H. Jerome Keisler’s research while affiliated with University of Michigan and other places

We prove analogues of the Craig interpolation theorem for the continuous model theory of metric structures.

In this paper, we provide an epistemic characterization of iterated admissibility (IA), i.e., iterated elimination of weakly dominated strategies. We show that rationality and common assumption of rationality (RCAR) in complete lexicographic type structures implies IA, and that there exist such structures in which RCAR can be satisfied. Our result is unexpected in light of a negative result in Brandenburger, Friedenberg, and Keisler (2008) (BFK) that shows the impossibility of RCAR in complete continuous structures. We also show that every complete structure with RCAR has the same types and beliefs as some complete continuous structure. This enables us to reconcile and interpret the difference between our results and BFK’s. Finally, we extend BFK’s framework to obtain a single structure that contains a complete structure with an RCAR state for every game. This gives a game-independent epistemic condition for IA.

Our map from full-support LPS's to CPS's takes LPS's to the set of CPS's with a fixed algebra of clopen conditions. But to obtain a map from the set of CPS's onto the set of LPS's, we must allow the algebra of conditions to vary. When CPS's are used in game theory, there is usually a natural fixed family of conditions—corresponding to the events that a player can observe in the play of the game. These notes, then, are best viewed as an exercise in probability theory: we examine the relationship between CPS's and LPS's in a general, non-game setting. Let Ω be a Polish space, and let A be the Borel σ-algebra on Ω .L etB denote a finite subalgebra of A. Since the elements of B will be used as conditions, we will adopt the convention of removing the empty set from B.T hat is,B is the set of nonempty elements of a finite subalgebra. Definition 1 A conditional probability system (CPS )o n(Ω,A,B) is a map p : A×B→ (0,1) such that: a. for all B ∈ B, p(B|B )=1 ; b. for all B ∈ B, p(·|B) is a probability measure on (Ω,A); c. for all A ∈ A and B,C ∈ B ,i fA ⊆ B ⊆ C then p(A|C )= p(A|B)p(B|C). Let CB be the set of all CPS's on (Ω,A,B) ,a nd letC be the set of all p such that p is a CPS on

In 1967 the author introduced a pre-ordering of all first order complete theories where T is lower than U if it is easier for an ultrapower of a model of T than an ultrapower of a model of U to be saturated. In a long series of recent papers, Malliaris and Shelah showed that this pre-ordering is very rich and gives a useful way of classifying simple theories. In this paper we investigate the analogous pre-ordering in continuous model theory.

Epistemic justifications of solution concepts often refer to type structures that are sufficiently rich. One important notion of richness is that of a complete type structure, i.e., a type structure that induces all possible beliefs about types. For instance, it is often said that, in a complete type structure, the set of strategies consistent with rationality and common belief of rationality are the set of strategies that survive iterated dominance. This paper shows that this classic result is false, absent certain topological conditions on the type structure. In particular, it provides an example of a finite game and a complete type structure in which there is no state consistent with rationality and common belief of rationality. This arises because the complete type structure does not induce all hierarchies of beliefs—despite inducing all beliefs about types. This raises the question: Which beliefs does a complete type structure induce? We provide several positive results that speak to that question. However, we also show that, within ZFC, one cannot show that a complete structure induces all second-order beliefs.

Answering a question of Cifú Lopes, we give a syntactic characterization of those continuous sentences that are preserved under reduced products of metric structures. In fact, we settle this question in the wider context of general structures as introduced by the second author.

In the paper Randomizations of Scattered Sentences, Keisler showed that if Martin’s axiom for aleph one holds, then every scattered sentence has few separable randomizations, and asked whether the conclusion could be proved in ZFC alone. We show here that the answer is “yes”. It follows that the absolute Vaught conjecture holds if and only if every Lω1ω$L_{\omega _1\omega }$-sentence with few separable randomizations has countably many countable models.