Cofinality spectrum theorems in model theory, set theory and general topology

By M. Malliaris, Saharon Shelah. 2012

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We connect and solve two longstanding open problems in quite different areas: the modeltheoretic question of whether SOP2 is maximal in Keisler’s order, and the question from general topology/set theory of whether p = t, the oldest problem on cardinal invariants of the continuum. We do so by showing these problems can be translated into instances of a more fundamental problem which we state and solve completely, using model-theoretic methods. By a cofinality spectrum problem s we essentially mean the data of a pair of models M M1 which code sufficient set theory, possibly in an expanded language, along with a distinguished set of formulas ∆s which define linear orders in M1. Let ts, the “treetops” of s, be the smallest regular cardinal λ such that one of a set of derived trees in M1 has a strictly increasing λ-sequence with no upper bound. Let C(s, ts) be the set of pairs of regular cardinals (κ1, κ2) such that κ1 ≤ κ2 λ,” or what is equivalent, such that (ω,<)λ/D contains no (κ, κ)-cuts for κ = cf(κ) ≤ λ, is λ+-good. We obtain several consequences, notably existence of a minimum Keisler class among the non-simple theories.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b5ec942206fd50c5f3c37fca670e62934d447562