# On a Theorem of Silver and Pcf Theory

Specialeforsvar ved Avgerinos Delkos

Titel: On a Theorem of Silver and Pcf Theorem

Abstract: The pcf theory was first introduced in 1978 by Saharon Shelah in a series of publications, culminating in his book [3]. It is a powerful theory with many applications and before its development our view of cardinal arithmetic was much more fundamental. One of the most well known consequences of this theory is that if @! is a strong limit cardinal then 2@! < @!4 . In this thesis we develop the necessary tools to begin our study of the pcf theory. In particular the first half is mostly focused on proving a variation of Silver's theorem which is another important result on cardinal arithmetic that sheds some light on the Singular Cardinal Hypothesis. That is, Silver proved in 1974 that if k is a singular cardinal of uncountable cofinality and the Generalized Continuum Hypothesis holds for every _ < k, then it also holds for k. It is considered one of the most remarkable theorems dealing with singular cardinals, before its discovery, many thought that it was possible for a singular cardinal to be the least counterexample of the GCH. The second half of this thesis explores more core elements of the pcf theory, the pcf function, the J<_ ideals and normality which are used in developing the theory needed to show the famous inequality mentioned above.

Vejleder:  Asger D. Tórnqvist
Censor:    Jesper Bengtsen, IT Universitetet