The Number of Atomic Models of Uncountable 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 ${\aleph }_1$.

arXiv comment22 pages