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$\leq_{SP}$ Can Have Infinitely Many Classes

Saharon Shelah, Danielle Ulrich

Abstract

Building off of recent results on Keisler's order, we show that consistently, $\leq_{SP}$ has infinitely many classes. In particular, we define the property of $\leq k$-type amalgamation for simple theories, for each $2 \leq k < ω$. If we let $T_{n, k}$ be the theory of the random $k$-ary, $n$-clique free random hyper-graph, then $T_{n, k}$ has $\leq k-1$-type amalgamation but not $\leq k$-type amalgamation. We show that consistently, if $T$ has $\leq k$-type amalgamation then $T_{k+1, k} \not \leq_{SP} T$, thus producing infinitely many $\leq_{SP}$-classes. The same construction gives a simplified proof of Shelah's theorem that consistently, the maximal $\leq_{SP}$-class is exactly the class of unsimple theories. Finally, we show that consistently, if $T$ has $<\aleph_0$-type amalgamation, then $T \leq_{SP} T_{rg}$, the theory of the random graph.

$\leq_{SP}$ Can Have Infinitely Many Classes

Abstract

Building off of recent results on Keisler's order, we show that consistently, has infinitely many classes. In particular, we define the property of -type amalgamation for simple theories, for each . If we let be the theory of the random -ary, -clique free random hyper-graph, then has -type amalgamation but not -type amalgamation. We show that consistently, if has -type amalgamation then , thus producing infinitely many -classes. The same construction gives a simplified proof of Shelah's theorem that consistently, the maximal -class is exactly the class of unsimple theories. Finally, we show that consistently, if has -type amalgamation, then , the theory of the random graph.

Paper Structure

This paper contains 6 sections, 20 theorems.

Table of Contents

  1. Introduction
  2. Background
  3. Forcing Axioms
  4. Non-Forking Diagrams
  5. Amalgamation properties
  6. Conclusion

Key Result

Theorem 1.1

Suppose $\theta \le \mu \le \lambda$ are infinite cardinals such that $\theta$ is regular, $\mu = \mu^{<\theta}$, and $\lambda \le 2^\mu$. Then there is a sequence $({\mathbf f}_\gamma:\gamma < \mu)$ from ${}^\lambda \mu$ such that for all partial functions $f$ from $\lambda$ to $\mu$ of cardinality

Theorems & Definitions (61)

  • Definition 2
  • Definition 3
  • Theorem 1.1
  • Definition 1.2
  • Example 1.3
  • proof
  • Theorem 1.4
  • proof
  • Theorem 1.5
  • proof
  • ...and 51 more