14. Cardinal Arithmetic - Springer Link

14. Cardinal Arithmetic Uri Abraham and Menachem Magidor

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . 2 Elementary Definitions . . . . . . . . . . . . . . . 2.1 Existence of Exact Upper Bounds . . . . . . . . 2.2 Application: Silver’s Theorem . . . . . . . . . . . 2.3 Application: A Covering Theorem . . . . . . . . 3 Basic Properties of the pcf Function . . . . . . . 3.1 The Ideal J μ+ in some generic extension. Using large cardinals Magidor proved the consistency of ℵω being the first cardinal κ for which 2κ > κ+ holds. For a long time it was believed that large cardinal and more complex applications of the forcing method should yield greater flexibility for values of the power set of singular cardinals. A first indication that there are possible limitations was the theorem of Silver (1974): If κ is a singular cardinal with uncountable cofinality and if 2δ = δ + for all cardinals δ < κ, then 2κ = κ+ . This result paved the way for further investigations by Galvin and Hajnal (1975). A representative

2. Elementary Definitions

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result of their work is the following: If ℵδ is a strong limit singular cardinal with uncountable cofinality, then 2ℵδ < ℵ(|δ|cf(δ) )+ . For example, if ℵω1 is a strong limit cardinal, then 2ℵω1 < ℵ(2ℵ1 )+ . The method of proof of these results relied in an essential way on the assumption that cf(δ) > ℵ0 . Shelah (1978) was able to prove similar results for singular cardinals with countable cofinality. For example, if ℵω is a strong limit cardinal, then (14.1) 2ℵω < ℵ(2ℵ0 )+ . In a series of papers culminating in his book [15], Shelah developed a powerful theory with many applications, pcf theory, which changed our view of cardinal arithmetic. A remarkable result of this theory is the following. If ℵω is a strong limit cardinal then 2ℵω < ℵω4 .

(14.2)

ℵ0

If 2 ≤ ℵ2 , then (14.1) is a better bound than (14.2), but in general, since (2ℵ0 )+ can be arbitrarily high, ω4 seems to be a firmer bound. The major definition in pcf theory is the set pcf(A) of possible cofinalities defined for every set  A of regular cardinals, as the collection of all cofinalities of ultraproducts A/D with ultrafilters D over A. This basic and rather simple definition appears in many places and is the basis of a very fruitful investigation. It is a basic definition also in the sense that while the power set can be easily changed by forcing, it is very hard to change pcf(A). Our aim in this chapter is to give a self-contained development of pcf theory and to present some of its important applications to cardinal arithmetic. Unless stated otherwise, all theorems and results in this chapter are due to Shelah. The fullest development of pcf theory is in Shelah’s book [15], and the interested reader can access newer articles (and the survey paper “Analytical Guide”) in the archive maintained at Rutgers University. In addition to this material, we have profited from expository papers (Burke-Magidor [2], Jech [7], and unpublished notes by Hajnal), and in particular a recently published book [6] which is very detailed, complete and carefully written. The authors thank Maxim R. Burke, Matt Foreman, Stefan Geschke, Peter Komj´ ath, John Krueger, Klaas Pieter Hart, Donald Monk, and Martin Weese for valuable corrections and improvements of earlier versions.

2. Elementary Definitions An ideal over a set A is a collection I ⊆ P(A) such that (1) I is closed under subsets, that is, X ∈ I and Y ⊆ X implies Y ∈ I, and (2) I is closed under

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finite unions, that is, X1 , X2 ∈ I imply X1 ∪ X2 ∈ I (and thus the union of any finite sequence of members of I is in I). If A ∈ I, then I is said to be proper. We do not require that ideals be proper (see the definition of J