Friday, November 11, 20163:45 pmnote new timeModel theory seminarGC 6417

# Straw into gold: Turning a c.c.c. forcing construction of a model into a ZFC proof

## Chris Laskowski

### University of Maryland

We use $M$-normal ultrapowers to give a new proof of a theorem of Keisler that if one can construct a standard model of a sentence of $L_{\omega_1,\omega}(Q)$ using a c.c.c. forcing, then a standard model already exists in V.

We use this to investigate the class of atomic models of a countable, first order theory $T$. In particular, we show that various `unsuperstable-like’ behaviors imply the existence of many non-isomorphic atomic models of size $\aleph_1$.

This is joint work with John Baldwin and Saharon Shelah.