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Cofinality quantifiers in Abstract Elementary Classes and beyond

Will Boney

TL;DR

Develops a framework to convert classes axiomatized by cofinality quantifiers into Abstract Elementary Classes by enforcing positive, deliberate use and applying a finitary abstract Skolemization. It proves that for theories $T$ in $L(Q^{cof}_C)(\tau)$, the class $K_T^+$ is an AEC with LS(K_T^+) = |\tau| + (sup C)^+, while highlighting that compactness does not guarantee amalgamation in these AECs. It further offers a uniform mu-AEC construction via L^+(Q^{cof}_C) yielding K^*_T with LS(K^*_T) = (|\tau|+\mu+|C|)^{<\mu} and undefinability of well-order. Collectively, the results unify several quantifier-based extensions under a general Skolemization framework, explaining how a range of positive, definable infinitary quantifiers give rise to AECs or mu-AECs and clarifying the limitations imposed by compactness.

Abstract

The cofinality quantifiers were introduced by Shelah as an example of a compact logic stronger than first-order logic. We show that the classes of models axiomatized by these quantifiers can be turned into an Abstract Elementary Class by restricting to positive and deliberate uses. Rather than using an ad hoc proof, we give a general framework of abstract Skolemization that can prove a wide range of examples are Abstract Elementary Classes.

Cofinality quantifiers in Abstract Elementary Classes and beyond

TL;DR

Develops a framework to convert classes axiomatized by cofinality quantifiers into Abstract Elementary Classes by enforcing positive, deliberate use and applying a finitary abstract Skolemization. It proves that for theories in , the class is an AEC with LS(K_T^+) = |\tau| + (sup C)^+, while highlighting that compactness does not guarantee amalgamation in these AECs. It further offers a uniform mu-AEC construction via L^+(Q^{cof}_C) yielding K^*_T with LS(K^*_T) = (|\tau|+\mu+|C|)^{<\mu} and undefinability of well-order. Collectively, the results unify several quantifier-based extensions under a general Skolemization framework, explaining how a range of positive, definable infinitary quantifiers give rise to AECs or mu-AECs and clarifying the limitations imposed by compactness.

Abstract

The cofinality quantifiers were introduced by Shelah as an example of a compact logic stronger than first-order logic. We show that the classes of models axiomatized by these quantifiers can be turned into an Abstract Elementary Class by restricting to positive and deliberate uses. Rather than using an ad hoc proof, we give a general framework of abstract Skolemization that can prove a wide range of examples are Abstract Elementary Classes.
Paper Structure (5 sections, 11 theorems, 13 equations)

This paper contains 5 sections, 11 theorems, 13 equations.

Table of Contents

  1. Introduction
  2. Cofinality quantifiers and background
  3. $\mathbb{L}(Q^\text{cof}_\mathcal{C})$ as an Abstract Elementary Class
  4. $\mathbb{L}\left(Q^\text{cof}_\omega\right)$ as an $\omega_1$-Abstract Elementary Class
  5. Abstract Skolemizations and a sufficient criteria to be an AEC

Key Result

Theorem 3.2

Fix a set of regular cardinal $C$ and set $\mathcal{L} = \mathbb{L}$, finitary first-order logic. Let $T \subset \mathcal{L}\left(Q^\text{cof}_\mathcal{C}\right)(\tau)$ be a theory where all instances of cofinality quantifiers appear positively. Then is an Abstract Elementary Class with $\text{LS}(\mathbb{K}_{T}^+) = |\tau|+(\sup C)^+$. Additionally, the same holds true if $\mathcal{L}$ is replac

Theorems & Definitions (21)

  • Definition 2.1
  • Remark 2.3
  • Example 2.4
  • Definition 2.5
  • Definition 3.1
  • Theorem 3.2
  • Proposition 3.3
  • Definition 3.4
  • Theorem 3.5
  • Definition 4.1
  • ...and 11 more