John T. Baldwin

On strongly minimal sets

The purpose of this paper is twofold. In §1 and §2 which are largely expository we develop the known theory of ℵ 1 -categoricity in terms of strongly minimal sets. In §3 we settle affirmatively Vaughts conjecture that a complete ℵ 1 -categorical theory has either just one or just ℵ 0 countable models, and in §4 we present an example which serves to illustrate the ideas of §3. As far as we know the only work published on strongly minimal sets is that of Marsh [3]. The present exposition goes beyond [3] in showing that any ℵ-categorical theory has a principal extension in which some formula is strongly minimal.

Logical stability in group theory

This paper investigates the logical stability of various groups. Theorem 1: If a group G is stable and locally nilpotent then it is solvable. Theorem 2: Every non-Abelian variety of groups is unstable.

Stable generic structures

Abstract Hrushovski originated the study of “flat” stable structures in constructing a new strongly minimal set and a stable ℵ 0 -categorical pseudoplane. We exhibit a set of axioms which for collections of finite structure with dimension function δ give rise to stable generic models. In addition to the Hrushovski examples, this formalization includes Baldwins almost strongly minimal non-Desarguesian projective plane and several others. We develop the new case where finite sets may have infinite closures with respect to the dimension function δ. In particular, the generic structure need not be ω-saturated and so the argument for stability is significantly more complicated. We further show that these structures are “flat” and do not interpret a group.

Stability theory, permutations of indiscernibles, and embedded finite models

We show that the expressive power of first-order logic over finite models embedded in a model M is determined by stability-theoretic properties of M . In particular, we show that if M is stable, then every class of finite structures that can be defined by embedding the structures in M , can be defined in pure first-order logic. We also show that if M does not have the independence property, then any class of finite structures that can be defined by embedding the structures in M , can be defined in first-order logic over a dense linear order. This extends known results on the definability of classes of finite structures and ordered finite structures in the setting of embedded finite models. These results depend on several results in infinite model theory. Let I be a set of indiscernibles in a model M and suppose (M, I) is elementarily equivalent to (M1, I1) where M1 is |I1|-saturated. If M is stable and (M, I) is saturated then every permutation of I extends to an automorphism of M and the theory of (M, I) is stable. Let I be a sequence of <-indiscernibles in a model M , which does not have the independence property, and suppose (M, I) is elementarily equivalent to (M1, I1) where (I1, <) is a complete dense linear order and M1 is |I1|-saturated. Then (M, I)-types over I are order-definable and if (M, I) is א1-saturated every order preserving permutation of I can be extended to a back-and-forth system.

Second-order quantifiers and the complexity of theories.

Introduction. Interpretations et codage. Decompositions de modeles. Decompositions dans les theories stables. Theories pur profondes. Quelques prototypes. Theories instables. Problemes

Counting models in universal Horn classes

DefinenK(λ) to be either ω, or the number of non-isomorphic models inK having cardinality α, whichever cardinal is larger. This paper contains a proof that for a congruence modular variety ⋎ of algebras of countable similarity type, there are only six possible functionsn⋎. It is also proved that ifnK(λ)≠2λ for some λ, andK is a universal Horn class of models for a countable language, thenK must satisfy two conditions, one of which is quite restrictive and requires that the members ofK are all in a certain sense Abelian.

ℵ0-Categoricity and stability of rings

This paper applies various techniques developed in logic to the study of associative rings (not necessarily commutative or with identity). A first-order formula of ring theory is a formula built up in the natural manner using only the logical connectives A (and), v (or), + (implies), 3, V (quantifiers over elements of the ring), the ring theoretic function symbols +, f, 0 and the variables V, , vi ,..., V, (cf. [16]). The theory of a ring R, Th(R), is the set of all first-order sentences (formulas with no free variables) which are true in R. We say R is &categorical if, up to isomorphism, Th(R) has at most one countably infinite model. An introduction to &,-categoric&y in an algebraic setting occurs in [15]. Note that any finite ring is X,-categorical. We have tried to make this paper accessible to anyone familiar with the Wedderburn-Artin structure theory for rings and the rudiments of model theory. Much of our work considers the relationship between the ascending or descending chain conditions usually studied in ring theory and certain ostensibly weaker notions which we define below. We say that the (left) ideal I of a ring R is strictly definable if there is a formula T(Q) of ring theory (with no constants other than 0) such that I is the set of members of R which satisfy v. It is an easy consequence of N,-categoricity that an N,,-categorical ring can have no infinite ascending or descending chain of strictly definable left ideals. In contrast, we show that there is no infinite &,-categorical ring which satisfies either the ascending or descending chain condition on (left) ideals. Thus, Qcategoricity

N⊥ as an abstract elementary class

Abstract In this paper we study abstract elementary classes of modules. We give several characterizations of when the class of modules A with Ext i ( A , N ) = 0 (for fixed N and all i ) is abstract elementary class with respect to the notion that M 1 is a strong submodel M 2 if the quotient remains in the given class.

FORMALIZATION, PRIMITIVE CONCEPTS, AND PURITY

We emphasize the role of the choice of vocabulary in formalization of a mathematical area and remark that this is a particular preoccupation of logicians. We use this framework to discuss Kennedy’s notion of ‘formalism freeness’ in the context of various schools in model theory. Then we clarify some of the mathematical issues in recent discussions of purity in the proof of the Desargues proposition. We note that the conclusion of ‘spatial content’ from the Desargues proposition involves arguments which are algebraic and even metamathematical. Hilbert showed that the Desargues proposition implies the coordinatizing ring is associative, which in turn implies the existence of a three-dimensional geometry in which the given plane can be embedded. With W. Howard we give a new proof, removing Hilbert’s ‘detour’ through algebra, of the ‘geometric’ embedding theorem. Finally, our investigation of purity leads to the conclusion that even the introduction of explicit definitions in a proof can violate purity. We argue that although both involve explicit definition, our proof of the embedding theorem is pure while Hilbert’s is not. Thus the determination of whether an argument is pure turns on the content of the particular proof. Moreover, formalizing the situation does not provide a tool for characterizing purity.

On the classifiability of cellular automata

Abstract Based on computer simulations Wolfram presented in several papers conjectured classifications of cellular automata into four types. We show a natural formalization of his rate of growth suggestion does not provide a classification (even probabilistically) of all cellular automata: For any rational p,q,0⩽p,q⩽=1 with p+q=1 , there is a cellular automata A p,q which has probability p of being in class 3, probability q of being in class 4. We also construct an automata which grows monotonically at rate log t , rather than at a constant rate.