The Number of Atomic Models of Uncountable Theories

TL;DR

This paper explores the existence and properties of atomic models in uncountable theories, demonstrating their existence under various set-theoretic assumptions and showing the diversity of atomic models in such theories.

Contribution

It establishes the existence of unique atomic models that are not constructible in certain uncountable theories and analyzes the number of atomic models under specific set-theoretic conditions.

Findings

Existence of a complete theory with a unique non-constructible atomic model in continuum size.

Consistency results showing theories with uncountably many atomic models of size .

Demonstrates the set-theoretic independence of the number of atomic models in certain theories.

Abstract

We show there exists a complete theory in a language of size continuum possessing a unique atomic model which is not constructible. We also show it is consistent with $ZFC + \aleph_1 < 2^{\aleph_0}$ that there is a complete theory in a language of size $\aleph_1$ possessing a unique atomic model which is not constructible. Finally we show it is consistent with $ZFC + \aleph_1 < 2^{\aleph_0}$ that for every complete theory $T$ in a language of size $\aleph_1$, if $T$ has uncountable atomic models but no constructible models, then $T$ has $2^{\aleph_1}$ atomic models of size $\aleph_1$.