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From: Ostap Bender on 8 Jun 2010 04:26
On Jun 8, 12:24 am, Tim Little <t...(a)little-possums.net> wrote:
> On 2010-06-08, |-|ercules <radgray...(a)yahoo.com> wrote:
>
> > What about this statement?
>
> > All possible digit sequences are computable to all, as in an
> > infinite amount of, finite lengths.
>
> What about it?  Apart from being an example of a mathematically
> ambiguous statement with at least 3 reasonable interpretations, 2 of
> which make it false and 1 makes it true, that is.

At least the number of reasonable interpretations is finite...

From: Graham Cooper on 8 Jun 2010 04:51
On Jun 8, 6:26 pm, Ostap Bender <ostap_bender_1...(a)hotmail.com> wrote:
> On Jun 8, 12:24 am, Tim Little <t...(a)little-possums.net> wrote:
>
> > On 2010-06-08, |-|ercules <radgray...(a)yahoo.com> wrote:
>
> > > What about this statement?
>
> > > All possible digit sequences are computable to all, as in an
> > > infinite amount of, finite lengths.
>
> > What about it?  Apart from being an example of a mathematically
> > ambiguous statement with at least 3 reasonable interpretations, 2 of
> > which make it false and 1 makes it true, that is.
>
> At least the number of  reasonable interpretations is finite...

I only get one interpretation which is true.

It's equivalent to George Greene's wording isn't it?

-Every digit sequence TO EVERY FINITE length is in the computable
list
-of reals.

I'm not disputing it could be ambiguous, but I'm calling your bluff,
what other interpretations?

Herc
From: Daryl McCullough on 8 Jun 2010 07:11
|-|ercules says...
>
>> "Jesse F. Hughes" <jesse(a)phiwumbda.org> wrote
>>> But I don't see how this analogy is supposed to make Cantor's theorem
>>> appear dubious.
>
>What about this statement?
>
>All possible digit sequences are computable to all, as in an infinite amount
>>of, finite lengths.

The correct statement is this:

I. For every real number r, for every natural number n, there
exists a computable real r' such that r agrees with r' in
the first n decimal places.

Note the logical form of this statement:

forall r, forall n, exists r' ...

The order of quantifiers makes a difference! If the change
the order of quantifiers we get a similar-looking but false
statement:

II. For every real number r, there exists a computable real r',
for every natural number n:
r agrees with r' in the first n decimal places.

This has the logical form:

forall r, exists r', forall n, ...

It differs from the first statement in that the order
of the quantifiers has been changed.

Statement I is true. Statement II is false. The claim that
"not all reals are computable" is equivalent to the claim
"Statement II is false". The diagonal argument proves that
Statement II is false, not that Statement I is false.

--
Daryl McCullough
Ithaca, NY

From: Jesse F. Hughes on 8 Jun 2010 09:38
"|-|ercules" <radgray123(a)yahoo.com> writes:

> "Jesse F. Hughes" <jesse(a)phiwumbda.org> wrote
>> "|-|ercules" <radgray123(a)yahoo.com> writes:
>>
>>> "Jesse F. Hughes" <jesse(a)phiwumbda.org> wrote
>>>> "|-|ercules" <radgray123(a)yahoo.com> writes:
>>>>
>>>>>>> Given a set of labeled boxes containing numbers inside them,
>>>>>>> can you possibly find a box containing all the label numbers of boxes
>>>>>>> that don't contain their own label number?
>>>>>
>>>>> Have a go mate!
>>>>
>>>> The answer is no, near as I can figure.
>>>>
>>>> Now, if you also knew that, for each set of numbers, there is a box
>>>> containing that set, then you'd have a paradox. Near as I can figure,
>>>> you *don't* know that.
>>>>
>>>> In set theory, on the other hand, we *do* know the analogous claim.
>>>
>>> So, no box ever containing the numbers of boxes not containing their own numbers
>>> means higher infinities exist?
>>
>> *Given* that every set of numbers is contained in some box, I guess
>> so.
>>
>> But I don't see how this analogy is supposed to make Cantor's theorem
>> appear dubious.
>
>
> So, as many have put it, the holy grail of mathematics, the infinite
> paradise is based on no box containing the numbers of boxes that
> don't contain their own number?

I wouldn't call Cantor's theorem a holy grail, nor claim that it is
*based on* your odd analogy, but in any case this discussion is pretty
pointless.

Your analogy does not refute the simple fact: Cantor's theorem is a
theorem of ZF.

--
Jesse F. Hughes
"Mistakes are big part of the discovery process.
I make lots of them. Kind of pride myself on it."
-- James S. Harris
From: Tim Little on 8 Jun 2010 20:34
On 2010-06-08, Daryl McCullough <stevendaryl3016(a)yahoo.com> wrote:
> Note the logical form of this statement:
>
> forall r, forall n, exists r' ...
>
> The order of quantifiers makes a difference! If the change
> the order of quantifiers we get a similar-looking but false
> statement

I think you're wasting your time. Like many cranks, Herc can't now
and probably won't ever be able to tell the difference.


- Tim
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