We connect and solve two longstanding open problems in quite different areas: the model-theoretic question of whether SOP2 is maximal in Keisler’s order, and the question from general topology/set… (More)

Cantor proved in 1874 [Cantor G (1874) J Reine Angew Math 77:258-262] that the continuum is uncountable, and Hilbert's first problem asks whether it is the smallest uncountable cardinal. A program… (More)

In the first edition of Classification Theory, the second author characterized the stable theories in terms of saturation of ultrapowers. Prior to this theorem, stability had already been defined in… (More)

The first part of this paper is an expository overview of the au-thors' recent work on Keisler's order, a far-reaching program of understanding basic model-theoretic structure through the lens of… (More)

We prove, in ZFC, that there is an infinite strictly descending chain of classes of theories in Keisler’s order. Thus Keisler’s order is infinite and not a well order. Moreover, this chain occurs… (More)

Chudnovsky, Kim, Oum, and Seymour recently established that any prime graph contains one of a short list of induced prime subgraphs [1]. In the present paper we reprove their theorem using many of… (More)

We prove a regularity lemma with respect to arbitrary Keisler measures μ on V , ν on W where the bipartite graph (V,W,R) is definable in a saturated structure M̄ and the formula R(x, y) is stable.… (More)

Our investigations are framed by two overlapping problems: finding the right axiomatic framework for so-called cofinality spectrum problems, and a 1985 question of Dow on the conjecturally nonempty… (More)

We show that every approximately differentially private learning algorithm (possibly improper) for a class H with Littlestone dimension d requires Ω ( log∗(d) ) examples. As a corollary it follows… (More)

Historically one of the great successes of model theory has been Shelah’s stability theory: a program, described in [17], of showing that the arrangement of first-order theories into complexity… (More)