We are interested in the question of how much the order of a non-standard model of PA can determine the model. In particular, for a model M , we want to characterize the complete types p(x, y) of non-standard elements (a, b) such that the linear orders {x : x < a} and {x : x < b} are necessarily isomorphic. It is proved that this set includes the complete types p(x, y) such that if the pair (a,...

The notion of a strongly determined type over A extending p is introduced , where p 2 S(A). A strongly determined extension of p over A assigns, for any model M A, a type q 2 S(M) extending p such that, if c realises q, then any elementary partial map M ! M which xes acl eq (A) pointwise is elementary over c. Examples are provided of many theories with strongly determined types, and some withou...

For a cardinal κ and a model M of cardinality κ let No(M) denote the number of non-isomorphic models of cardinality κ which are L∞,κequivalent to M . We prove that for κ a weakly compact cardinal, the question of the possible values of No(M) for models M of cardinality κ is equivalent to the question of the possible numbers of equivalence classes of equivalence relations which are Σ1-definable ...

Certain basic concepts of geometrical stability theory are generalized to a class of closure operators containing algebraic closure. A specific case of a generalized closure operator is developed which is relevant to Vaught’s conjecture. As an application of the methods, we prove Theorem A. Let G be a superstable group of U− rank ω such that the generics of G are locally modular and Th(G) has f...

Suppose λ is a singular cardinal of uncountable cofinality κ. For a model M of cardinality λ, let No(M) denote the number of isomorphism types of models N of cardinality λ which are L∞λ-equivalent to M. In [She85] Shelah considered inverse κ-systems A of abelian groups and their certain kind of quotient limits Gr(A)/Fact(A). In particular Shelah proved in [She85, Fact 3.10] that for every cardi...

The Zeaman-House-Lovejoy (ZHL) model for discrimination learning involves interacting perceptual and response learning processes, with rate parameters v and 0, respectively. This paper describes a number of predictions of the ZHL model for the optional shift paradigm. Kendler and Kendler showed that the proportion of subjects who learn an optional shift as a reversal increases with subject’s ag...

Jensen formulates classes of notions of forcing in [1]. We take subproper among those. This class is wider than proper, may differ from semiproper of Shelah and is iterable under the revised countable support a la Donder according to [1]. We formulate a class of notion of forcing whose definition involves less parameters than Jensen’s subproper and show that it iterates under a type of revised ...

Proof. If T were not complete, there would a sentence ψ such that neither ψ and ¬ψ would follow from T . But then T ∪ {ψ} and T ∪ {¬ψ} would have infinite models. Since λ ≥ |L|, both would actually have models of cardinality λ by the theorems of Skolem and Löwenheim. But these cannot be isomorphic, because they are not elementarily equivalent, contradicting the λ-categoricity of T . Theorem 1.2...