Re: Infinity does exist?

From: |-|erc (gotcha_at_beauty.com)
Date: 07/01/04 

Date: Thu, 01 Jul 2004 00:42:09 GMT

"Jack Dominey" <look@my.sig> wrote in
> >Therefore, the only conclusion they could think of was there must be HIGHER INFINITY
> >than countable infinity.
>
> Let it be noted to all who killfiled |-|erc that he said something
> coherent and AFAICT, correct.

AFAICT? Now that is conservative, I reached someone!

Will Twentyman accepts the following :

> Try this: what does it mean for a sequence to represent its diagonal?
>

3
3.1
3.14
3.141

the diagonal is 3.141

All sets of this incremental type, every sequence of digits on the diagonal is present in the set
as a member also.

So if *some* diagonal on the set of computable numbers form the anti_diag,
every sequence of digits in anti_diag is present in members of the set.

Say this is the computables list :

0.1234
0.2111
0.1211
0.1121

The diagonal is 0.1111...

We know the set can't contain 0.2222...

but is can have all the finite prefixes.

0.2
0.22
0.222
0.2222
..

the diagonal of this *incremental* subset of the computable numbers *is* anti_diag hence
there is no finite sequence of digits of anti_diag not present on the list
of computable numbers.

Barb however thinks there are <<<unique digit sequences>>> somewhere on anti_diag.

>> TO THIS QUESTION
>>
>> > >How many (leading) digits of anti-diag appear on the countable infinite
>> > >list
>> > >of computable numbers?
>>
>> BARB WRITES
>>
>> > Any finite number. All of which, of course, are < Aleph_0.
>
>
>So anti_diag has an infinite number of digits. And only a finite number of
>those
>are present in the countable infinite list of computable numbers. Right?
>
>So that means there are {infinite set of digits} - {finite set of digits}
>= {infinite set of digits}
>
>there are an infinite set of digits of anti_diag not present on the list of
>computable numbers. Is that right?

Yes, if by "set" you actually mean "sequence";

> > > the diagonal of this *incremental* subset of the computable numbers *is* anti_diag hence
*********************************************************************

WILL WRITES
> > > there is no finite sequence of digits of anti_diag not present on the list
> > > of computable numbers.
> >
> > So what?

> BARB WRITES
> >there are an infinite set of digits of anti_diag not present on the list of
> >computable numbers. Is that right?
>
> Yes, if by "set" you actually mean "sequence";

> So which is it,
> all digit strings from anti_diag are on the list of computable numbers?
> OR
> there are digit strings from anti_diag which are not on the list of computable numbers?
>
>
> ********* answer goes here ********

   there is no finite sequence of digits of anti_diag not present on the list
   of computable numbers.
   Will : yes

   there are an infinite set of digits of anti_diag not present on the list of
   computable numbers.
   Barb : Yes, if by "set" you actually mean "sequence"

Herc

Relevant Pages

  • Re: Infinity does exist?
    ... >>than countable infinity. ... All sets of this incremental type, every sequence of digits on the diagonal is present in the set ... Say this is the computables list: ... >are present in the countable infinite list of computable numbers. ...
    (sci.logic)
  • Re: Infinity does exist?
    ... >>than countable infinity. ... All sets of this incremental type, every sequence of digits on the diagonal is present in the set ... Say this is the computables list: ... >are present in the countable infinite list of computable numbers. ...
    (comp.theory)
  • Re: WELL WHICH IS IT... ?
    ... >> be on it to an infinite number of digits, ... apparently Ghost and I interpretted the above differently. ... interpretted it as "Given the list of computables, and a real number X, ... X can have all its digits as a single entry on the list, ...
    (comp.theory)
  • Re: WELL WHICH IS IT... ?
    ... >> be on it to an infinite number of digits, ... apparently Ghost and I interpretted the above differently. ... interpretted it as "Given the list of computables, and a real number X, ... X can have all its digits as a single entry on the list, ...
    (sci.math)
  • Re: WELL WHICH IS IT... ?
    ... >> be on it to an infinite number of digits, ... apparently Ghost and I interpretted the above differently. ... interpretted it as "Given the list of computables, and a real number X, ... X can have all its digits as a single entry on the list, ...
    (sci.logic)
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