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A Complete Bounded Theory with Unbounded Types

Hongyu Zhu

Abstract

One measure of the complexity of a first-order theory, and similarly a type, is the complexity of the formulas required to axiomatize it. We say a theory is bounded if there is an axiomatization involving only $\forall_n$-formulas for some finite $n$, and unbounded otherwise. One might expect bounded theories to have only bounded types. In fact, an analogue holds in infinitary logic, where the complexity of a Scott sentence roughly agrees with the complexity of the most complicated automorphism orbit. Our main result, however, shows this is not the case in the first-order setting: Namely, there can be a bounded theory, in fact $\forall_1$-axiomatizable, which has unbounded types.

A Complete Bounded Theory with Unbounded Types

Abstract

One measure of the complexity of a first-order theory, and similarly a type, is the complexity of the formulas required to axiomatize it. We say a theory is bounded if there is an axiomatization involving only -formulas for some finite , and unbounded otherwise. One might expect bounded theories to have only bounded types. In fact, an analogue holds in infinitary logic, where the complexity of a Scott sentence roughly agrees with the complexity of the most complicated automorphism orbit. Our main result, however, shows this is not the case in the first-order setting: Namely, there can be a bounded theory, in fact -axiomatizable, which has unbounded types.
Paper Structure (12 sections, 17 theorems, 3 equations, 1 figure)

This paper contains 12 sections, 17 theorems, 3 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Base Theory
  4. Constructing the Theory
  5. Completeness of the Theory
  6. The Invariants
  7. Winning Strategies for $\exists$
  8. Finite-height Trees
  9. Completeness
  10. Analysis of Types
  11. Complicated Types
  12. Open Questions

Key Result

Proposition 4.2

There is a unique model $\mathcal{C}\vDash T$ whose domain is equal to $C^{\mathcal{C}}$, i.e. every element is (the interpretation of) a constant. In addition, $\mathcal{C}$ embeds into every model of $T$.

Figures (1)

  • Figure 1: $\mathcal{N}$

Theorems & Definitions (52)

  • Definition 1.1
  • Definition 2.1
  • Definition 2.3
  • Definition 3.1
  • Remark 3.3
  • Definition 3.4
  • Definition 4.1
  • Proposition 4.2
  • proof
  • Definition 4.3
  • ...and 42 more