**March 4**

Alexander Van Abel,
CUNY

**Omitting Classes of Elements**

In this talk, we will review Morley's 1963 article 'Omitting Classes of Elements' (in modern parlance, omitting types). In this paper, Morley investigates the question of how large a structure's cardinality must be before it is forced to realize a type (for example, an ordered field with cardinality greater than the continuum must contain non-Archimedean elements). Given a fixed language, there is a cardinal $\kappa$ such that for every theory $T$ and any type $\Sigma$, if $T$ has a model of size $\kappa$ omitting $\Sigma$ then it has such a model in every cardinality. Morley's main result is that given a countable language $L$, if for every $\alpha < \omega_1$ there is a model of $T$ of size $\geq \beth_\alpha$ which omits $\Sigma$, then there is such a model in every cardinality (and if $L$ has at most $\aleph_\beta$ symbols, we replace '$\omega_1$' in this statement by '$\omega_{\gamma + 1}$' where $2^{\aleph_\beta} = \aleph_\gamma$). The proof uses the Erdős–Rado partition theorem and indiscernible sequences.