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From: Jesse F. Hughes on 11 Jan 2010 07:58
Andrew Usher <k_over_hbarc(a)yahoo.com> writes:

> On Jan 9, 6:21 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote:
>
>> > You keep saying this but it's plainly false in real logic. If ZFC has
>> > a model where 'R' is countable - and LS says it must - then ZFC can
>> > not establish its uncountability as that would make a contradiction.
>> > Sure, we can _write_ a proof in ZFC but it can't be sound.
>>
>> Wrong.  The proof is clearly sound and it's conclusion true.
>>
>> The statement "R is not countable" is true iff there is no surjection
>> N -> R.  In models of ZFC, there is no surjection (in the model) from
>> the interpretation of N to the interpretation of R.  
>
> Yes, but the surjection (in countable models) still _exists_, we just
> can't find it in ZFC. The conclusion that *N < *R is then false.

No.

The proposition |N| < |R| is true iff there is no surjection in the
model from the interpretation of N onto the interpretation of R.
Hence, this proposition is true in every model.

>> In any case, what the hell would you mean that "we can write a proof
>> in ZFC but it can't be sound"?  The only way in which a deductive
>> argument is unsound is if it is invalid or one of its premises false.
>> Surely you agree that the argument R is uncountable is valid, so which
>> premise do you doubt?
>
> In logic, when you prove that A -> B, exhibiting just one possible
> world where A is true and B false contradicts the proof. Countable
> models do the same thing for ZFC's proof, even though the conclusion
> is true: the real real numbers are not countable.

You didn't answer my question. An argument is unsound only if either
it is invalid or at least one of its premises is false. So, which is
it?

(In any case, the fact is that |N| < |R| is true in every model of ZFC
in the requisite sense, namely, that in each model, there is no
surjection [[N]] -> [[R]].)

--
Jesse F. Hughes
"Most people don't even know what a rootkit is, so why should they
care about it."
-- Thomas Hesse, sony executive defends DRM-by-rootkit.
From: Jesse F. Hughes on 11 Jan 2010 07:59
Andrew Usher <k_over_hbarc(a)yahoo.com> writes:

>> (or rather mapping what M thinks is N onto what M thinks
>> is R). The fact that M itself is countable does not contradict
>> this - M is countable, so there is a mapping from the natural
>> numbers onto what M thinks is R. But that mapping is not
>> an element of M.
>
> This just shows the incompleteness of ZFC if such situations can
> exist.

Nothing at all to do with incompleteness.

--
Jesse F. Hughes

"You see 300 of something, anything, and you go `[Man], that's a lot of
stuff.'" -- Jim Bigler, quoted in the Pittsburgh Post-Gazette.
From: David C. Ullrich on 11 Jan 2010 08:49
On Sun, 10 Jan 2010 20:46:40 -0800 (PST), Andrew Usher
<k_over_hbarc(a)yahoo.com> wrote:

>On Jan 9, 9:04�am, David C. Ullrich <ullr...(a)math.okstate.edu> wrote:
>
>> "The real numbers are larger than any countable set"
>> is a theorem of ZFC.
>>
>> And hence, in any model of ZFC it is true that there is no function
>> mapping N onto R. Now. Say M is a countable model of ZFC.
>> It's true that there is function _in M_ mapping N onto R
>
>I suppose you meant a function not in M.

What I actually intended in that sentence was "there is no function
in M...". Yes, it was certainly a typo.

>> (or rather mapping what M thinks is N onto what M thinks
>> is R). The fact that M itself is countable does not contradict
>> this - M is countable, so there is a mapping from the natural
>> numbers onto what M thinks is R. But that mapping is not
>> an element of M.
>
>This just shows the incompleteness of ZFC if such situations can
>exist.

ZFC us certainly incomplete. But that has nothing to do with
what we're talking about here - if T is a complete extension
of ZFC (which of course cannot be recursively axiomatizable)
then everything we've said applies equally well to T, even
though T is complete.

>Andrew Usher

David C. Ullrich

"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)
From: Jesse F. Hughes on 12 Jan 2010 10:08
Andrew Usher <k_over_hbarc(a)yahoo.com> writes:

> On Jan 11, 6:58 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote:
>
>> >> The statement "R is not countable" is true iff there is no surjection
>> >> N -> R.  In models of ZFC, there is no surjection (in the model) from
>> >> the interpretation of N to the interpretation of R.  
>>
>> > Yes, but the surjection (in countable models) still _exists_, we just
>> > can't find it in ZFC. The conclusion that *N < *R is then false.
>>
>> No.
>>
>> The proposition |N| < |R| is true iff there is no surjection in the
>> model from the interpretation of N onto the interpretation of R.
>
> That statement is absurd and offensive to logic. The notion of one-to-
> one correspondence, and therefore equinumerosity, precedes any formal
> theory.

Er, right. So, modern mathematical semantics generally is absurd and
offensive to logic. After all, the interpretation of the proposition
|N| < |R| follows directly from the definition of model.

>> Hence, this proposition is true in every model.
>
> True in ZFC but not True philosophically.

Whatever that means. The fact is that mathematical theories are
interpreted in structures and inherit their meaning via their
interpretations. That's simply how things are.

>
>> >> In any case, what the hell would you mean that "we can write a proof
>> >> in ZFC but it can't be sound"?  The only way in which a deductive
>> >> argument is unsound is if it is invalid or one of its premises false.
>> >> Surely you agree that the argument R is uncountable is valid, so which
>> >> premise do you doubt?
>>
>> > In logic, when you prove that A -> B, exhibiting just one possible
>> > world where A is true and B false contradicts the proof. Countable
>> > models do the same thing for ZFC's proof, even though the conclusion
>> > is true: the real real numbers are not countable.
>>
>> You didn't answer my question.  An argument is unsound only if either
>> it is invalid or at least one of its premises is false.  So, which is
>> it?
>
> The (implied) premiss that is false is that N and R in ZFC are the
> real N and R. If you say that we should not use this premiss, then you
> are admitting that ZFC can say nothing about the real N and R, and
> that therefore the proof 'R is larger than any countable set' can not
> be formalised in ZFC - which was my original claim.

No, this isn't the conclusion. Let's suppose there is a real (i.e.,
intended) model for ZFC, in which [[N]] is the "real" N and [[R]] is
the "real" R. Then the proof |N| < |R| shows that there is no
surjection from the "real" N onto the "real" R -- which is just what
you wanted to know.

As it happens, it also shows that there is no surjection from N to R
in a countable model of ZFC, but so what?

In any case, your "implied" premise is nonsense. It is not part of
the formal argument, and it is the formal argument you called
unsound. So, try again.

--
"This is based on the assumption that the difference in set size is what
makes the important difference between finite and infinite sets, but I think
most people -- even the mathematicians -- will agree that that probably
isn't the case." -- Allan C Cybulskie explains infinite sets
From: Jesse F. Hughes on 12 Jan 2010 11:36
Andrew Usher <k_over_hbarc(a)yahoo.com> writes:

> On Jan 12, 9:08 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote:
>
>> No, this isn't the conclusion.  Let's suppose there is a real (i.e.,
>> intended) model for ZFC, in which [[N]] is the "real" N and [[R]] is
>> the "real" R.
>
> OK, I'll accept that. But ...
>
>> Then the proof |N| < |R| shows that there is no
>> surjection from the "real" N onto the "real" R -- which is just what
>> you wanted to know.
>
> This is wrong, I think. The proof only shows that there is no
> surjection within ZFC; I don't see why we have to believe that the
> standard model includes all possible mappings, given that non-standard
> models don't.

The so-called standard model is supposed to include all the sets that
"really" exist. If so, then it includes all functions which "really"
exist and hence there is no surjection from the "real" N onto the
"real" R.

> The reason we know that N < R is the informal or non-
> firstorderisable argument; a formal argument within set theory can
> only be less certain.

I don't know what you mean by "the non-firstorderisable argument"
(though I'm mighty fond of the adjective).

In any model of ZFC, there is no surjection from [[N]] onto [[R]].
If we suppose that there is a universe containing every set, it is
surely a model of ZFC and hence it follows that in this universe (the
"real" universe of sets), there is no surjection from [[N]] (the
"real" N) onto [[R]] (the "real" R).

And that's what you wanted to know.

Your complaint, it seems, is not that ZFC fails to prove what you
want, but that it also proves other claims which don't interest you.
Namely, we see that there is no surjection [[N]] -> [[R]] in any model
of ZFC. Since the universe of sets is a model of ZFC, the fact you're
interested in follows. But you seem to be bugged by the observation
that this fact also follows in non-standard models, even though there
is nothing at all controversial about its truth in non-standard
models, once one realizes that the claim is relativized to the
functions existing in such models.

--
Jesse F. Hughes
"I'm not good at math(s) - but even I know zero is nothing. To suggest
zero isn't nothing is madness."
-- Phantom scojocup, on why 0 * 4 = 4
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Next: definition of ellipsis to define Finite in the old math and why the Peano Axioms are inconsistent Re: #281 mathematics ends at about 10^500 Re: Powerset