Introduction to Cardinal ArithmeticThis book is an introduction to modern cardinal arithmetic, developed in the frame of the axioms of Zermelo-Fraenkel set theory together with the axiom of choice. It splits into three parts. Part one, which is contained in Chapter 1, describes the classical cardinal arithmetic due to Bernstein, Cantor, Hausdorff, Konig, and Tarski. The results were found in the years between 1870 and 1930. Part two, which is Chapter 2, characterizes the development of cardinal arith metic in the seventies, which was led by Galvin, Hajnal, and Silver. The third part, contained in Chapters 3 to 9, presents the fundamental investigations in pcf-theory which has been developed by S. Shelah to answer the questions left open in the seventies. All theorems presented in Chapter 3 and Chapters 5 to 9 are due to Shelah, unless otherwise stated. We are greatly indebted to all those set theorists whose work we have tried to expound. Concerning the literature we owe very much to S. Shelah's book [Sh5] and to the article by M. R. Burke and M. Magidor [BM] which also initiated our students' interest for Shelah's pcf-theory. |
Contents
III | 5 |
IV | 15 |
V | 20 |
VI | 30 |
VII | 40 |
VIII | 58 |
IX | 70 |
XI | 79 |
XXIV | 199 |
XXV | 202 |
XXVI | 209 |
XXVII | 213 |
XXVIII | 218 |
XXIX | 221 |
XXX | 225 |
XXXI | 233 |
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Introduction to Cardinal Arithmetic Michael Holz,Karsten Steffens,E. Weitz No preview available - 1999 |
Common terms and phrases
assertion assume without loss assumption axiom axiom of choice bijection called cf a/D choose club cofinal conclude continuum function contradiction Corollary countable define definition denote dom(f element elementary substructure equivalent exists filter finite fld(R formula function f Further let Furthermore hence Hint inaccessible cardinal inductive hypothesis infer infinite cardinal number injective limit ordinal max pcf(a min(a modulo natural number nonempty set normal sequence obtain order type ordinal functions partial ordering pcf(a pcf(b pcf(c principal number progressive set Proof Let Prove ran(ƒ recursion regular cardinals satisfying says sequence f sequence of members set of regular Shelah singular cardinal strictly increasing modulo substructure of H successor ordinal sup(a supremum tcf S/I Theorem transfinite induction transitive ultrafilter unbounded upper bound well-ordering yields