Talk:Hilbert's problems/Archive 1

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On problem 6

6th problem, "non-mathematical"?

I'm not sure I understand, or agree with, the status given for the 6th problem: "Non-mathematical". As I understand it, Hilbert was intending the same thing for physics which had already occured for Euclidean geometry. Quoting Hilbert:

Geometry is a science which essentially has developed to such a state that all its facts may be derived by logical deduction from previous ones. Comptely different from e.g. electricity theory or optics where even today new facts are still being discovered."

and

Now also all other sciences are to be treated following the model of geometry, first of all mechanics, but then also optics and electricity theory.

How is such a program "non-mathematical"? (See Sauer [1]).

Paul August ☎ 21:09, 10 March 2006 (UTC)

It requires empirical observations. General relativity could be axiomatized, for instance, but it's not a full and correct theory of physics. We don't have one. Thus, the problem is basically one of a lack of empirical observations, insofar as the problem is to axiomatize the correct laws of physics. —Simetrical (talk • contribs) 02:16, 23 May 2006 (UTC)

Isn't it impossibly naive to expect solving all physical problems in the near future(even more so back then). I mean... most of the mathematical problems are proving one theorem or the like whereas this is discovering all the hidden laws of physics and essentially completing all of science since all knowledge about all systems could then be deduced straightforwardly. Elentirmo 15:52, 24 August 2006 (UTC)

Axiomatizing physics, does not (to me) mean solving all physical problems, it simply means axiomatizing all accepted physical theories (correct or not, complete or not), The analogy with geometry is exact I think. For example Euclidean geometry was axiomatized, but this did not depend on Euclidean geometry being a correct or complete theory of the universe we live in (which of course it apparently isn't). I would also hesitate to call Hilbert naive. Paul August ☎ 17:00, 24 August 2006 (UTC)

Problem 7

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

I think the obvious extension is false if b is not algebraic. (counterexample: a=10, b=log 2, a^b=2) What's the undecided question? Ralphmerridew 04:13, 16 February 2006 (UTC)

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

Problem 16

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.)

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 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)

3rd problem

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)

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.

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 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 (Aftenposten)
Recent activity regarding Hilbert's 16th problem: Elin Oxenhielm's proposed proof (Nature, 9 December 2003).
Oxenhielm, E. On the second part of Hilbert's 16th problem, in: Nonlinear Analysis: Theory, Methods and Applications

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.

Well, large cardinals of the sort we know today can't settle CH, because they're preserved under small forcing. Steel suggests that there might be things we would still recognize as large cardinal axioms--"natural markers of consistency strength"--that could settle it. --Trovatore 08:22, 20 July 2005 (UTC)

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)

Damn straight! --Trovatore 08:22, 20 July 2005 (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 [2], 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.

What is the 1st problem, exactly?

So I followed the link to Hilbert's speech, and it appears to me that (at least in the speech; I haven't found the subsequent written list), Hilbert's question was not "is CH true?" but rather "is CH provable?". (He doesn't seem to have explicitly considered that it might be refutable.) Provable from what exactly, he doesn't specify (note that ZFC had not been formulated at the time). --Trovatore 03:43, 27 July 2005 (UTC)

One possibility, I guess, is that he wanted to know if CH could be proved by some elementary combinatorial argument along the lines of the ones that had characterized the study of the transfinite up to that time. I must concede that we are now pretty confident that the answer to that question is "no" (though the question is not well enough formulated to admit a truly definitive answer). --Trovatore 03:43, 27 July 2005 (UTC)

If this analysis of what Hilbert meant can be substantiated, then it might be appropriate to do the following:

Change the status to "partially solved", with a footnote similar to the current one
Important! Change the description to "provability of the continuum hypothesis"
Make it clear somewhere that the question of the truth of the continuum hypothesis still has no satisfactory answer.

--Trovatore 03:43, 27 July 2005 (UTC)

Hilberts 24th problem

I've added a reference I found on Rüdiger Thieles Homepage. Somebody interested in checking this out? Maybe the link to the pdf can be removed as it only contains a few lines and isn't of much interest. cheers --Cyc 11:55, August 8, 2005 (UTC)

12th problem

The table states that the problem was solved, yet the page Hilbert's twelfth problem states that this is an open problem. So which one is correct? Samohyl Jan 20:44, 9 August 2005 (UTC)

The specific case of complex quadratic fields is solved; the general, broad question is wide open. Charles Matthews 08:25, 3 November 2005 (UTC)

10 vs. 23 problems

There is a discussion on german wikipedia if hilbert did present 10 or 23 problems at first. In a german translation of a speach given by him in Paris 1900 he talks about all 23 problems. If someone has any clues to where the 10 comes from, he may share it on our discussion there. You're welcome to participate in english. Maybe someone even has the original french paper. Chrislb --84.163.201.189 18:33, 11 August 2005 (UTC)

An as I may cite from David Hilbert: He put forth a most influential list of 23 unsolved problems at the International Congress of Mathematicians in Paris in 1900. So his speach was actually the first publication - the original paper should then point out the correct number. Chrislb --84.163.201.189 18:43, 11 August 2005 (UTC)

2nd problem again

The 2nd problem now reads:

Prove that the axioms of arithmetic are consistent (that is, is arithmetic a formal system that does not contain a contradiction?. Resolution: This is not possible; Gödel's incompleteness theorems show that sufficiently strong proof theories cannot prove their own consistency.)

The characterisation of the problem is fine, but, as I have said before, to regard Gödel's incompletness theorems as resolving this problem neagatively is silly: Gödel's theorem only tells us that sufficiently strong formulations of arithmetic cannot prove their own inconsistency, not that no proof of the result is possible. In fact we have such a proof: Gentzen's 1938 consistency proof for arithmetic by means of ordinal induction.

There is widespread mistrust of this result, and I'd like to spend a bit of time showing why the mistrust is based on misapprehension of what mathematical demonstration is all about. Gentzen's proof uses structural proof theory to transform the kind of opaque combinatorics of axiomatics into the transparent combinatorics of ordinal notation systems that is directly amenable to mathematical intuition. Later work has shown that the well-foundedness of the particular ordinal Gentzen used, $\epsilon _{0}$ , is equivalent to the orderability of the extended polynomials (that is the polynomials that use, beyond addition and multiplication, also exponentiation) with respect to eventual domination, an even more transparently accessible concept.

It is logically possible to doubt the means used in this proof, but this kind of radical doubt could be applied to many, in not all of the proofs of Hilbert's problems. If we are going to say this problem has been negatively resolved, then we should say that we have proven none of the problems.

I should expand on the above at consistency proof, I've been meaning to for almost a year now... --- Charles Stewart 15:40, 14 October 2005 (UTC)

In the context of this discussion, it seems that we need to decide, not just whether we're convinced by a particular argument, but whether Hilbert would have accepted it as a proof. It strikes me as possible that Hilbert would have considered Gödel's result as resolving the problem negatively.

Now for a slightly OT question: I've always wondered about why the Gentzen proof is considered to have philosophical import as establishing the consistency of PA. It seems to me that if you believe in the existence of N as a completed whole, then it's hard to doubt the consistency of PA, because N is obviously a model of PA. On the other hand, if you don't believe in N, then why on Earth would you believe in ε₀? --Trovatore 21:56, 15 October 2005 (UTC)

On the first point, I'd favour the consensus of the mathematical community, but there's no doubt that Hilbert's reaction to Gentzen's result would be an interesting data point, if indeed he did study it. There's no doubt that Goedel's result derailed Hilbert's program, I'd be surprised if Hilbert thought it settled his 2nd question.

On the second point, the existential import of a model of PA is a lot more than the natural numbers alone: it comes with a lot of functions over the natural numbers: all of the dizzyingly fast-growing functions in the Ackermann hierarchy are provably total in PA. You could say that the existence of recursive models of PA is an alternative to Gentzen's proof, but the pieces of that mathematics fell into place rather later. --- Charles Stewart 01:31, 17 October 2005 (UTC)

So the totality of the Ackermann function as an isolated proposition might not be intuitively obvious, but it follows from axioms that are, once you accept the existence of N.

Let me put it another way: If the natural numbers go on forever and induction is reliable, then how can PA prove a contradiction, since it can prove only truths, and contradictory statements can't both be true? And if the natural numbers don't go on forever or induction is not reliable, then how can you rely on an inductive proof of consistency, up to an ordinal that's greater than infinitely many naturals? --Trovatore 01:42, 17 October 2005 (UTC)

Reliability of induction is entirely the problem. How exactly do you prove that the induction axiom schema is satisfied in your model? There are nice proofs by appeal to regular induction in the meta-theory, or by appeal to epsilon-induction, but if you are concerned with validating induction these are not going to cut it. The problem with induction lies in the complexity of the schema (Sigma_4 formulae give me headaches), the nice thing about Gentzen's proof is it lays out what the complexity assocaited with proofs is by means of a structure, admittedly more complex than N, but where we are only concerned with very simple operations. --- Charles Stewart 13:50, 17 October 2005 (UTC)

It seems to me to be "directly accessible to intuition" that if N exists, then induction on N is valid. That's induction for any property, not just first-order definable ones, so the complexity of the formula doesn't come up. What exactly is the intuitive motivation for believing in induction for simple formulas, if you don't believe in just-plain-induction? --Trovatore 15:50, 17 October 2005 (UTC)

If you find the validity of induction to be blindingly obvious, then the 2nd problem is going to seem rather like having lengthy philosophical arguments about the existence of the external world. But consider this: there are predicates for which induction is problematic, such as those in the Sorites paradox. One might at least be able to sympathise with someone who is worried that there is something funny going on with induction with universal quantifers, since your step cases involve formulae that are quantifying over numbers your induction hasn't reached yet (I think this is what is going on with Ed Nelson's argument that regular arithmetic is impredicative). If you restrict to the nice, decidable case of induction with Sigma_1 sentences then you are free of this worry. --- Charles Stewart 18:10, 17 October 2005 (UTC)

It's interesting that you should mention Nelson, because he's actually a fine example of what I'm getting at. As you may know, he does not in fact believe that PA is consistent (more: he believes that it's inconsistent, or at least has believed this). Indeed, he does not accept the (completely finitistic!) proof of the consistency of Q (Robinson arithmetic).

And I think from his perspective, this position (or at least the skeptical as opposed to "strong atheist" part of it) is entirely rational. If natural numbers are not "real objects", "out there", but simply products of our imagination, then it is hard to see why arguments should be reliable if they are based on the assumption of the existence of naturals far beyond our experience of numeration of real objects. --Trovatore 18:57, 17 October 2005 (UTC)

Nelson's brand of skepticism really undermines normal mathematical intuition; to paraphrase what I said above, this is something that has to be in place when we come to think about Hilbert's problems, otherwise the exercise makes no sense. To get back to what should be in the article: is there anything in particular the matter with what I wrote for the 2nd problem? The consistency proof article can get into the waters we've been treading here, I've been thinking for a while that a proper discussion of the 2nd problem would provide a good focus for that article. --- Charles Stewart 19:45, 17 October 2005 (UTC)

I think it's all right (though there's an unmatched paren and the epsilon didn't come out as a symbol). As I said somewhere, we do now know rather precisely what you have to assume to prove PA consistent, and that's probably good enough to call the problem "resolved". --Trovatore 02:37, 18 October 2005 (UTC)

I like that way of characterising it. --- Charles Stewart 13:50, 18 October 2005 (UTC)

I've cleaned up the punctuation and added a footnote; see what you think. --Trovatore 18:16, 18 October 2005 (UTC)

wvBailey here to Trovatore: It's very clear from reading Godel's own proof that he proved something. I'll quote him here (from the English translation in The Decidable. If you have a better one, lemme know): "It is reasonable therefore to make the conjecture that these axioms and rules of inference [PM and ZFS] are also sufficient to decide all questions which can be formally expressed in the given systems. In what follows it will be shown that this is not the case, but rather that, in both of the cited systems [PM and ZFS], there exist relatively simple problems of the theory of ordinary whole numbers which cannot be decided on the basis on the axioms--4 (his footnote 4: "More precisely, there exist undecideable sentences....etc.") "This situation does not depend upon the special nature of the constructed systems, but rather holds for a very wide class of formal systems, among which are included, in particular, all those which arise from the given systems by addition of infinitely nmany axioms (his footnote 5: "in PM only those axioms are considered distinct which do not arise from each other by a change of types.") If Godel had doubts, please cite the reference. Thanks, wvbaileyWvbailey 01:22, 6 January 2006 (UTC)

I'm not sure I take your point. ~~Of course Gödel had no doubts that the consistency of arithmetic could not be established in ZFC or in the system of Principia, granted that those theories are consistent.~~ The inference he disclaimed was that said consistency could not be established by finitistic methods. There might have been, per Gödel, finitistic methods too strong to be formalized in the system of Principia or even in ZFC. Since to my knowledge no one has ever said precisely what a finitistic method is, I don't see any way that question even could be settled. --Trovatore 02:49, 6 January 2006 (UTC)

Whoops, said something silly--comes of trying to respond too fast. The consistency of arithmetic certainly can be established in ZFC, and probably in PM (I don't know that theory very well). What I mean is, just because arithmetic (I'm assuming Peano Arithmetic) can't prove its own consistency doesn't prove (for Gödel) that there aren't more powerful, but still finitistic, methods that could prove the consistency of Peano Arithmetic. --Trovatore 02:54, 6 January 2006 (UTC)

parentheses

There are too many sentences in parentheses. Things should be reworded to not need this. --MarSch 12:11, 3 November 2005 (UTC)

Hilbert's first problem page

Charles Matthews suggested, over at Wikipedia talk:WikiProject Mathematics that redirects like Hilbert's first problem should be filled out in some sort of "buffer page". Whether this is necessary or not in the general case, I certainly agree in the specific case of the first problem.

If you look at Hilbert's speech (there's a link on the main page), he raises two points in the first problem: the continuum hypothesis (or possibly, the provability of the continuum hypothesis), and the issue of the existence of a wellordering of the reals. The latter question is not currently touched here. I see a wide range of possible positions as to its status:

Solved affirmatively by Zermelo's proof, using the axiom of choice.
Solved negatively by Cohen's proof of independence from ZF
Open, because Hilbert wanted to "find" a wellordering, which implies definability, and the existence of a definable wellordering of R is independent of ZFC
Solved as neither "yes" nor "no", for exactly the reason stated above
Solved negatively, because no definable wellordering can be proved to be a wellordering, using just ZFC
Solved negatively, because Hilbert would have wanted something much better than just "definable"; he would have expected something definable in some natural combinatorial way, and if we specify try to pin down that notion according to descriptive set theory, we can prove that no such simple wellordering exists.

The page could also discuss what's consistent with large cardinals--it can be shown that large cardinals of the sort understood today cannot exclude definable wellorders, though they can push the lower bound on their complexity up pretty high, but not higher than Δ²₂, by a result of Saharon Shelah. --Trovatore 17:41, 3 November 2005 (UTC)

Thanks. It's pretty convincing on the basic point, which is that the unmediated redirect is untenable. Charles Matthews 17:56, 3 November 2005 (UTC)

Both this page and the second problem would benefit from this, although another fix would be to have a section in the existing target page that explicitly deals with these problems of interpretation. For the 2nd problem, I already plan on writing a "2nd problem" section, though it's on the sometime/maybe section of my tasks list at the moment. --- Charles Stewart 18:57, 3 November 2005 (UTC)

There is a question here as regards the speech: The section is apparently entitled "Cantors Problem von der Mächtigkeit des Continuums", which doesn't mention wellordering the reals. That's from the text here. Question: is this section title from Hilbert's notes, or from some later compilation or something? Hilbert definitely mentions the wellordering problem in the section, towards the end. --Trovatore 19:57, 3 November 2005 (UTC)

IIRC Hilbert published his address, so the section headings should be from that publication. --- Charles Stewart 21:01, 3 November 2005 (UTC)

Then perhaps among the ambiguities to be discussed in Hilbert's first problem is whether or not the question of wellordering the reals is in fact part of the problem. --Trovatore 21:15, 3 November 2005 (UTC)

The way that the well orfering of the real numbers comes into this has to do with two versions of the Continuum Hypothesis:

The cardinal number of the set of real numbers is $\aleph _{1}$ .
Every infinite set of real numbers is either countable or has the same cardinality as the set of all real numbers.

The point is that these two versions are equivalent only if there is a well ordering of the set of real numbers. (Otherwise the cardinality of the set of real numhers is not even an aleph.) Also, there is no reason to think that Hilbert was not satisfied with Zermelo's proof of the well ordering theorem. Hilbert was a prime exponent of set theoretic mathematics. His later talk of finitary poofs was just to enable a proof of the consistency of foraml systems in which set theoretic reasoning is encapsulated, a proof using such clear and explicit methods that it would convince everybody.

German texts

Look here for original German versions of the published paper. Charles Matthews 22:39, 3 November 2005 (UTC)

Three requests/questions

Could we have the original German quotes for Hilbert?
Where the problems are yes/no answers can we put the answer?
Should we add a spoiler template?

Rich Farmbrough 18:32, 8 December 2005 (UTC)

As a very regular 'pedia user I missed the fact that the numbers in the table are links to the problems. I'll se if I can improve this somehow. Meanwhile anyone can suggest or make an improvemnet go ahead. 213.48.182.7 18:46, 8 December 2005 (UTC)

Re point 1: maybe it would be a good idea to put the German up on wikisource. The text accompanying each problem is quite long, though, and there isn't always a single sentence or two where one says this states the problem. For the second, the answer is explained a little for some of the problems, it would be nice if this were done for all of them. Third, no - we're not giving away plot details here! --- Charles Stewart 18:51, 8 December 2005 (UTC)

1. That would be good, I slipped from the David Hilbert page to here without realising it, (must be getting old) I really meant

"About a year later, he attended a banquet, and was seated next to the new Minister of Education, Bernhard Rust. Rust asked, "How is mathematics in Göttingen now that it has been freed of the Jewish influence?" Hilbert replied, "Mathematics in Göttingen? There is really none any more" (Reid, 205)."

2. E.G. "Is there a non-regular, space-filling polyhedron?" this is listed as resolved, and I would like to know the answer!

3. I didn't think so....

Rich Farmbrough 00:15, 9 December 2005 (UTC)

I don't know the source for (1), though I've heard the quote several times. If you have Reid's book, can you check if she says anything about my question at Wikipedia:Reference desk/Mathematics#Source for Hilbert's Hotel? --- Charles Stewart 00:21, 9 December 2005 (UTC)

Problem 18

Has a superscript 6, but the footnote refers to problem 16 as well. Is this a typo or should there by another footnote? njh 04:05, 5 January 2006 (UTC)

I've cut out the comment about the 16th, since it contains nothing crucial. The dedicated page is a better place for updates. Charles Matthews 09:43, 5 January 2006 (UTC)

In original German text (see the second external link), the 18th question asks "ob es auch im n-dimensionalen Euklidischen Raume nur eine endliche Anzahl wesentlich verschiedener Arten von Bewegungsgruppen mit Fundamentalbereich giebt." To translate roughly and modernize some of the jargon, he's asking whether there are finitely many different types of groups acting properly discontinuously on Euclidean n-space. This isn't included in the two aspects of the question mentioned in the list here, but it's historically important because it was the motivation for Bieberbach's theorems on crystallographic groups. I think some mention of this question should be added. Matt Day 13:39, 26 April 2006 (UTC)

Proposed format change for footnotes

Would anyone object if I changed the formatting of the footnotes. I can't stand those "up arrow thingies". I think this looks better:

Tabulated information

Hilbert's twenty-three problems are:

Problem	Brief explanation	Status
1st	The continuum hypothesis (that is, there is no set whose size is strictly between that of the integers and that of the real numbers).	No consensus.¹
2nd	Prove that the axioms of arithmetic are consistent (that is, that arithmetic is a formal system that does not prove a contradiction).	Partially resolved: Some hold it has been shown impossible to establish in a consistent, finitistic axiomatic system² - However, Gentzen proved in 1936 that consistency of arithmetic followed from the well-foundedness of the ordinal $\epsilon _{0}$ , a fact amenable to combinatorial intuition.
3rd	Can two tetrahedra be proved to have equal volume (under certain assumptions)?	Resolved - no, using Dehn invariants.
4th	Construct all metrics where lines are geodesics.	Too vague.³
5th	Are continuous groups automatically differential groups?	Resolved.
6th	Axiomatize all of physics.	Non-mathematical.
7th	Is a^b transcendental, for algebraic a ≠ 0,1 and irrational algebraic b?	Resolved - Yes - Gelfond's theorem or the Gelfond–Schneider theorem).
8th	The Riemann hypothesis (the real part of any non-trivial zero of the Riemann zeta function is ½) and Goldbach's conjecture (every even number greater than 2 can be written as the sum of two prime numbers).	Open.⁴
9th	Find most general law of the reciprocity theorem in any algebraic number field.	Partially resolved.⁵
10th	Determination of the solvability of a Diophantine equation.	Resolved - Matiyasevich's theorem implies that this is impossible.
11th	Solving quadratic forms with algebraic numerical coefficients.	Resolved.
12th	Extend Kronecker's theorem on abelian extensions of the rational numbers to any base number field.	Open.
13th	Solve all 7-th degree equations using functions of two parameters.	Resolved.
14th	Proof of the finiteness of certain complete systems of functions.	Resolved
15th	Rigorous foundation of Schubert's enumerative calculus.	Resolved.
16th	Topology of algebraic curves and surfaces.	Open.
17th	Expression of definite rational function as quotient of sums of squares.	Resolved.
18th	Is there a non-regular, space-filling polyhedron? What is the densest sphere packing?	Resolved.⁶
19th	Are the solutions of Lagrangians always analytic?	Resolved.
20th	Do all variational problems with certain boundary conditions have solutions?	Resolved.
21st	Proof of the existence of linear differential equations having a prescribed monodromic group	Resolved.
22nd	Uniformization of analytic relations by means of automorphic functions.	Resolved.
23rd	Further development of the calculus of variations.	Resolved.

Notes

^1 Cohen's independence result, showing the continuum hypothesis to be independent of ZFC (Zermelo-Frankel set theory, extended to include the axiom of choice) is often cited to justify the assertion that the first problem has been solved. One contemporary view is that it may be the case that set theory should have additional axioms, capable of settling the problem.

^2 A matter of opinion, not shared by all. Gentzen's result shows rather precisely how much needs to be assumed to prove that Peano arithmetic is consistent. It is widely held that Gödel's second incompleteness theorem shows that there is no finitistic proof that PA is consistent (though Gödel himself disclaimed this inference [this needs a reference-- wvb]). The following is from Nagel and Newman, pp. 96 and 97: "This imposing result of Godel's analysis should not be misunderstood: it does not exclude a meta-mathematical proof of the consistency of arithmetic. What it excludes is a proof of consistency that can be mirrored by the formal deductions of arithmetic- Footnote 29.[Footnote 29 gives an example of the trisection of an angle-- it is possible, although not with straight-edge and compass]. Meta-mathematical proofs of the consistency of arithmetic have, in fact, been constructed, notably by Gerhard Gentzen, a member of the Hilbert school, in 1936, and by others since then-footnote 30" [Footnote 30: Describes Gentzen's proof, which uses transfinite induction; "30: "Gentzen's proof depends on arranging all the domonstrations of arithmetic in a linear order according to their degree of 'simplicity'...but Gentzen's argument cannot be mapped onto the formalism of artimetic. Moreover, although most students do not question the cogency of the proof, it is not finitistic in the sense of Hilbert's original stipulations for an absolute proof of consistency."[italics added]...."But these [meta-mathematical] proofs cannot be irepresented within the arithmetical calculus; and, since they are not finitistic, they do not achieve the proclaimed objectives of Hilbert's original program."

^3 According to Rowe & Gray (see reference below), most of the problems have been solved. Some were not completely defined, but enough progress has been made to consider them "solved"; Rowe & Gray lists the fourth problem as too vague to say whether it has been solved.

^4 Problem 8 contains two famous problems, both of which remain unsolved. The first of them, the Riemann hypothesis, is one of the seven Millennium Prize Problems, which were intended to be the "Hilbert Problems" of the 21st century.

^5 Problem 9 has been solved in the abelian case, by the development of class field theory; the non-abelian case remains unsolved, if one interprets that as meaning non-abelian class field theory.

^6 Rowe & Gray also list the 18th problem as "open" in their 2000 book, because the sphere-packing problem (also known as the Kepler conjecture) was unsolved, but a solution to it has now been claimed (see reference below).

Paul August ☎ 14:09, 9 February 2006 (UTC)

Comments

I prefer this to the up arrows, too. --- Charles Stewart^(talk) 16:00, 9 February 2006 (UTC)
Well, OK. The footnote thing is really very cramping, and it is hard to pack in all the necessary qualifying statements. Charles Matthews 16:21, 9 February 2006 (UTC)
I don't believe the MathML problem has been fixed with Paul-style footnotes. Every now and then I still see a page where the Wikipedia typewriter ball (or whatever that thing is supposed to be) jammed on top of the text. --Trovatore 16:41, 9 February 2006 (UTC)

Hmm … Trov, last I recall the problem occurred only if we had "use MathML if possible (experimental)" selected in "Math" preferences, and then only for certain files containing TeX and certain combinations of the "ent" templates. We had created a test page which produced the problem. However when I now look at that test page I no longer see a problem. Do you? Paul August ☎ 20:18, 9 February 2006 (UTC)

No. But it seems to be an intermittent bug. Sometimes I see the problem, then I don't see it on the same page minutes later. --Trovatore 23:57, 9 February 2006 (UTC)

2nd problem's back

Thanks to Paul's playing with footnotes, I've had another look at the 2nd footnote, and, naturally enough, found something else to object to. What sort of an objection to Gentzen's solution is it that the solution is not finitistic, when, from what I can gather (Richard Zach's PhD on Hilbert’s Finitism: Historical, Philosophical, and Metamathematical Perspectives (PDF)) that the earliest evidence of Hilbert's notion of finitism was only coming together around 1920 (his Logik-Kalkül lecture notes) and so can't possibly be what he meant in his address? --- Charles Stewart^(talk) 20:58, 9 February 2006 (UTC)

Nagel and Newman are published: as in "peer-reviewed-printed-ink-on-paper-and-not-on-the-web; if there are alternatives, show us the beef (web-references have a value of 0-). Places to start: Torkel Franzen Gödel's Theorem A.K. Peters, Wellesley Mass., 2005; and John W. Dawson Jr. Logical Dilemmas, The Life and Work of Kurt Gödel, A. K. Peters, Wellesley Mass, 1997. wvbaileyWvbailey 23:53, 26 April 2006 (UTC)

So I gather that in this connection Hilbert had some sort of idea of "absolute proof" and wanted a proof, full stop, that arithmetic was consistent, as opposed to a proof from assumptions. What that could mean I don't really know. Maybe he wanted to show that the consistency of arithmetic was a validity, or a logical necessity, or something like that? I'd just be speculating.

It seems to me that his ideas on finitistic proofs, even if they were not formulated until later, were at least his, and so give some indication that that may have been something like what he had in mind. This is quite different from the ZFC/CH case; since Hilbert had nothing to do with the formulation of ZFC, the case for a connection between Hilbert's 1st problem and independence of CH from ZFC seems pretty weak; the strongest position for someone taking the point of view that Cohen's result answers Hilbert's first is to say that Hilbert's "direct proof" would have had to be from much less than ZFC. (That would be somewhat analogous to saying that a positive answer to the second problem would be showing the consistency of arithmetic from logic alone.)

But maybe none of this matters very much. The signal contribution of the list of problems was the research that they stimulated; maybe the emphasis should be more on the results that came out of that research, and less on exegesis of Hilbert's intent. --Trovatore 06:24, 11 February 2006 (UTC)

That's at the very least an interesting point to go into. Charles Matthews 09:48, 11 February 2006 (UTC)

inconsistency with millenium prize problems page

This article claims that 8 and 12 are unsolved. The Millenium Prize Problems page says that only one of the problems is unsolved...Surely one of them should be changed... —The preceding unsigned comment was added by 128.135.226.69 (talk • contribs) 12:11, 27 March 2006 (UTC)

I'm guessing this page is correct and changing Clay Mathematics Institute. —Simetrical (talk • contribs) 02:17, 23 May 2006 (UTC)

Summary inconsistency

The "summary" section says that problem 23 is too vague to be considered solved. The table lists it as solved. This is inconsistent. Also, the summary lists problems 5, 15, 21, and 22 as having solutions with partial acceptance, but the table just lists them as "resolvd" with no qualifications. Ken Arromdee 19:07, 31 March 2006 (UTC)

Removed "P.O.V." from the references

Whoever you are, you just did a disservice to the readers. I'm sure you are aware of this, but just forgot: This form of bibliographic reference is called an annotated bibliography. And, you erased an annotation by none other than Martin Davis. Good show, dude.wvbaileyWvbailey 13:38, 12 September 2006 (UTC)

I removed some POV words and phrases from references added by User:Martin Davis and others. Whoever User:Martin Davis is in real life, he seems to be a new contributor to Wikipedia (judging from his contributions list), and so may not be aware of Wikipedia:Neutral point of view. Subjective terms such as "excellent", "wonderful" and "foolish" are unencyclopaedic, and are as out of place in references as they would be in the body of a Wikipedia article. To quote from Wikipedia:What Wikipedia is not, "Wikipedia is not a place to publish your own thoughts and analyses". Gandalf61 15:23, 12 September 2006 (UTC)

Herein lies the ultimate example of why distinguished academics (among others) avoid contributing to the warped and wonky world of wiki. We are wiki and we determine its policies and guidelines -- wiki-world aka the-work-in-progress needs some work. 'nuff said. wvbaileyWvbailey 16:46, 12 September 2006 (UTC)