H. Jerome Keisler

Fundamentals of Model Theory

Publisher Summary Model theory may be described as the union of logic and universal algebra. This chapter presents some of the methods of both western and eastern model theory. Many notions and results have two distinct versions—a western version dealing with arbitrary formulas and an eastern version dealing with quantifier-free formulas. The deeper proofs in model theory usually depend on the construction of a model with certain properties. The constructions almost always use elementary chains, diagrams and other expansions of the language, compactness theorem, downward Lowenheim–Skolem theorem, omitting type theorem, forcing, ultraproducts, and homogeneous sets. The chapter concentrates on the simplest nontrivial language, first order logic. For some applications, many-sorted logic is more natural.

GOOD IDEALS IN FIELDS OF SETS

In this paper we shall study some problems which are purely settheoretical in character, but which arose in connection with a problem in the theory of models. The principal results of this paper form an important part of the proofs of the series of results in the theory of models which are announced in [1], [2], [3], and in the appendix of [4]. The present article, however, is entirely free of any notions from the theory of models. It is well-known that, for every infinite cardinal a, there exist nonprincipal prime ideals in the field S(X) of all subsets of a set X of power a, and furthermore there are exactly 221 such prime ideals. The object of this paper is to show that non-principal prime ideals with certain additional properties exist. Let X be a set of power a, and let S( Y) be the set of all non-empty finite subsets of a set Y. If f, g map S.( Y) into the field of subsets of X, we write g > f if, for all a E S.( Y), g(a) -f(a). Let us say that an ideal I in the field S(X) is SJ( Y)-good if, for every monotonic function f on S,(Y) into I, there exists an additive function g > f on S,( Y) into I. If Y is countable, then every ideal I is S.( Y)-good (cf. Corollary 4.2). Our two main results are as follows.

Ultraproducts which are not Saturated

In this paper we continue our study, begun in [5], of the connection between ultraproducts and saturated structures. If D is an ultrafilter over a set I , and is a structure (i.e., a model for a first order predicate logic ℒ ), the ultrapower of modulo D is denoted by D-prod . The ultrapower is important because it is a method of constructing structures which are elementarily equivalent to a given structure (see Frayne-Morel-Scott [3]). Our ultimate aim is to find out what kinds of structure are ultrapowers of . We made a beginning in [5] by proving that, assuming the generalized continuum hypothesis (GCH), for each cardinal α there is an ultrafilter D over a set of power α such that for all structures , D-prod is α + -saturated.

Theory of Models with Generalized Atomic Formulas

We shall prove the following theorem, which gives a necessary and sufficient condition for an elementary class to be characterized by a set of sentences having a prescribed number of alternations of quantifiers. A finite sequence of relational systems is said to be a sandwich of order n if each is an elementary extension of ( i ≦ n—2 ), and each is an extension of ( i ≦ n—2 ). If K is an elementary class, then the statements (i) and (ii) are equivalent for each fixed natural number n .

On the strength of nonstandard analysis

It is often asserted in the literature that any theorem which can be proved using nonstandard analysis can also be proved without it. The purpose of this paper is to show that this assertion is wrong, and in fact there are theorems which can be proved with nonstandard analysis but cannot be proved without it. There is currently a great deal of confusion among mathematicians because the above assertion can be interpreted in two different ways. First, there is the following correct statement: any theorem which can be proved using nonstandard analysis can be proved in Zermelo-Fraenkel set theory with choice, ZFC, and thus is acceptable by contemporary standards as a theorem in mathematics. Second, there is the erroneous conclusion drawn by skeptics: any theorem which can be proved using nonstandard analysis can be proved without it, and thus there is no need for nonstandard analysis. The reason for this confusion is that the set of principles which are accepted by current mathematics, namely ZFC, is much stronger than the set of principles which are actually used in mathematical practice. It has been observed (see [F] and [S]) that almost all results in classical mathematics use methods available in second order arithmetic with appropriate comprehension and choice axiom schemes.

Elementary extensions of models of set theory

Model Theoretic methods are used to extend models of set theory while leaving specified sets fixed. In particular, every countable modelU of ZF has: (i) an extension leaving every set inU fixed, and (ii) for each (inU) regular cardinala an extension enlarginga but leaving each cardinal less thana fixed.

The Hyperreal Line

The aim of this article is to explain that the hyperreal line is, what it looks like, and what it is good for. Near the beginning of the article we shall draw pictures of the hyperreal line and sketch its construction as an ultrapower of the real line. In the middle part of the article, we shall survey mathematical results about the structure of the hyperreal line. Near the end, we shall discuss philosophical issues concerning the nature and significance of the hyperreal line.

On the number of homogeneous models of a given power

It is shown that, For each complete theoryT, the nomberhT(m) of homogeneous models ofT of powerm is a non-increasing function of uncountabel cardinalsm Moreover, ifhT(ℵ0)≦ℵ0, then the functionhT is also non-increasing ℵ0 to ℵ1.

On cardinalities of ultraproducts

Introduction. In the theory of models, the ultraproduct (or prime reduced product) construction has been a very useful method of forming models with given properties (see, for instance, [2]). I t is natural to ask what the cardinality of an ultraproduct is when we are given the cardinalities of the factors. In this paper we obtain some new results in that direction; however, the questions stated explicitly in [2, p. 208], are still open. Let us first mention briefly some of the known results. Throughout this note we shall let D be a nonprincipal ultrafilter over a set / of infinite power X. Additional notation is explained in §1 below. 1. a^at/D^a* [2, p. 205]. 2. If D is not countably complete, then H i e r cti/D is either finite or of power at least 2 [2, p. 208]. 3. If D is uniform, then \/D>\; moreover, (2<>)7# = 2 \ where 2 ( X ) -E^<x2^ [2, p. 206]. 4. There exists a D such that if a is infinite, then a/D~a [2, p. 207], [l, p. 399], and [3, p. 838], (Two more general versions for products of cardinals are given in [l].) We shall prove the following results.

Measures and forking

Abstract Shelahs theory of forking (or stability theory) is generalized in a way which deals with measures instead of complete types. This allows us to extend the method of forking from the class of stable theories to the larger class of theories which do not have the independence property. When restricted to the special case of stable theories, this paper reduces to a reformulation of the classical approach. However, it goes beyond the classical approach in the case of unstable theories. Methods from ordinary forking theory and the Loeb measure construction from nonstandard analysis are used.