Re: when laypersons look smarter than math professors Re: a question for the anti-Cantorians

The Ghost In The Machine wrote:
>>>
>> One of the more interesting issues is ordering a list of
semicomputable
> numbers. (I call a number "semicomputable" if it can be approximated
> to any given degree of precision -- or any number of digits --
> with a Turing machine using a sufficiently large amount of tape
space.
> A fully computable number in this context would have a finite number
> of digits. There are admittedly a number of quibbles here; 1/3's
> decimal expansion is infinite but its duodecimal expansion is not;
> 1/3 is a finite representation for a certain number in Q with an
> infinite decimal expansion.)
>
> Now postulate a UTM and an encoding scheme, and list the
semicomputable
> numbers in order:
>
> UTM(1) = ...
> UTM(2) = ...
> UTM(3) = ...
>

I think we can compute this list (or at least up to the
diagonal) because each digit in each semicomputable number is
'fixed' at some finite time. BTW, what exactly do you imply by UTM?
Well, I probably wouldn't understand EXACTLY what you imply by UTM,
but what do you sort-of mean by UTM?

> (We ignore issues such as whether a TM requires a prepopulated tape,
> and whether UTM(n) properly generates a number, freezes up, or
> jiggles.)
>
> Now construct an antidiagonal. Is the antidiagonal associated
> with a UTM? Probably not, but I'm not sure there's an elegant
> way to prove it; the best I can do is show it's not on this list.
>
> If one restricts the range of a "real" function to that of the
> semicomputables, I for one cannot be certain as to precisely
> how many numbers there are (there are only a denumerably infinite
> number of machines, though if one includes input tapes one might
> get an uncountable number of run-sets -- but that's cheating
> in a way since the input tapes would probably comprise all
> real numbers in that case).
>

Is it true that a union of a denumerably infinite set of denumerably
infinite sets is denumerably infinite?

> Fortunately, mathematicians are under no such restrictions.
> If one has a list f : N -> R, then one can prove the existence
> of (and approximate to any desired precision) an antidiagonal
> number which is not on that list, or in the set
> representing the actual range (as opposed to the specified
> range -- the reals) of that list. Cantor found two methods,
> in fact, only one of which depends on an antidiagonal (the
> other depends on various properties of sequences and the
> "continuumness" or contiguity of the reals).
>
> --
> #191, ewill3@xxxxxxxxxxxxx
> It's still legal to go .sigless.

.

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