Some Prime Elements in the Lattice of Interpretability Types

A general theorem is proved which implies that the types of PA (Peano Arithmetic), ZF (Zermelo-Fraenkel Set Theory) and GB (Gödel-Bernays Set Theory) are prime in the lattice of interpretability types. 0. Introduction. One of the important goals of mathematical logic is to investigate the strength of theories. A good approximation to the intuitive concept of strength is the quasiordering "7 is interpretable in S". We shall consider interpretation in a very general sense which was introduced by Mycielski in [5]. Mycielski has shown (among other things) that after canonical factorization of that quasiordering a distributive lattice is obtained. The elements of the lattice will be called types; thus every theory determines a type—the type to which it belongs. An important task is to determine the prime elements of the lattice. (An element is prime or join-irreducible if it is not the join of two smaller elements.) Following Mycielski we shall call a theory connected if its type is prime. We introduce the concept of sequential theory; roughly speaking a theory is sequential iff it permits some coding of any finite sequence of elements. The main theorem of this paper (Theorem 4.2) says that every sequential theory is connected. This answers a question of [1] since PA, ZF, GB and Th(w; +, • ) are sequential theories. (Actually we arrived at the concept of sequential theories when trying to generalize a former proof of connectedness for these theories.) In §§2 and 3 we prove some lemmas and theorems about sequential theories which are prerequisites for the proof of the main theorem. Though we prove only a little bit more general statements than we need for the main theorem, they already show that interesting mathematics can be developed in every sequential theory. §4 is devoted to the rest of the proof of the main theorem. We use the sufficient condition for being connected found by Mycielski in [1]. Related problems are discussed in the last section. Using our theorem we shall show that if equality is treated as a logical symbol then the lattices of interpretability types are different from Mycielski's lattice and that a concept of sequentiality introduced by Vaught [10] cannot be used for generalizing our theorem. I am grateful to P. Hájek, J. Krajicek, J. Paris, A. Wilkie and especially to J. Mycielski for helpful discussions and suggestions. Received by the editors November 26, 1980 and, in revised form, September 10, 1982. 1980 Mathematics Subject Classification. Primary 03B10; Secondary 03H15. ©1983 American Mathematical Society 0002-9947/83 $1.00 + $.25 per page 255 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use