A Hanf number for saturation and omission
Volume 213 / 2011
Fundamenta Mathematicae 213 (2011), 255-270
MSC: 03C52, 03C85.
DOI: 10.4064/fm213-3-5
Abstract
Suppose $\boldsymbol t =(T,T_1,p)$ is a triple of two countable theories $T\subseteq T_1$ in vocabularies $\tau \subset \tau_1$ and a $\tau_1$-type $p$ over the empty set. We show that the Hanf number for the property `there is a model $M_1$ of $T_1$ which omits $p$, but $M_1 {\restriction} \tau$ is saturated' is essentially equal to the Löwenheim number of second order logic. In Section 4 we make exact computations of these Hanf numbers and note some distinctions between `first order' and `second order quantification'. In particular, we show that if $\kappa$ is uncountable, then $h^3(L_{\omega,\omega}(Q), \kappa) = h^3(L_{\omega_1,\omega}, \kappa)$, where $h^3$ is the `normal' notion of Hanf function (Definition 4.12).
