Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Amalgamation and Keisler's Order

Danielle Ulrich

Abstract

Malliaris and Shelah famously proved that Keisler's order $\trianglelefteq$ has infinitely many classes. In more detail, for each $2 \leq k < n < ω$, let $T_{n, k}$ be the theory of the random $k$-ary $n$-clique free hypergraph. Malliaris and Shelah show that whenever $k+1 < k'$, then $T_{k+1, k} \not \trianglelefteq T_{k'+1, k'}$. However, their arguments do not separate $T_{k+1, k}$ from $T_{k+2, k+1}$, and the model-theoretic properties detected by their ultrafilters are difficult to evaluate in practice. We uniformize the relevant ultrafilter constructions and obtain sharper model-theoretic bounds. As a sample application, we prove the following: suppose $3 \leq k < \aleph_0$, and $T$ is a countable low theory. Suppose that every independent system $(M_s: s \subsetneq k)$ of countable models of $T$ can be independently amalgamated. Then $T_{k, k-1} \not \trianglelefteq T$. In particular, for all $k < k'$, $T_{k+1, k} \not \trianglelefteq T_{k'+1, k'}$.

Amalgamation and Keisler's Order

Abstract

Malliaris and Shelah famously proved that Keisler's order has infinitely many classes. In more detail, for each , let be the theory of the random -ary -clique free hypergraph. Malliaris and Shelah show that whenever , then . However, their arguments do not separate from , and the model-theoretic properties detected by their ultrafilters are difficult to evaluate in practice. We uniformize the relevant ultrafilter constructions and obtain sharper model-theoretic bounds. As a sample application, we prove the following: suppose , and is a countable low theory. Suppose that every independent system of countable models of can be independently amalgamated. Then . In particular, for all , .

Paper Structure

This paper contains 12 sections, 62 theorems, 1 equation.

Table of Contents

  1. Introduction
  2. A refined set mapping theorem
  3. Independent Systems of Models
  4. Coloring Partial Types
  5. Consequences of $\mathcal{P}^-(k)$-amalgamation of models
  6. When $\lambda$ is Small
  7. Boolean Algebras and Forcing
  8. Full Boolean-Valued Models
  9. Colorability is Preserved Under Forcing Iterations
  10. The Ultrafilter Constructions
  11. The $\mathbf{s}$-extension property
  12. Conclusion

Key Result

Theorem 1.1

Suppose $T$ is a complete countable theory, and $\mathcal{U}$ is a $\lambda$-regular ultrafilter on $\mathcal{P}(\lambda)$, and $M_0, M_1 \models T$. Then $M_0^\lambda/\mathcal{U}$ is $\lambda^+$-saturated if and only if $M_1^\lambda/\mathcal{U}$ is.

Theorems & Definitions (143)

  • Theorem 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Corollary 1.5
  • Conjecture 1.6
  • Theorem 1.7
  • Definition 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 133 more