Wednesday, November 19, 20144:50 pmModels of PAGC 6300

# Forcing over models of arithmetic

## Kameryn Williams

### The CUNY Graduate Center

I will talk about forcing over models of arithmetic. Our primary application will be the following theorem, due to Simpson: if a model M of PA is countable, then M has a subset U such that that (M,U) is a pointwise definable model of PA*. Time permitting, we will see that the MacDowell–Specker theorem fails for uncountable languages: for M countable and nonstandard, there are U_alpha for alpha < omega_1 such that (M, U_alpha)_{alpha < omega_1} is a model of PA* and has no elementary end extensions.

Kameryn Williams is a graduate student in mathematics at the CUNY Graduate

Center, specializing in set theory and mathematical logic. He received a

bachelor’s degree in mathematics from Boise State University in 2012.