the link Jim. :^)

On Sunday, October 2, 2022 at 7:55:43 PM UTC-4, Jim Burns wrote:
> On 10/2/2022 7:08 PM, Timothy Golden wrote:
>
> > I can't help that some of your own bitterness
> > is due to
> > yet as we see you dodge the halting problem of
> > the ellipsis,
> The ellipsis does not have a halting problem.
> The ellipsis replaces description that is
> too well-known to need repeating.

Well, why not just insert it this once since it is so boring?

>
> For example,
> |
> | ℕ = {0,1,2,3,...}

Right here in this line above, eh?
As you correctly point out we are dealing in a problem of human psychology.
The falsification of your statement both precedes you and succeeds you.
As it stares you in the face; your direct transgression as denial; what is one to do?
"The ellipsis replaces description that is too well-known to need repeating.."
"The ellipsis does not have a halting problem."
I believe what we witness here is a psychological effect that I will call 'the untouchable'.
It is as if to say that as a status quo mathematician you may touch, but all else, you keep your hands off.
This is a faith position for you, I suppose. No doubt it is an entrained position as well.
To confront such details is to challenge not so much you as a person, but the schooling that you have absorbed.
The notion that we might start from scratch and do it all again from naught: will it come out the same?

And by the way, if I give a large value like x=33333333333334, are you really going to review its successors to prove that this one is natural?
This is the proof, isn't it? If so then I can determine with first order logic that this is unproven to be a natural number, or rather it will prove to not be a natural number due to the probability of human error or failing within the span of the review. Especially under a mathematical court, let's say, where more than two individuals are involved. The argument may sound silly, but by Peano you literally have to count up to this value in order to verify its natural qualities. Digital analysis, and by this I mean pertaining to the digits and their mechanistic qualities, does not require this sort of toil. Indeed we get there quite fast simply by reviewing the digits in the 'string'. Fourteen of them. Of course an assumption that this is a modulo ten value.

The concern of working large numbers has been a theme here on usenet's sci.math ever since I started spending time here. Already at a large finite value we've put Peano out of business. Literally in your tongue: you can't get there from here. Oh, and the ellipsis has no interaction with this detail, right? Yet onward toward infinity is where most feel compelled to go. That the digital analysis gets us there the fastest: it's an inconvenient fact that old Peano just didn't want to touch. Really, who am I to poke fun at the old man. Given this value that I've presented you, he might still be working on it to this day. Thus we do not even have to require the halting problem in its fullest form. Simply put a large enough value halts the system for a long enough time that the 'system' is dysfunctional. Absurdity ensues.. Now, taking these issues seriously: the value x=33333333333334 is natural, and we won't be needing to compute it as a successor for we can witness that this is not humanly possible. We will have to rely upon some other mechanism of logic to account for it. Oddly enough, this same or similar sort of logic will allow us to work x=333...34, but for the ellipsis and their meaning, which I'm afraid have really gotten put out of whack if your rendition holds up.

Claims of an inductive N and a non-inductive N are a fraud I suspect. N were already inductive. In the paragraphs above we see that instantiation confronts the language of abstraction. As well, the slow demise of the incremental progression in the face of digital analysis exposes a weakness of natural analysis. That the discussion is one of philosophy as much as it is of mathematics is feasible. Bringing physics into the fold as well since numbers are of interest to all of these professions, and here the tenets of the natural value are coming under scrutiny. I had no idea that this was possible.

>
> stands for
> |
> | Inductive(ℕ) ∧
> | ~∃𝐵 ⫋ ℕ : Inductive(𝐵)
> |
> | Inductive(𝐵) <->
> | 0 ∈ 𝐵 ∧ ∀k ∈ 𝐵 : k+1 ∈ 𝐵
>
> The set ℕ of natural numbers does not halt.
>
> I suppose that is considered a problem by
> people who do not want the natural numbers
> to be the natural numbers.
>
> But now, we've left mathematics and
> entered psychology. I am less interested in
> why you do not want the natural numbers
> to be the natural numbers.

Well, I am starting to wonder myself. If the ellipsis requires the grand dodge that you've laid out then the whole deal is crikey.
That ambiguity can be exposed in the basis of mathematics is ultimately a criterion of openness perhaps.
At least though you've come around to even discussing those little dots.

On Monday, October 3, 2022 at 6:30:02 AM UTC-7, timba...@gmail.com wrote:
> On Sunday, October 2, 2022 at 7:55:43 PM UTC-4, Jim Burns wrote:
> > On 10/2/2022 7:08 PM, Timothy Golden wrote:
> >
> > > I can't help that some of your own bitterness
> > > is due to
> > > yet as we see you dodge the halting problem of
> > > the ellipsis,
> > The ellipsis does not have a halting problem.
> > The ellipsis replaces description that is
> > too well-known to need repeating.
> Well, why not just insert it this once since it is so boring?
> >
> > For example,
> > |
> > | ℕ = {0,1,2,3,...}
> Right here in this line above, eh?
> As you correctly point out we are dealing in a problem of human psychology.
> The falsification of your statement both precedes you and succeeds you.
> As it stares you in the face; your direct transgression as denial; what is one to do?
> "The ellipsis replaces description that is too well-known to need repeating."
> "The ellipsis does not have a halting problem."
> I believe what we witness here is a psychological effect that I will call 'the untouchable'.
> It is as if to say that as a status quo mathematician you may touch, but all else, you keep your hands off.
> This is a faith position for you, I suppose. No doubt it is an entrained position as well.
> To confront such details is to challenge not so much you as a person, but the schooling that you have absorbed.
> The notion that we might start from scratch and do it all again from naught: will it come out the same?
>
> And by the way, if I give a large value like x=33333333333334, are you really going to review its successors to prove that this one is natural?
> This is the proof, isn't it? If so then I can determine with first order logic that this is unproven to be a natural number, or rather it will prove to not be a natural number due to the probability of human error or failing within the span of the review. Especially under a mathematical court, let's say, where more than two individuals are involved. The argument may sound silly, but by Peano you literally have to count up to this value in order to verify its natural qualities. Digital analysis, and by this I mean pertaining to the digits and their mechanistic qualities, does not require this sort of toil. Indeed we get there quite fast simply by reviewing the digits in the 'string'. Fourteen of them. Of course an assumption that this is a modulo ten value.
>
> The concern of working large numbers has been a theme here on usenet's sci.math ever since I started spending time here. Already at a large finite value we've put Peano out of business. Literally in your tongue: you can't get there from here. Oh, and the ellipsis has no interaction with this detail, right? Yet onward toward infinity is where most feel compelled to go. That the digital analysis gets us there the fastest: it's an inconvenient fact that old Peano just didn't want to touch. Really, who am I to poke fun at the old man. Given this value that I've presented you, he might still be working on it to this day. Thus we do not even have to require the halting problem in its fullest form. Simply put a large enough value halts the system for a long enough time that the 'system' is dysfunctional. Absurdity ensues. Now, taking these issues seriously: the value x=33333333333334 is natural, and we won't be needing to compute it as a successor for we can witness that this is not humanly possible. We will have to rely upon some other mechanism of logic to account for it. Oddly enough, this same or similar sort of logic will allow us to work x=333...34, but for the ellipsis and their meaning, which I'm afraid have really gotten put out of whack if your rendition holds up.
>
> Claims of an inductive N and a non-inductive N are a fraud I suspect. N were already inductive. In the paragraphs above we see that instantiation confronts the language of abstraction. As well, the slow demise of the incremental progression in the face of digital analysis exposes a weakness of natural analysis. That the discussion is one of philosophy as much as it is of mathematics is feasible. Bringing physics into the fold as well since numbers are of interest to all of these professions, and here the tenets of the natural value are coming under scrutiny. I had no idea that this was possible.
> >
> > stands for
> > |
> > | Inductive(ℕ) ∧
> > | ~∃𝐵 ⫋ ℕ : Inductive(𝐵)
> > |
> > | Inductive(𝐵) <->
> > | 0 ∈ 𝐵 ∧ ∀k ∈ 𝐵 : k+1 ∈ 𝐵
> >
> > The set ℕ of natural numbers does not halt.
> >
> > I suppose that is considered a problem by
> > people who do not want the natural numbers
> > to be the natural numbers.
> >
> > But now, we've left mathematics and
> > entered psychology. I am less interested in
> > why you do not want the natural numbers
> > to be the natural numbers.
> Well, I am starting to wonder myself. If the ellipsis requires the grand dodge that you've laid out then the whole deal is crikey.
> That ambiguity can be exposed in the basis of mathematics is ultimately a criterion of openness perhaps.
> At least though you've come around to even discussing those little dots.

Sometimes it's written "{... infinitely-many ...}".

It's written to indicate "Really, infinitely many. No really.".

On Monday, October 3, 2022 at 8:29:00 AM UTC-7, Ross A. Finlayson wrote:
> On Monday, October 3, 2022 at 6:30:02 AM UTC-7, timba...@gmail.com wrote:
> > On Sunday, October 2, 2022 at 7:55:43 PM UTC-4, Jim Burns wrote:
> > > On 10/2/2022 7:08 PM, Timothy Golden wrote:
> > >
> > > > I can't help that some of your own bitterness
> > > > is due to
> > > > yet as we see you dodge the halting problem of
> > > > the ellipsis,
> > > The ellipsis does not have a halting problem.
> > > The ellipsis replaces description that is
> > > too well-known to need repeating.
> > Well, why not just insert it this once since it is so boring?
> > >
> > > For example,
> > > |
> > > | ℕ = {0,1,2,3,...}
> > Right here in this line above, eh?
> > As you correctly point out we are dealing in a problem of human psychology.
> > The falsification of your statement both precedes you and succeeds you.
> > As it stares you in the face; your direct transgression as denial; what is one to do?
> > "The ellipsis replaces description that is too well-known to need repeating."
> > "The ellipsis does not have a halting problem."
> > I believe what we witness here is a psychological effect that I will call 'the untouchable'.
> > It is as if to say that as a status quo mathematician you may touch, but all else, you keep your hands off.
> > This is a faith position for you, I suppose. No doubt it is an entrained position as well.
> > To confront such details is to challenge not so much you as a person, but the schooling that you have absorbed.
> > The notion that we might start from scratch and do it all again from naught: will it come out the same?
> >
> > And by the way, if I give a large value like x=33333333333334, are you really going to review its successors to prove that this one is natural?
> > This is the proof, isn't it? If so then I can determine with first order logic that this is unproven to be a natural number, or rather it will prove to not be a natural number due to the probability of human error or failing within the span of the review. Especially under a mathematical court, let's say, where more than two individuals are involved. The argument may sound silly, but by Peano you literally have to count up to this value in order to verify its natural qualities. Digital analysis, and by this I mean pertaining to the digits and their mechanistic qualities, does not require this sort of toil. Indeed we get there quite fast simply by reviewing the digits in the 'string'. Fourteen of them. Of course an assumption that this is a modulo ten value.
> >
> > The concern of working large numbers has been a theme here on usenet's sci.math ever since I started spending time here. Already at a large finite value we've put Peano out of business. Literally in your tongue: you can't get there from here. Oh, and the ellipsis has no interaction with this detail, right? Yet onward toward infinity is where most feel compelled to go. That the digital analysis gets us there the fastest: it's an inconvenient fact that old Peano just didn't want to touch. Really, who am I to poke fun at the old man. Given this value that I've presented you, he might still be working on it to this day. Thus we do not even have to require the halting problem in its fullest form. Simply put a large enough value halts the system for a long enough time that the 'system' is dysfunctional. Absurdity ensues. Now, taking these issues seriously: the value x=33333333333334 is natural, and we won't be needing to compute it as a successor for we can witness that this is not humanly possible. We will have to rely upon some other mechanism of logic to account for it. Oddly enough, this same or similar sort of logic will allow us to work x=333...34, but for the ellipsis and their meaning, which I'm afraid have really gotten put out of whack if your rendition holds up.
> >
> > Claims of an inductive N and a non-inductive N are a fraud I suspect. N were already inductive. In the paragraphs above we see that instantiation confronts the language of abstraction. As well, the slow demise of the incremental progression in the face of digital analysis exposes a weakness of natural analysis. That the discussion is one of philosophy as much as it is of mathematics is feasible. Bringing physics into the fold as well since numbers are of interest to all of these professions, and here the tenets of the natural value are coming under scrutiny. I had no idea that this was possible.
> > >
> > > stands for
> > > |
> > > | Inductive(ℕ) ∧
> > > | ~∃𝐵 ⫋ ℕ : Inductive(𝐵)
> > > |
> > > | Inductive(𝐵) <->
> > > | 0 ∈ 𝐵 ∧ ∀k ∈ 𝐵 : k+1 ∈ 𝐵
> > >
> > > The set ℕ of natural numbers does not halt.
> > >
> > > I suppose that is considered a problem by
> > > people who do not want the natural numbers
> > > to be the natural numbers.
> > >
> > > But now, we've left mathematics and
> > > entered psychology. I am less interested in
> > > why you do not want the natural numbers
> > > to be the natural numbers.
> > Well, I am starting to wonder myself. If the ellipsis requires the grand dodge that you've laid out then the whole deal is crikey.
> > That ambiguity can be exposed in the basis of mathematics is ultimately a criterion of openness perhaps.
> > At least though you've come around to even discussing those little dots..
> Sometimes it's written "{... infinitely-many ...}".
>
> It's written to indicate "Really, infinitely many. No really.".

Usually in set theory the entire set of an infinite sequence of numbers is
infinite, while of course it's just usually that each initial segment is finite.

I.e. constructivists have to work their way up, it's easier.

This then just makes little machines of mathematics in numerical resources,
but there are intuitionists' machines that provide constructive results
that achieve limits, and result for deconstructions for constructivism.

I.e. the intuitionists can make a much "stronger" theory with results over
deductive analysis, look at for example the success of set theory that
it's intuitive that sets are well-founded and even well-ordered.

There's just for example another, ..., stronger theory, weak in its alternatives,
each strong, thus strong.

In numbers, ....

On Wednesday, September 28, 2022 at 11:13:40 PM UTC-7, Ross A. Finlayson wrote:
> On Wednesday, September 28, 2022 at 10:51:05 PM UTC-7, Chris M. Thomasson wrote:
> > On 9/28/2022 10:28 PM, Ross A. Finlayson wrote:
> > > On Wednesday, September 28, 2022 at 10:23:37 PM UTC-7, Chris M. Thomasson wrote:
> > >> On 9/28/2022 10:13 PM, Chris M. Thomasson wrote:
> > >>> On 9/28/2022 9:32 PM, Ross A. Finlayson wrote:
> > >>>> On Wednesday, September 28, 2022 at 9:28:37 PM UTC-7, Chris M.
> > >>>> Thomasson wrote:
> > >>>>> On 9/24/2022 9:57 AM, Sergi o wrote:
> > >>>>>> On 9/21/2022 2:17 PM, mitchr...@gmail.com wrote:
> > >> [...]
> > >>> Humm... Interesting. Wrt 13/3 - 1 is = 3.333..., so we can scoot over
> > >>> the endless three's, so to speak... Just like my direct solution does
> > >>> for 333...34. Add one, to the the ending 4. Fair enough?
> > >> Ouch. I mean add one to the ending three to make the four, in 333...34.
> > >>
> > >> The floor operations basically strips off the infinite threes as in
> > >> floor(1/3) = 0
> > >> i[n] = floor((1/3) * 10^(n+2) + 1)
> > >
> > > Oh you mean storage?
> > >
> > No. I mean getting rid of the string of infinite 3's (think base 10
> > representation), 1/3 = .3333333....
> >
> > Fine.
> >
> > floor(1/3) = 0
> > floor(.3333333...) = 0
> >
> > However, floor(123456789.1011121314) = 123456789
> >
> > To get 333...34 in a direct function, well, exploiting 1/3 sure seems to
> > work! ;^)
> > i[n] = floor((1/3) * 10^(n+2) + 1)
> Subtracting 1/3 twice?
>
> Figuring out how many bits in the floating point,
> here the idea is to, ..., "subtract 1/3 twice, get
> .333...333... somewhere 3444...", that the "somewhere"
> was where the floating point unit, would extend.
> I.e., it might be fair, "the subtracted 1/3 twice
> each extended all the way down .333..., leaving .3444...,
> only wherever the floating point unit is beyond precision".
>
> I figure it would be close or same to any other usual calculations,
> also "for all the bits in the FPU value what result its bits is
> its representation".
>
> I.e. the Floating Point Unit is much bigger than Fixed Point,
> the register width usually enough 32/64 doubles and 80 bit intel floats.
> Then though keeping the guarantees of fixed point, for floating point,
> I mostly look at floating point for how to implement fixed point arithmetic,
> and mostly I use the vector registers and about having that the MMX is
> used for shifting and so on instead of multimedia along ... multiplying transforms.
>
> Then, though again the floating point unit is bigger, it doesn't have to
> do integer arithmetic, I'd like to think that the floating point units, are
> given to statistics, for where words are small in small alphabets, in words,
> while floating point values are "80 bits".
>
> Which is good to have defined as "10 8 bits".
>
> Also I forget that it's a real number in [0, infinity) not just [0,1],
> where "infinity" excuse me is, "Not A Number", "NaN", Not-a-number.
>
> I guess the 754 and 854 kind of define this, IEEE floats,
> with platforms "do or don't", IEEE float.
>
>
>
>
> Now I've also written a scritto and if you'll excuse me I'll write it here.
>
> You know, with a C runtime, and socket, and, ..., a C runtime, all sorts C then C++ code compile, ....
>
> Then, some, event machine, is that let's say that there is JavaScript runtime, and, there is
> "static JavaScript runtime", in terms of, say, "C runtime, ...".
>
> This is enough library most tools make a POSIX.
>
> "Why old POSIX?" Yeah, "why old POSIX", ....
>
> Tooling down POSIX, basically is for reflecting the derivation of POSIX,
> in the line terminal and the setting. Then, there is for making in tooling,
> what are the environment and configuration, what makes terminal,
> results a terminal.
>
> The console and the terminal, here for kernel mode user land,
> reflects a usual enough runtime quite unremarkable.
>
> Then though it results for the framework and platform, for basically giving off
> the hardware access, what results that resources are actually well-defined again,
> putting nice conventions all over what results "cooperative multithreading under
> constant bounds with immediate suspend is nice", what would result under routine
> that for example it's usually organized in for example "an overall adaptive resource",
> all sorts what running emulation makes run inspecting code under running it, in
> terms of binary objects then running them.
>
> So, a terminal, then for a shell, would be usually enough for the filesystem,
> what results "as a computer, it looks like a shell, ...". The framebuffer is usually
> considered a single area, in what is for example full frame console emulation,
> that must provide a shell console and terminal, then with a usual user's minimal
> semantics, simplest tools laying around, and homes.
>
> Here the console emulation itself is often where the framebuffer, under console-only,
> writes console to the framebuffer, for example where otherwise "what is the framebuffer"
> is a "virtual framebuffer" if it's made virtual, there's no patent reasons not to drive output
> directly to RAM besides the view adapter.
>
> So, this operating system should be along the lines of "the system has 4GB of RAM and
> 1GB Quad CPU and an entire Terabyte of disk", then to run "the system has 32 bit address
> virtual RAM according to the runtime", 1 Megabyte of Disk, and Zero RAM.
>
>
> Then for driving full HD and especially for split-screen HD, or 30/60 refresh, it's usually
> to be expected that the drivers are in a DMA convention must be. There's also the disk
> controller, far be it for me to queue the spool to the disk controller anyways, "putting
> all the I/O through the disk controller".
>
> (What results a disk controller.)
>
> Here that's the idea, the I/O, to work the I/O, in terms of: basically cooperating with
> the disk controller network I/O, and, the fact that disk "read-often" or "read-random",
> makes "memory-mapped files" for the C runtime, disk controller cache and virtual memory.
>
> So, the I/O, is broken down into protocols, the Internet protocols these days about
> result to "compression", and, "encryption", and catalog, what result block coders
> that all computer nework I/O code executes in their block coding, with "TLS session"
> and "Deflate/Gzip un-encumbered compression", or data format.
>
> Then protocols vary for example their routine all transfer, for making:
>
> JPEG protocols <- there are them
> "Internet protocols", ..., data transfer in data formats
>
> This way there is for making organization, "here are your videos", "here is your code",
> "this is your shell", ....
>
> I'd love to think I could make a computer, that I could program by tapping at the shell.
>
> Probably best start with "disk controller and DMA".

GeneraL Processing Unit: PC

PC operation

power and the PC
peripherals and the PC

display
input

power
board
bus
chip <- central processing unit

bus chip port

board bus chip port

operation:

1) power
2) "firmware" (BIOS, OpenFirmware, EFI, ...)

The "OEM"
each integrator is an "OEM", ....
Original Equipment Manufacturer

The entry point to the system passing control from the executive out of firmware, here the point is that the difference between
an executive and an operating system is the modularity, where here the point is for example "build a system with modules for
a given configuration, that's flexible under USB insertion order, then write that to boot media, or boot from media". The point
is to boot from media what with respect to bootloaders in their organization firmware or "loading images", there results
"small, modular media", printing out the modular drivers for the configuration from a configuration--reading boot adapter,
that builds an image.

The "operating system" is the protected memory, the executive is just "real-time scheduler".