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 , proved its basic properties and showed that .