Talk:Hilbert's problems

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Archive: before 2007

Contents 1 Second problem and consistency proof

[edit] Second problem and consistency proof

The current table entry for the second problem is quite odd; it has several long footnotes that really belong as content in the article about Hilbert's second problem. But there isn't any article that describes the second problem, and the article that should (Hilbert's second problem) redirects to consistency proof, which is about consistency proofs in general but not about Hilbert's second problem.

Here is what I would like to do:

1. Change table to say:

2nd

Prove that the axioms of arithmetic are consistent.

Partially resolved: Gentzen proved in 1936 that the consistency of arithmetic follows from the well-foundedness of the ordinal ε0. Gödel's second incompleteness theorem shows that no proof of its consistency can be carried out within arithmetic itself.

2. Move the footnotes from this page to Hilbert's second problem (no longer redirected to consistency proof) and use them to write an article about the second problem.

Thoughts? CMummert 14:59, 5 January 2007 (UTC)

That would definitely be an improvement on the current situation. --Zundark 15:31, 5 January 2007 (UTC)

I was the guy who added all these footnotes. The reason is, from my reading -- that is, from the published stuff I've encountered -- I would say that the matter is actually "resolved" -- but some assert "in the affirmative", some assert "in the negative". Thus it has become a "matter of debate" -- at least in the literature. Kleene "resolves" it as a resounding NO ... given certain assumptions/qualifications. (And Kleene 1952 has a long discussion at the end about Gentzen's proof, so he isn't ignoring the issue). And the quote by Nagel and Newmann is particularly damning with respect to Gentzen's proof -- they do not accept it as a resolution: period. The problem is: this is what the literature is saying, not us. If it were left to me, I would word it as below. I agree that some of this could be moved to another article, but the summary here requires a more sharply-pointed "A matter of debate; some hold it as resolved in the affirmative, some in the negative" for the casual reader.

2nd

Prove that the axioms of arithmetic are consistent.

A matter of debate. Given the results of the subsequent 50 years, Hilbert's question is too vaguely worded to resolve the matter with a simple yes or no. Gentzen proved in 1936 that consistency of arithmetic follows from the well-foundedness of the ordinal ε0. Gödel's second incompleteness theorem shows that no proof of its consistency can be carried out within arithmetic itself.

wvbaileyWvbailey 16:02, 5 January 2007 (UTC)

Yes, some people think that it "is" resolved (in each direction) and some think it is not; that is what I want to cover in depth in the new article on Hilbert's second problem. Since you have read up on it, you might be interested in helping with that article.

The wording you suggest overstates the extent of "debate". Everyone understands everyone else's position about the interpretation of Godel and Gentzen's proofs, and the "debate" boils down to which side you are sympathetic towards. How about this version, parallel to the first problem:

2nd

Prove that the axioms of arithmetic are consistent.

Gentzen proved in 1936 that the consistency of arithmetic follows from the well-foundedness of the ordinal ε0. Gödel's second incompleteness theorem, proved in 1931, shows that no proof of its consistency can be carried out within arithmetic itself. There is not consensus on whether these results give a solution to the problem as stated by Hilbert.

It is true that we must reflect what the literature says, but this issue requires extra care because a good understanding of these issues took a while to form and so some literature from the early to mid 20th was written before the issues were as well understood as they are now. CMummert 16:31, 5 January 2007 (UTC)

The write-up of Gentzen's proof which I read in Mendelson also uses a kind of infinitary logic which I find more problematical than the well-foundedness of epsilon zero. A separate article would be good. I changed the "see also" section of "consistency proof" accordingly. JRSpriggs 05:46, 6 January 2007 (UTC)

I haven't read Mendelson's exposition. It is possible to recast the Gentzen's proof in a way that uses an ω-rule to deal with the induction axiom, but the proof can also be done without infinitary inference rules. CMummert 16:43, 6 January 2007 (UTC)

I would be satisfied if the entry were more like this (put the no? not?-consensus first and then in time order Godel followed by Gentzen):

2nd

Prove that the axioms of arithmetic are consistent.

There is not consensus on whether the results of Gödel 1931 and Gentzen 1936 give a solution to the problem as stated by Hilbert: Gödel's second incompleteness theorem shows that no proof of its consistency can be carried out within arithmetic itself. Gentzen proved that the consistency of arithmetic follows from the well-foundedness of the ordinal ε0.

I expect the sub-article will result in quite a snarl (which should be quite interesting). Kleene's quote is in context of the intuitionists: the issues around what Kleene calls "the completed infinite" and Brouwer's fundamental objection to it, and around Church's thesis (cf p. 318, §62 Church's Thesis, Chapter XII Partial Recursive Functions). There is (at least one) very large gorilla in the room. wvbaileyWvbailey 16:21, 6 January 2007 (UTC)

That order isn't parallel to the box for the first problem. Do you favor reversing that one as well?

I believe they are fine as they appear now excepting a link to the new article should be added within the box (there are no footnotes now, so the reader needs a link).wvbaileyWvbailey 18:49, 7 January 2007 (UTC)

As with all the other problems, the problem number 2nd links to the article with details. CMummert 18:52, 7 January 2007 (UTC)

Yes you are correct; it links nicely. Thanks. wvbaileyWvbailey 19:26, 7 January 2007 (UTC)

As for the other article, it isn't a "subarticle" (same issue as with the algorithm articles), it's independent. The issue is not as contentious as you make it out to be; what is required to prove the consistency of PA is well understood. The situation is quite similar to the status of the continuum hypothesis. CMummert 16:43, 6 January 2007 (UTC)

Actually, I don't think the situation is very like that of CH, and for precisely the reason you state: The status of the 2nd problem is well understood, and is unlikely to change, barring some spectacular development (say, the success of Edward Nelson's project to prove that arithmetic is actually inconsistent). CH is not like that. It is reasonably possible that, thirty years from now, it will be widely agreed, based on Woodin's, work, that the continuum is . Oh, not a full consensus I think, but maybe the sort of weak "it's the natural thing to assume" sort of consensus that we see now for large cardinals. --Trovatore 07:37, 7 January 2007 (UTC)

Since there was no vocal opposition, I have split out the article Hilbert's second problem. The discussion should move to that article's talk page. CMummert 17:10, 6 January 2007 (UTC)