April 22
Corey Switzer, CUNY
Hanf Numbers of Arithmetics
Recall that given a complete theory $T$ and a type $p(x)$ the Hanf number for $p(x)$ is the least cardinal $\kappa$ so that any model of $T$ of size $\kappa$ realizes $p(x)$ (if such a $\kappa$ exists and $\infty$ otherwise). The Hanf number for $T$, denoted $H(T)$, is the supremum of the successors of the Hanf numbers for all possible types $p(x)$ whose Hanf numbers are $\lt\infty$. We have seen so far in the seminar that for any complete, consistent $T$ in a countable language $H(T) \leq \beth_{\omega_1}$ (a result due to Morley). In this talk I will present the following theorems: (1) The Hanf number for true arithmetic is $\beth_{\omega}$ (Abrahamson-Harrington-Knight) but (2) the Hanf number for False Arithmetic is $\beth_{\omega_1}$ (Abrahamson-Harrington)