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A QUICK PROOF OF THE PCF THEOREM

The following note provide a short proof of Shelah’s pcf theorem via the new Theorem 7 below. The following theorems are used: (1) Shelah’s theorem on the existence of stationary sets in I[λ]; (2) Shelah’s trichotomy theorem; (3) the characterization [2] of sequences with exact upper bounds. A proof of Shelah’s trichotomy theorem is in [3],II,1.2 and a shorter proof is in the appendix to [2]. The existence of stationary sets in I[λ] was proved in [4]. Proofs of this theorem and of the eub theorems are included in the Preliminaries Section below for the reader’s convenience. More on I[λ] can be found in [1]. I first presented a development of pcf theory using bounding numbers in Hattingen, Germany, in 1998, and used the same approach in a sequence of lectures on pcf theory in the Rényi institute in Budapest in 2002, where Theorem 7 was first used. The present note is based on the lecture I gave in the workshop on Cardinal Arithmetic in Jerusalem in March 2004.