Professor Steve Jackson invites you to attend the Masters thesis project defense of Jared Holshouser on Wednesday, March 27th at 5:00 pm in GAB 461.

"The Prime Ideal Theorem and the Axiom of Choice"

Abstract:

It is easy to establish that the axiom of choice implies the prime ideal theorem. Both statements are equivalent to variety of important theorems, and it is natural to wonder if the statements are themselves equivalent. We will show that the prime ideal theorem is strictly weaker than the axiom of choice. This proof will take us through a forcing construction and some inner model theory. Furthermore, to complete the proof, we will require the Halpern-Lauchli theorem, a combinatorial result which is a strengthening of Ramsey's theorem. That the prime ideal theorem does not imply the axiom of choice was originally proved by Halpern and Levy in 1971. Our work here constitutes a modernization of their original proof.