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Date: Thu, 04 Jul 24 13:49:47 +0000
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From: WM <wolfgang.mueckenheim@tha.de>
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Le 04/07/2024 à 05:20, Jim Burns a écrit :
> On 7/3/2024 4:00 PM, WM wrote:

>> But it is impossible to
>> construct or define or find or recognize or communicate
>> a non-terminating decimal without using a finite formula,
>> and be it as simple as "0.111...".
> 
> Answer the questions
> "What is a non.terminating.decimalⁿᵒᵗᐧᵂᴹ?"  and
> "What is a line.pointⁿᵒᵗᐧᵂᴹ?"
> and one can prove from very reasonable assumptions
> that non.terminating.decimalsⁿᵒᵗᐧᵂᴹ _exist_
> and line.pointsⁿᵒᵗᐧᵂᴹ _exist_
> and
> a non.terminating.decimalⁿᵒᵗᐧᵂᴹ determines
> no less than one real number (line.pointⁿᵒᵗᐧᵂᴹ) and
> no more than one real number (line.pointⁿᵒᵗᐧᵂᴹ).

Yes, the line point is the limit of an infinite sequence.
Example: 0.999... --> 1.
But the sequence cannot be given other than by its defining formula like 
"0.999...".
> 
> If it is impossible to constructᵂᴹ or defineᵂᴹ or findᵂᴹ or
>   recognizeᵂᴹ or communicateᵂᴹ a 
> non.terminating.decimalⁿᵒᵗᐧᵂᴹ

It is not impossible to construct every desired digit of the sequence. But 
it is impossible to do so without a finite formula. And it is impossible 
to desire all digits. ℵo are always missing.

>> because it cannot be handled with enough precision.
> 
> Then it _exists_ and is unable to be
>   handled.with.enough.precisionᵂᴹ.

It is handled by a formula, sometimes even with absolute precision like 
0.999... --> 1.
> 
>> Always almost all digits are dark.
> 
> All.nonempty.digit.setsⁿᵒᵗᐧᵂᴹ hold a first member.
> All.digitsⁿᵒᵗᐧᵂᴹ have a first.after.digit

Most are unknown.

Regards, WM