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Shelah’s pcf theory and its applications. (English) Zbl 0713.03024

Sei a eine unendliche Menge von regulären Kardinalzahlen, so daß \(\min (a)>| a|\) gilt. Sei d ein Ultrafilter auf a und \(\Pi\) a das Cartesische Produkt von a. Dann ist \(\Pi\) a/D eine Ordnung von \(\Pi\) a, wo für f,g\(\in \Pi a\), \(f=_ Dg\) genau dann, wenn \(\{\iota\in a:\) \(f(\iota)=g(\iota)\}\in D\), bzw. \(f\leq_ Dg\) genau dann, wenn \(\{\iota\in a:\) f(\(\iota\))\(\leq g(\iota)\}\in D\). cf(\(\Pi\) a/D) ist die Ordinalzahl, die die Kofinalität von \(\Pi\) a/D darstellt. \[ pcf(a)=\{\lambda:\;\lambda =cf(\Pi a/D)\quad bei\quad irgendeinem\quad D\}. \] Das Hauptresultat ist das folgende Theorem: Sei \(\lambda =cf(\Pi a/D)\) und \(\mu =\lim_ Da\) die einzige Kardinalzahl, für die bei jedem \(\beta <\mu\), \(\{\alpha\in a:\beta <\alpha \leq \mu \}\in D\) gilt. Dann gibt es für jedes reguläre \(\lambda '\) mit \(\mu <\lambda '<\lambda\) eine Menge \(a'\) von regulären Kardinalzahlen, \(| a'| =| a|\), und einen Ultrafilter \(D'\) auf \(a'\), so daß \(\lim_{D'}a'=\mu\) und \(cf(\Pi a'/D')=\lambda '.\)
Diese Theorie wird für die Konstruktion von Jonsson-Algebren auf verschiedenen Kardinalzahlen verwendet, wobei \((A,(f_ i)_{i<\omega})\) eine Jonsson-Algebra genannt wird, wenn es keine echte Teilalgebra \((B,(f_ i| B)_{i<\omega})\) gibt.
Von weiteren Resultaten der Kardinalzahlarithmetik sind die folgenden nennenswert: Theorem: Sei a ein Intervall von regulären Kardinalzahlen, \(a=[\min (a),\sup (a)]\), so daß \(\min (a)^{| a|}<\sup (a)\). Dann gilt \(\max (pcf(a))=| \Pi a|\). Theorem: Für jede Limesordinalzahl \(\delta\) gilt \(\aleph_{\delta}^{cf(\delta)}<\aleph_{(| \delta |^{cf(\delta)})^+}\).
Reviewer: A.Tauts

MSC:

03E10 Ordinal and cardinal numbers
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