Re: In a Bad Mood

From: The Ghost In The Machine (ewill_at_aurigae.athghost7038suus.net)
Date: 06/12/04 

Date: Sat, 12 Jun 2004 08:01:42 GMT

In sci.logic, |-|erc
<gotcha@beauty.com>
 wrote
on Sat, 12 Jun 2004 02:17:51 GMT
<jJtyc.5299$sj4.3021@news-server.bigpond.net.au>:
> "Barb Knox" <see@sig.below> wrote in
>> "|-|erc" <gotcha@beauty.com> wrote:
>>
>> >> 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"; we're dealing with sequences of
>> digits here, not a sets (since order is important, and multiple instances of
>> the same digit can occur).
>>
>> All finite-length subsequences of the diagonal sequence could be members of
>> the list, but the entire infinite-length diagonal sequence can not be a member
>> (by its construction).
>>
>> You yourself indicated in another post that you pretty-much accept the fact
>> that the method of constructing the diagonal sequence does mean it can not be
>> any particular member of the list, but it seems that somehow you nevertheless
>> consider the diagonal sequence to be "on" the list even though it is different
>> from every member!
>
> A fair interpretation, the only conclusion being its not a valid number.
>
> There is no unique sequence of digits to anti_diag.
> If there was there must be some finite digit position in the sequence of
> anti_diag where this unique sequence begins.
>
> What is the value of that digit position. It can't be 1, 2, 3, ..
> Given any Suffix Starting Point of anti_diag I can index a
> computable number that covers the prefix and the 1st digit of the
> suffix combined.
>
> Therefore, the digit sequence of anti_diag not present on the
> list of computable numbers begins at digit infinity. Right?

What "digit infinity"? I'm not sure "digit omega" (which would
at least be a proper ordinal!) makes any sense in this context.

>
> Its much better than the paradox you have to realise, since *all*
> digits are provably in sequence on the list.
>
> The *set*
> 0.9
> 0.99
> 0.999
> ...
>
> Is equivalent to the number 0.99..

One could admittedly make a case therefor; one valid extension
of Q to R is via Cauchy sequences. However, the number defined
thereby is not in the set, as can trivially be shown in this case:

s_i = 1 - 10^(-i), integer i > 0
S = 1

Or one can define sqrt(a), a > 0, by using the following recursion:

g_1 = 1
g_{i+1} = ((a / g_i) + g_i) / 2

which results in the sequence (for a = 2)

1, 3/2, 17/12, 577/408, 665857/470832, 886731088897/627013566048, ...

It turns out that this sequence converges very quickly,
to sqrt(2), even if one's initial guess is way off.
It's not monotonic but that doesn't matter too much here,
and one can take alternate terms in a pinch.

Is sqrt(2) rational? Turns out that it's not, yet every
term in the sequence is.

> You treat countable infinity like its *big*, its not its infinite,
> therefore every sequence is covered.

Highly debatable, given Cantor's first proof.

> Anti_diag is merely an exponential complexity issue on the
> size of the numbers being larger than the combinations to represent them.
>
> 345
> 678
> 123
> ..
>
> 373 is the diag, you'd think 484.. would never appear on the list.
> Stretch the grid out to infinity, 484.. isn't on the list? Yes it
> is, it doesn't even matter what digits are part of ..

Any finite prefix of ".484..." is of course on the list, by construction.
This does not mean that the number itself is.

>
> The question is, does the infinite list map to the entire number line?
> Given its proper interpretation the answer is yes. like
> {0.9, 0.99, 0.999, ...} maps to the number 1 on the number
> line.

Therefore all sets are closed? An interesting theory, but what
does one do with a set such as

{0.3, 0.4, 0.33, 0.44, 0.333, 0.444, ... }

Does that map to two points, namely 1/3 and 4/9?

{0.2, 0.3, 0.4, 0.22, 0.33, 0.44, 0.222, 0.333, 0.444, ... }

Does that map to three points, 2/9, 1/3, and 4/9?
What if one scatters the items?

How about Q itself? Does that map to every real number? Certain
infinite proper subsets of Q can be used to define any real number;
in fact, that's absurdly easy if one can compute floor(r * N) / N,
where N > 0 is any integer. A faster function would be
floor(r * 10^N) / 10^N, of course.

Not every infinite proper subset of Q converges: the
subset J of all integers, for instance, defines no real number
as a limit point.

>
> Computationally atleast, uncountable infinity is not even a concept.

Correct, but mathematics isn't usually concerned regarding
computability except for computing problems.

> If you construct a anti_diag there was some mechanism to do so,
> such it can be listed as an algorithm, a row in the infinite list
> it hasn't reached yet.
>
> The list is complete, but the LISTS INFINITE FORM is required to
> map to every number.
>
> Computer Theorist : This is my infinite countable list of reals
> Mathematician : I examined the list and it missed this number,
> this number, this number...
> Computer Theorist : You can't examine an infinite list.

The mathematician is correct, but the resulting function is not
computable. Say one ordered the list in a way such that
the complexity of each function increases as one goes down the list.
The function defining the diagonal would then require a complexity
greater than any function on the list -- an absurdity as far as
computability is concerned, but not an impediment to Cantor's two proofs.

>
> Herc
> 

-- 
#191, ewill3@earthlink.net
It's still legal to go .sigless.

Relevant Pages

  • Re: Well Ordering the Reals
    ... >>> rightmost zero in an unending sequence of ever more rightward ... >> naturals, you might as well call it something, I suppose. ... > In TO's system of "whole numbers", there is a most significant digit and ... >> infinite unending string of bits, even if most are generally ignored. ...
    (sci.math)
  • Re: Cantor Confusion
    ... Cantor's diagonal proof is about infinite sequences. ... > You do not require that one digit represents the number 1/3 in Cantor's ... > infinite sequence? ... I do no want not extend anything to nodes representing numbers. ...
    (sci.math)
  • Re: Warning signal
    ... But if the irrational or transcendental binary sequence is transmitted ... Infinite aperiodic streams with finite aperiodic sub-streams of ... Suppose without loss of generality that the first digit is 0. ...
    (sci.math)
  • Re: Warning signal
    ... But if the irrational or transcendental binary sequence is ... Infinite aperiodic streams with finite aperiodic sub-streams of ... Suppose without loss of generality that the first digit is 0. ...
    (sci.math)
  • Re: abundance of irrationals!)
    ... No such "sum" exists except as, and only as, the limit of the infinite ... If an infinite sequence does not have to "achieve" its limit, ... > It is only for lists not initially known not to be surjective that ...
    (sci.math)
Loading