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An analogue of U-rank for atomic classes

John T. Baldwin, Michael C. Laskowski, Saharon Shelah

Abstract

For a countable, complete, first-order theory $T$, we study $At$, the class of atomic models of $T$. We develop an analogue of $U$-rank and prove two results. On one hand, if some tp(d/a) is not ranked, then there are $2^{\aleph_1}$ non-isomorphic models in $At$ of size $\aleph_1$. On the other hand, if all types have finite rank, then the rank is fully additive and every finite tuple is dominated by an independent set of realizations of pseudo-minimal types.

An analogue of U-rank for atomic classes

Abstract

For a countable, complete, first-order theory , we study , the class of atomic models of . We develop an analogue of -rank and prove two results. On one hand, if some tp(d/a) is not ranked, then there are non-isomorphic models in of size . On the other hand, if all types have finite rank, then the rank is fully additive and every finite tuple is dominated by an independent set of realizations of pseudo-minimal types.

Paper Structure

This paper contains 11 sections, 37 theorems, 27 equations.

Table of Contents

  1. Context
  2. The new rank and statements of the main results
  3. Properties of the rank, chains, and additivity
  4. Finite and infinite chains
  5. Additivity
  6. Pseudo-minimal types and finitely ranked classes
  7. Few atomic models implies ranked
  8. A class of linear orders and weakly striated models
  9. The forcing
  10. Proof of Proposition \ref{['forcing']}
  11. Many non-isomorphic models in ${\bf At_T}$

Key Result

Theorem 2.5

Let $T$ be a complete theory in a countable language for which there is an uncountable atomic model. Then:

Theorems & Definitions (78)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Example 2.4
  • Theorem 2.5
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 68 more