Applications of the proper forcing axiom to models of Peano Arithmetic
V. Gitman, Applications of the proper forcing axiom to models of Peano arithmetic. ProQuest LLC, Ann Arbor, MI, 2007, p. 149.
This is the abstract of my PhD Thesis completed under the supervision of Joel David Hamkins at the CUNY Graduate Center in 2007.
In Chapter 1, new results are presented on Scott's Problem in the subject of models of Peano Arithmetic. Some forty years ago, Dana Scott showed that countable Scott sets are exactly the countable standard systems of models of ${\rm PA}$, and two decades later, Knight and Nadel extended his result to Scott sets of size $\omega_1$. Here it is shown that assuming the Proper Forcing Axiom, every arithmetically closed proper Scott set is the standard system of a model of ${\rm PA}$. In Chapter 2, new large cardinal axioms, based on Ramsey-like embedding properties, are introduced and placed within the large cardinal hierarchy. These notions generalize the seldom encountered embedding characterization of Ramsey cardinals. I also show how these large cardinals can be used to obtain indestructibility results for Ramsey cardinals.