## 116c- Lecture 12

We completed the proof of Silver’s theorem. Silver’s argument, as understood by Baumgartner and Prikry, started a whole new series of results that culminated in Shelah’s celebrated pcf theory. See

T. Jech, Singular cardinals and the pcf theory, The Bulletin of Symbolic Logic 1(4) (1995), 408-424

for an introduction (without proofs) and historical remarks, or

M. Burke, M. Magidor, Shelah’s pcf theory and its applications, Ann. Pure Appl. Logic 50 (3) (1990), 207-254

for a more technical introduction , including proofs. Jech’s paper is available through JSTOR.

We defined the cumulative hierarchy ${sf WF}=bigcup_{alphain{sf ORD}}V_alpha$, proved its basic properties and showed that $V={sf WF}$.