1
$\begingroup$

Why was there an approximately thirty year gap between the discovery of the compactness theorem (for countable theories) in 1930 and Robinson's elucidation of nonstandard analysis in the early 1960s? Perhaps my question is naive, but my reasoning is thus: Once one has the compactness theorem, the existence of nonstandard models of $\mathbb{N}$ and $\mathbb{R}$ follows straightforwardly: Add a new constant symbol $c$ and an axiom $c > n$ for each natural number $n$ (the full details can be found in chapter 1 of Richard Kaye's book Models of Peano Arithmetic). Building up a thorough, polished theory of NSA once the existence of nonstandard models of the reals has been established is of course hardly a trivial task, but I wonder why such a project wasn't already undertaken in the 1930s.

(Note this tangentially relevant question.)

$\endgroup$
2
  • 2
    $\begingroup$ Why is there a 2,000+ year gap between Euclid's lemma and Gauss's prime factorization theorem? After all, the latter "follows straightforwardly" from the former, as presented in modern textbooks. History does not work the textbook way, especially when conceptual leaps in abstraction are involved. And in this case you needed someone with mastery of recent developments in mathematical logic and interest in history of "discredited" infinitesimals on top of it. $\endgroup$ – Conifold Apr 6 at 20:35
  • 1
    $\begingroup$ It seems we have a lot of such questions here. Some advance seems obvious, why wasn't it made earlier? I think the answers to all these questions are similar: it was not obvious ahead of time. $\endgroup$ – Gerald Edgar Apr 6 at 21:33
5
$\begingroup$

The understanding of the "obvious" corollary of Gödel Completeness Theorem now called Compactness Theorem and its consequences was a "slow" process.

See John Dawson, The Compactness of First-order Logic From Gödel to Lindström (HPL,1993), page 18:

After 1934, [...] the only person who seems to have recognized the importance of compactness was the Russian A. I. Maltsev. Beginning in 1936, he published 'a seminal run' of papers in what would now be called model-theoretic algebra, papers in which he 'obtained applications [of logic] to group theory of greater technical virtuosity than the possibly more basic applications to algebra later on found [...] by Henkin and Abraham Robinson' (Sabbagh 1991). Three of those papers (1936,1940 and 1941) are of interest here.

In his first published work, written in German, Maltsev 1936 [Maltsev, A.I. 1936 'Untersuchungen aus dem Gebiete der mathematischen Logik', Matematicheskii Sbornik, n.s., 1, 323-336.] devoted his efforts to generalizing two theorems, one for the propositional calculus and the other for the restricted functional calculus. The theorems in question were Godel's compactness theorem and Skolem's result that no denumerable set of formulas of first-order logic can completely characterize the structure of the natural numbers.

See English translation into: A.I. Mal'cev, The Metamathematics of Algebraic Systems: Collected Papers 1936-1967 (North Holland, 1971), page 1:

This article is devoted to generalizing two theorems, one for propositional calculus (PC) and the other for first-order predicate logic (FOPL). The first theorem is due to Gödel [K. Gödel, Die Vollsthdigkeit der Axiome des logischen Funktionenkalkuls (1936),] and can be formulated as follows:

For any countable system of formulas of PC to be consistent, it is sufficient that every finite part of the system be consistent.

Finally, see also On a General Method for Obtaining Local Theorems in Group Theory by A. Mal'cév, Review by Leon Henkin and Andrzej Mostowski (Jsl, 1959):

Historical note. The formulation and proof of the "general local theorem" [If every finite subset of a given (possibly non-denumerable) set of first-order sentences is satisfiable, then so is the whole set] for denumerable sets of first-order sentences is of course due to Gödel in 1930. The corresponding result for non-denumerable sets of formulas of propositional calculus was given by Mal'cev in 1936.

See Abraham Robinson, Nonstandard arithmetic (1967):

"In 1934 it was pointed out by Thoralf Skolem that there exist proper extensions of the natural number system which have, in some sense, "the same properties" as the natural numbers. [...] he did not concern himself further with the properties of the structures whose existence he had established. n due course these and similar structures became known as nonstandard models of arithmetic and papers concerned with them appeared in the literature (e.g. Henkin, Kemeny, Mendelson, Robinson). Beginning in the fall of 1960, the application of similar ideas to analysis led to a rapid development in which nonstandard models of arithmetic played an auxiliary but vital part."

IMO it is important to observe that in Abraham Robinson, On the Metamathematics of Algebra (1951) there is no Compactness Th, while in the revised and enlarged edition: Abraham Robinson, Introduction to Model Theory and the Metamathematics of Algebra (1963) the theorem is there (page 21).

What happened in between?: the "slow" process described above that occurred during the 50s and 60s.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.