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From: David C. Ullrich on 7 Jan 2010 10:58
On Wed, 6 Jan 2010 10:59:36 -0800 (PST), "porky_pig_jr(a)my-deja.com"
<porky_pig_jr(a)my-deja.com> wrote:

>On Jan 6, 1:03�pm, David C. Ullrich <ullr...(a)math.okstate.edu> wrote:
>> On Tue, 05 Jan 2010 13:44:36 -0500, "Jesse F. Hughes"
>>
>> <je...(a)phiwumbda.org> wrote:
>> >Charlie-Boo <shymath...(a)gmail.com> writes:
>>
>> >> What is so nice about the statement "ZFC (whatevuh) can prove
>> >> everything provable." is: [...]
>>
>> >You act as if this is a claim that people have made. �I've never heard
>> >anyone make such a claim and it seems trivially false on the plainest
>> >interpretation.
>>
>> Well of course it's false. He refuted it!
>>
>> Some people are never satisfied... CB says utterly wrong things
>> and people complain. He finally says something true, and not
>> only that it's something obviously true, and people complain
>> about that too.
>>
>> Giggle.
>>
>>
>>
>
>Even the broken clock shows the correct time twice a day. shall we
>rejoice every time it happens?

whoosh...




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 7 Jan 2010 15:38
Charlie-Boo <shymathguy(a)gmail.com> writes:

> On Jan 5, 1:44 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote:
>> Charlie-Boo <shymath...(a)gmail.com> writes:
>> > What is so nice about the statement "ZFC (whatevuh) can prove
>> > everything provable." is: [...]
>>
>> You act as if this is a claim that people have made.  I've never heard
>> anyone make such a claim and it seems trivially false on the plainest
>> interpretation.  (ZFC does not prove the theorems of non-well-founded
>> set theory, for instance.)
>>
>> Can you find a single mathematical text that has claimed this?  Or
>> any other source that makes this claim?
>> --
>> Jesse F. Hughes
>>
>> "C is for Cookie.  That's good enough for me."  
>>                                      Cookie Monster
>
> Make that "in Ordinary Mathematics".
>
> http://groups.google.com/group/sci.logic/msg/3b4c60e5742d025f?hl=en

No, even Lord Boetian didn't explicitly say that. Instead, he tried
to get your claim somewhat closer to the truth.

In any case, Lord Boetian isn't really the sort of citation I was
looking for. Is there any case in which this claim has been published
(you know, where editors are involved)? Or any case in which a
professional mathematician (preferably with some recognition in the
field) has said this?

--
"Now for once I might actually have an audience that realizes that
[my proof of Fermat's Last Theorem is correct], because you see,
they'll finally know what's in it for them--cold, hard cash."
--James Harris embarks on a new mathematical strategy.
From: Jesse F. Hughes on 8 Jan 2010 07:53
Charlie-Boo <shymathguy(a)gmail.com> writes:

> �The development of mathematics towards greater exactness has, as is
> well-known, lead
> to formalization of large areas of it such that you can carry out
> proofs by following a few
> mechanical rules. The most comprehensive current formal systems are
> the system of
> Principia Mathematica (PM) on the one hand, the Zermelo-Fraenkelian
> axiom-system
> of set theory on the other hand. These two systems are so far
> developed that you
> can formalize in them all proof methods that are currently in use in
> mathematics.�
>
> On formally undecidable propositions of Principia Mathematica and
> related systems I - Kurt Godel, 1931

Yes, all proof methods can be formalized in those systems.

That does *not* mean that they prove everything provable. What he
means is that all of the usual bits of informal mathematical reasoning
that was being used in 1931 could be made formal in these two formal
systems.

--
Jesse F. Hughes
"Have we learned nothing, nothing, from the downfall of Vanilla Ice?"
-- Time Magazine columnist Lev Grossman on
James Frey's /A Million Pieces/.
From: Jesse F. Hughes on 8 Jan 2010 09:31
Andrew Usher <k_over_hbarc(a)yahoo.com> writes:

> On Jan 8, 6:53 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote:
>
>> Yes, all proof methods can be formalized in those systems.
>>
>> That does *not* mean that they prove everything provable.  What he
>> means is that all of the usual bits of informal mathematical reasoning
>> that was being used in 1931 could be made formal in these two formal
>> systems.
>
> How are these two statements not contradictory? If all proofs can be
> formalised in ZFC, then every proof is a proof in ZFC, etc.

I suppose it depends on what you mean. Let's take ZFA, for instance,
the theory of non-well-founded sets. Admittedly, Goedel was *not*
talking about this theory (since no one was talking about ZFA in
1931), but let's see in what sense "you can formalize in them all
proof methods" that are used in ZFA.

You can do so in this way, as I recall: interpret the sets of ZFA as
particular kinds of graphs. Graphs can easily be represented in ZFC.
This re-interpretation induces a translation of the language of ZFA
into the language of ZFC (where the epsilon relation of ZFA is *not*
the epsilon relation of ZFC). Under this interpretation, the axioms
of ZFA are mapped to theorems of ZFC. Since the underlying logic
(namely FOL=) is the same for both theories, it follows that every
theorem of ZFA is mapped to a theorem of ZFC.

In this sense, the reasoning of ZFA can be formalized in ZFC. Really,
this is not so different than our usual interpretation of PA in ZFC.

This is what I think that Goedel had in mind. I don't see any
contradiction here, nor do I think that this formalization is
adequately captured by saying that "ZFC proves everything that is
provable in ordinary mathematics." It seems to me that this latter
statement is very misleading.

--
Jesse F. Hughes
"To be honest, I don't have enough interest in math to spend the time
it would take to clean up the mess that I believe has been created in
the past 100 or so years." -- Curt Welch lets the world down.
From: Jesse F. Hughes on 8 Jan 2010 09:33
Andrew Usher <k_over_hbarc(a)yahoo.com> writes:

> On Jan 8, 7:13 am, Charlie-Boo <shymath...(a)gmail.com> wrote:
>
>> >  But, he
>> > (Goedel) proved that no first order theory can 'prove everything
>> > provable'.
>>
>> Then you can answer my question: How do you define "everything
>> provable"?
>
> I would say that it must be taken in an informal sense, to mean
> everything that we can write a proof for. For example, we can prove
> that the real numbers are larger than any countable set, but
> Loewenheim-Skolem says we can't do it in ZFC (or any first order
> theory).

Loewenheim-Skolem says *what* now?

You're plainly mistaken. ZFC *does* prove that R is uncountable.

--
Jesse F. Hughes
"If the world weren't rather strange, by now I should at least be with
some research group talking about my number theory research."
-- James S. Harris learns the world is a funny place
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Prev: Mathematics can only cover a validity-range from 10^500 to 10^-500 Re: #272 mathematics ends at about 10^500 Re: Powerset
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