Hanf numbers and presentation theorems in AECs

Abstract

This paper addresses a number of fundamental problems in logic and the philosophy of mathematics by considering some more technical problems in model theory and set theory. The interplay between syntax and semantics is usually considered the hallmark of model theory. At first sight, Shelah's notion of abstract elementary class shatters that icon. As in the beginnings of the modern theory of structures ([Cor92]) Shelah studies certain classes of models and relations among them, providing an axiomatization in the Bourbaki ([Bou50]) as opposed to the Gödel or Tarski sense: mathematical requirements, not sentences in a formal language. This formalism-free approach ([Ken13]) was designed to circumvent confusion arising from the syntactical schemes of infinitary logic; if a logic is closed under infinite conjunctions, what is the sense of studying types? However, Shelah's presentation theorem and more strongly Boney's use [Bon] of aec's as theories of L κ,ω (for κ strongly compact) reintroduce syntactical arguments. The issues addressed in this paper trace to the failure of infinitary logics to satisfy the upward Löwenheim-Skolem theorem or more specifically the compactness theorem. The compactness theorem allows such basic algebraic notions as amalgamation and joint embedding to be easily encoded in first order logic. Thus, all complete first order theories have amalgamation and joint embedding in all cardinalities. In contrast *

Hanf numbers for Extendibility and related phenomena

“…All the exotica mentioned here and described in more detail in [BB17] occurs below ω1 . Baldwin and Boney [BB17] have shown that the Hanf number for amalgamation is no more than the first strongly compact cardinal. This immense gap motivated the current paper.…”

“…Further results by [BKS09, KLH16,BKS16], where the hypothesis are weakened to allow incomplete sentences of L ω1,ω or even AEC (Abstract Elementary Classes (K, ≤) where the properties of strong substructure, ≤ are defined axiomatically) are placed in context in [BB17]. Analogous results were proved earlier for incomplete sentences by [BKS16] who code certain bipartite graphs in way that determine specific inequalities between the cardinalities of the two parts of the graph; in this case all models have cardinality less than ω1 .…”

Accessible categories, set theory, and model theory: an invitation

“…Large cardinals imply a simple answer (earlier results used model-theoretic techniques, see e.g. [BB17]):…”

Kurepa trees and spectra of $\mathcal{L}_{ω_1,ω}$-sentences

“…This cardinal µ(K) is called the Hanf number for amalgamation. Baldwin and Boney in [20] proved the existence of a Hanf number for amalgamation, but their definition of a Hanf number is different than the one that Grossberg conjectured. Their result states that if µ is a strongly compact cardinal, K is an AEC with LS(K) < µ, and K satisfies amalgamation cofinally below µ, then K satisfies amalgamation in all cardinals ≥ µ.…”