Re: Gödel's theorems in Wikipedia

On 23 Jun 2005 20:31:22 GMT, Chris Menzel
<cmenzel@xxxxxxxxxxxxxxxxxxxx> wrote:

>Aatu Koskensilta <aatu.koskensilta@xxxxxxxxx> wrote:
>>I took the time to rewrite the three first sections of the rather sorry
>>Wikipedia entry on Gödel's incompleteness theorems
>>(http://en.wikipedia.org/wiki/G%F6del%27s_incompleteness_theorem). Any
>>corrections or suggestions would be appreciated.
>
>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.

This seems better, because it avoids the "for any property P", which
it seems to me could be read in at least two ways. (If I were in
charge the word "any" would be banned.) But if you want to make
it easy to follow you could break it into more than one sentence:

"Suppose P is a property such that...; then T is not omega-consistent.
If there is no such property P then P is omega-consistent."

(Decreases the size of the stack the reader needs to maintain in his
head, because it doesn't use the "if, for no property P...".)

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


************************

David C. Ullrich
.