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In his famous "23 problems" speech, Hilbert gave his second problem as follows:

The axioms of arithmetic are essentially nothing else than the known rules of calculation, with the addition of the axiom of continuity. I recently collected them and in so doing replaced the axiom of continuity by two simpler axioms, namely, the well-known axiom of Archimedes, and a new axiom essentially as follows: that numbers form a system of things which is capable of no further extension, as long as all the other axioms hold (axiom of completeness). I am convinced that it must be possible to find a direct proof for the compatibility of the arithmetical axioms, by means of a careful study and suitable modification of the known methods of reasoning in the theory of irrational numbers.

Now, I'm not sure what he's referring to in the "recently" but it might be his paper "On the concept of number" published also at 1900. In this paper Hilbert gives an axiomatic system for the real numbers (with order).

Now, I vaguely (very vaguely...) know that the theory for real closed fields behaves quite differently from the theory of arithmetic for the natural numbers - the former is not subject to Gödel's incompleteness theorem while the latter, of course, is. So it seems to be (and I might be wrong) that if Hilbert meant something in the spirit of the former, than Gödel has not answered Hilbert's second problem at all (of course, there is not denying the importance of Gödel's results to Hilbert's program; I am interested here only in Hilbert's 2nd problem).

Now, everywhere I look it seems that Hilbert's 2nd problem is interpreted as a question of the axioms of the arithmetic of natural numbers, e.g. Peano arithmetic. For example, Wikipedia states that

It is now common to interpret Hilbert's second question as asking for a proof that Peano arithmetic is consistent (Franzen 2005:p. 39).

(I looked at Frenzen' book; I have to admit I didn't see anything that sounded like the above quotation there but I might have simply missed).

So, what was Hilbert's 2nd problem about? Is it correct to interpret it as a question about Peano arithmetic? Is it correct to claim that Gödel's theorem had a major impact on the question? Or is it a confusion between Hilbert's program and the 2nd question?

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1 Answer

The universal understanding is that a positive solution to Hilbert's second problem requires a convincing proof of the the consistency of some adequate set of axioms for the natural numbers. The history of the problem is laid out in the Stanford Encyclopedia entry on Hilbert's program, section 1.1. That explanation seems to resolve the issue of what Hilbert was referring to in the quote in the question above:

Hilbert provided such an axiomatization in (1900b), but it became clear very quickly that the consistency of analysis faced significant difficulties, ... . Hilbert thus realized that a direct consistency proof of analysis, i.e., one not based on reduction to another theory, was needed. He proposed the problem of finding such a proof as the second of his 23 mathematical problems in his address to the International Congress of Mathematicians in 1900 (1900a) and presented a sketch of such a proof in his Heidelberg talk (1905).

Note that the term "analysis" in that article is the traditional term for the theory of natural numbers and sets of natural numbers, which is now called second-order arithmetic. In the first decades of the 20th century, the study of formal logic and model theory was still in its infancy, and many basic facts which we now take for granted were not known to researchers in that era. In particular, Hilbert would have had no reason to expect that the theory of the real numbers as a field would behave differently from the theory of second order arithmetic. Tarski's work on the decision procedure for real closed fields came much later, in 1951.

Hilbert's work on his second problem led to the development of Hilbert's program, which sought to give a "finitistic" consistency proof of arithmetic. This was later shown to be impossible by the incompleteness theorems.

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This is a good answer, but I am still not convinced that Hilbert's 2nd problem is about natural numbers, as opposed to real numbers, and I actually disagree with what the Encyclopedia says, based on my reading of 1900a where Hilbert doesn't say anything whatsoever about reduction to another theory or the need to talk about natural numbers. –  Gadi A Oct 25 '11 at 13:05
It seems remarkable that somehow the entire mathematical logic community could have misunderstood Hilbert's intentions for the last 111 years. –  Carl Mummert Oct 25 '11 at 13:35
Not quite; it's obvious that one of Hilbert's intentions WAS to prove the consistency of Peano arithmetic; however, this is not my question. My question is more about a specific historical detail, and Mathematics has many similar small errors (see "Pell's" equation, "Burnside's" lemma etc). –  Gadi A Oct 25 '11 at 16:32
@Gadi A: I think it would be better if you quoted whole section. I think treating PA as a first-order theory is a later event, at the time people would have referred to the original Peano axioms which is a second order theory (sometimes called $PA^2$ these days). So he is basically asking for a consistency of $PA^2$. Also remember that at that time he was alive and many famous logicians were also, including Bernays. The possibility of misunderstanding the question is very unlikely. –  Kaveh Nov 2 '11 at 14:17
As I said, it seems to me he was referring to "his" axiom system for the real numbers, not to a first or second order theory of natural numbers. –  Gadi A Nov 8 '11 at 10:03
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