A Hanf number for saturation and omission

John T. Baldwin¹ ; Saharon Shelah²

¹ Department of Mathematics, Statistics, and Computer Science M//C 249 University of Illinois at Chicago 851 S. Morgan Chicago, IL 60607, U.S.A.
² Einstein Institute of Mathematics Hebrew University of Jerusalem Givat Ram, Jerusalem 91904, Israel and Department of Mathematics Rutgers University Piscataway, NJ 08854-8019, U.S.A.

Fundamenta Mathematicae, Tome 213 (2011) no. 3, pp. 255-270.

Résumé

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).

Détail

DOI : 10.4064/fm213-3-5

Keywords: suppose boldsymbol triple countable theories subseteq vocabularies tau subset tau tau type empty set hanf number property there model which omits nbsp restriction tau saturated essentially equal wenheim number second order logic section nbsp make exact computations these hanf numbers note distinctions between first order second order quantification particular kappa uncountable omega omega kappa omega omega kappa where normal notion hanf function definition nbsp