Talk:Hilbert's problems

Transcendental_number says #7 is "unresolved". This pages says that it is solved. Which is right?

The Gelfond-Schreiber theorem states that if a and b are algebraic numbers, and b is also irrational, then a^b is transcendental. The more general case (where b is not algebraic) is still unproved according to Mathworld. Chas zzz brown 22:25 Feb 24, 2003 (UTC)

Thanks :) Martin

Shouldn't that be "Schneider" rather than "Schreiber"? Michael Hardy 22:21 Apr 25, 2003 (UTC)

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It looks like Elin Oxenhielms recent work have been presented in the media as a general solution of the second part of Hilberts 16th. Actually she solves it for a special case and asserts that her method can be used for the general case. While interesting it is certainly not a general solution (unless of course she or another is able to prove her assertion.)

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I notice that the 16^th Problem has been spelt out, whereas the 2^nd Problem simply redirects to Gödel's incompleteness theorem. Would it be possible to have the actual problems as posed by Hilbert spelt out as per the 16^th? In other words, an article on each problem spelling it out and then directing the reader to (for example) Gödel's incompleteness theorem by way of an answer. This would seem to me to be more encyclopædic. Phil 11:33, Nov 28, 2003 (UTC)

Here is a link to the original speak (in german) http://www.mathematik.uni-bielefeld.de/~kersten/hilbert/rede.html. Feel free to translate :) Here is a translation, but that is probably copyrighted http://aleph0.clarku.edu/~djoyce/hilbert/problems.html.

The most widely used English translation of Hilbert's text is the translation by Mary Winston Newson published in 1902. The

HTML version of this translation (http://aleph0.clarku.edu/~djoyce/hilbert/problems.html) provided by David Joyce (mentioned above and under external links in the main article) is widely referenced on the web, and pops up quickly if you do a Google search on "Hilbert problems" (I don't know where the translation at Hilbert's sixteenth problem came from, but it is not as clear as the Newson translation). I don't think reproducing an English translation of the original Hilbert text in Wikipedia adds any value, plus there is a Wikipedia style guideline that says don't include copies of primary sources. I think articles that present a summary of the history and current status of each problem - like the Hilbert's third problem article - are much more useful. -- Gandalf61 14:13, Nov 28, 2003 (UTC)

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3rd problem summary seems really wrong to me, almost reverse of the actual problem... However I'd let someone more competent than me on the subject matter sort that out... JidGom 00:36, 8 Dec 2003 (UTC)

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It's the title Hilbert used in his speech. Then he proceeded to state that he expected it to be proven impossible by using a particular instance of tetrahedra.

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I think it would be appropriate to remove all references to Oxenhielm from this page given the apparant fact that her paper has been universally disproven through peer review. Further coverage of related events may be more appropriate on a page relevant to 'mathematical sociology,' where her situation deserves serious study, but to continue to relate it to Hilberts' problems will probably just feed the issue's flames in the wrong way. --12/29

I agree. I have removed them all (copied below for reference). --Zundark 10:35, 30 Dec 2003 (UTC)

and in late November 2003, Swedish mathematician Elin Oxenhielm (http://www.oxenhielm.com/) was said to be able to solve the second part of problem 16 for a special case; the debate is still going on.

• Recent activity regarding Hilbert's 16th problem: Elin Oxenhielm's proposed proof (http://www.aftenposten.no/english/world/article.jhtml?articleID=685217) (Aftenposten)
• Recent activity regarding Hilbert's 16th problem: Elin Oxenhielm's proposed proof (http://www.nature.com/nsu/031208/031208-4.html) (Nature, 9 December 2003).
• Oxenhielm, E. On the second part of Hilbert's 16th problem (http://dx.doi.org/10.1016/j.na.2003.10.002), in: Nonlinear Analysis: Theory, Methods and Applications

Contents

1 Problem #1 2 Problem #2 3 Proposal for problems 1 & 2 4 More on the 1st problem

Problem #1

I'd say the way most set theorists see the independence of CH from ZFC is that we need better axioms for set theory, namely set theory needs compelling large cardinal axioms capable of settling this question. And there, my impression is that most set theorosts lean towards thinking CH is false: good large cardinal axioms will probably reveal structure of intermediate cardinality.

The current claim, that the independence from ZFC settles Hilbert's problem is a bit silly, since the ZFC axiom scheme was not proposed until more than twenty years later. ---- Charles Stewart 07:55, 8 Oct 2004 (UTC)

Problem #2

To summarise Hilbert's 2nd problem "Are the axioms of arithmetic consistent?" and then claim it is 'solved' by Gödel's incompleteness theorem is silly (if you think arithmetic might be consistent) and shows a fundamental failure to understand what is at stake. Goedel's result only shows that the kind of finitary proof Hilbert sought was impossible; in fact combinatorial principles that go beyond standard arithmetic axiomatisations such as first-order Peano arithmetic can be used to prove consistency of arithmetic, such as Gerhard Gentzen achieved with his consistency proof.

Both of these links need to be changed, and the claim that the first is settled should be changed, possibly also the second ---- Charles Stewart 15:37, 21 Oct 2004 (UTC)

Proposal for problems 1 & 2

Following an exchange with User:Gandalf61 [1] (http://en.wikipedia.org/wiki/User_talk:Chalst#Hilbert.27s_problems), I have:

• Changed status of first problem to open (ie. no generally satisfactory solution to the problem has been proposed, but, contrary to previous version, widespread hope is held that large cardinal axioms can solve the problem, so independence results do not close problem);
• Left status of second problem; changed redirect to page Consistency proof, where I will add a brief discussion of the problem, Gentzen's consistency proof and the status of the problem.

More on the 1st problem

So, I am relatively new to the Wikipedia, and I am not sure what the protocol is for making a factual change after their has been a discussion and I disagree with the outcome. So, I appologize is this isn't the right procedure (and if so please let me know what the right one is).

The reason I have changed the status of the 1st problem to solved is that it is commonly accepted in the mathematical community that the background universe of discourse is always a set theory which is essentially ZFC. Some might argue that there are slightly better axiomizations of set theory (e.g. Godel Bernays Set Theory) or that there might be a better base system (like some category theorists), but for the most part no one believes that it is necessary to have a system with much greater consistency strength that ZFC to do mathematics. And, in all of these systems forcing can easily be codified and hence the independence of CH can be show. (In fact it has been shown than no large cardinal axiom can settle CH.)

While there are many set theorists who study extensions of ZFC (usually by adding large cardinal assumptions) there are very few if any that would say that those extensions are necessary to do mathematics or that they are the only "correct" extension.

ZFC is accepted as the basis for mathematics in a similar way as the Church-Turing Thesis is accepted. Some might argue over the subtleties of the right way to express the basic universe of set theory, but for the most part they are all the same (and all the versions commonly in use agree on the independence of CH)

Also, it is worth mentioning that while some people do hope that there will be axioms which can be added which settle CH (the most notable of them being Woodin), they do not believe that these axioms will be necessary for the rest of mathematics, but rather that they will be so intrinsically beautiful that people want to accept them. And, while some of the work of Woodin suggests that there are such nice axioms which settle CH, there is no reason to believe that mathematics as a whole will embrace them.

Any how, if you have any comments or thoughts, please let me know. Aleph0 02:30, 20 Nov 2004 (UTC)

Yes, a few...

1. What mathematicians in general think about the status of ZFC does not really settle the point; the experts are the small number of mathematicians and logicians who specialise in set theory.
2. I think Harvey Friedman has successfully demonstrated that some mathematics does need axioms beyond ZFC.
3. If I am not mistakened about the "independence of CH from LCAs" result it only applies to LCAs based on Cohens's technique of inner models. There are new, more powerful, forcing techniques, such as core models, to which I think this result does not apply. I plan to check this, but it will not be in the next couple of weeks, since I will be very busy.
4. There was a discussion on the FOM mailing list recently about what are LCAs: you might check it out. IIRC, the consensus is that LCAs might settle CH.
5. If you are not happy with my summary, you could mail the FOM list to see whether they regard the 1st problem as settled.

I propose to 'split the difference': leave it as settled, but with a footnote saying there are grounds to dispute this. I will apply the edit shortly. ---- Charles Stewart 22:49, 20 Nov 2004 (UTC)