Re: Gödel's theorems in Wikipedia

Chris Menzel wrote:

Aatu, I have a suggestion for the definition of omega-consistency in the
first section of the article; the one you provided seems to me a bit
hard to parse.  Currently, it reads:

  Omega-consistency is a technical concept which applies to a theory T if
  T does not prove for any property P that there exists a number with a
  property P and simultaneously prove for every natural number n that n
  does not have the property P.

I suggest:

Omega-consistency is a technical concept which applies to a theory T if,
for no property P, (i) T proves the general proposition that there
exists some number with the property P, but (ii) for every specific natural number n, T proves that n does not have the property P. 

Changed.

It also strikes me that the sentence that follows in that paragraph
(which is intended to clarify the second part of the definition) becomes
unnecessary with this change.


I agree, it's now gone.

Also, the second paragraph in that section might fit better in the
section "Misconceptions about Gödel's Theorem."


Moved it there.

And I still think someone ought to just clobber the section on "The
Meaning of Gödel's Theorem", which is a mess, especially in contrast
with your sections on the incompleteness theorem and the nice new
section (yours also?) on Gentzen's Theorem.

Yes, I rewrote the section on Gentzen's Theorem. I just couldn't stand the previous version which included such interesting claims as

"Gentzen's theorem implies both of Gödel's theorems, and was the natural consequence of wondering whether Gödel's narrow and negative results depended on the assumption of using a theory self referentially as its own proof theory."

!

I'm thinking of ditching "The Meaning of Gödel's Theorem" and adding something about proof theoretical reflection and inexhaustibility in its place, but that'll have to wait.

Thanks for the suggestions.

--
Aatu Koskensilta (aatu.koskensilta@xxxxxxxxx)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
 - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
.

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