Descriptive Set Theory and Infinitary Languages

John P. Burgess

Zbornik radova, Tome 2 (1977) no. 10, p. 9

Résumé

Kurepa trees, partitions, lensen's principles, large cardinals, and other notions from combinatorial set theory play an enormous role in the model theory of generalized-quantifier languages. (See e.g. [29].) Borel and analytic sets, Polish group actions, and notions from descriptive set theory can play almost as large a role in the model theory of certain infinitary languages. (See [31] and [32].) The present paper is a study, by the methods of descriptive set theory, of the class of strong first-order languages. These, roughly, are the infinitary languages which are strong enough to express wellfoundedness, at least over countable structures, yet weak enough that the satisfaction relation is AI-definable. Examples, culled from the literature of exotic model theory, are present in § 1. The set- theoretic machinery for their study is set up in §§ 2-4. §§ 5 and 6 are devoted to an exposition of the properties shared by all strong first-order languages. Most notably: There is a quasiconstructive complete proof procedure involving rules with $N_1$ premisses for any strong first-order language, and even the weak version of Beth's Definability Theorem fails for every such language. Many of the results in this paper date from the author's days as a student in R.L. Vaught's seminar at Berkeley, 1972-73. At that time I had the benefit of correspondence with Profs. Barwise and Moschovakis, and especially of frequent discussions with Prof. Vaught and D. E. Miller. Most of this work was included in [6], and a few items have appeared in print ([5]; [8], § 2). More recent discussions with Miller led to the discovery of the proof procedure and the counterexample to Beth's Theorem alluded to above, and to the writing of this paper.