Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 212.32502
Autor: Erdös, Paul; Tarski, A.
Title: On some problems involving inaccessible cardinals (In English)
Source: Essays Found. Math., dedicat. to A.A. Fraenkel on his 70th Anniv. 50-82 (1962).
Review: [For the entire collection see Zbl 128.24103.]
In this paper, several properties of infinite cardinals are investigated. Let P1, ... ,P4,Q,R be the following properties: P1(\lambda): There is a set with power \lambda which is simply ordered by a relation \leq in such a way that no subset of it with power \lambda is well ordered by the same relation \leq or by the converse relation \geq . P2(\lambda): There is a complete graph on a set of power \lambda that can be divided in two subgraphs neither of which includes a complete subgraph on a set of power \lambda. P3(\lambda): Every \lambda-complete prime ideal in the set algebra formed by all subsets of \lambda is principal. P4(\lambda): There is a \lambda-complete and \lambda-distributive Boolean algebra which is not isomorphic to any \lambda-complete set algebra. Q(\lambda): There is a ramification system < A, \leq > of order \lambda such that (1) the set of all elements x in A of order \xi has power < \lambda for every \lambda, (2) every subset of A well-ordered by \leq has power \lambda. R(\lambda): there is \lambda-distributive Boolean algebra which is \lambda-generated by a set of power \lambda and which is not isomorphic to any \lambda-complete set algebra.
By S1 {D ––>} S2 we mean that for every infinite cardinal in D property S1 implies S2. And let C, AC, SL, IC be the class of all infinite, accessible, singular strong limit and inaccessible cardinals, respectively. We write S1 ––> S2 instead of S1 {C ––>} S2.
The main result of this paper is a diagram of implications. It is also obtained that the property R applies a very comprehensive class of inaccessible cardinals. Most of implications in the opposite direction, and the existence of an inaccessible cardinal which does not have the property R still remain open.
Reviewer: K.Namba
Classif.: * 03E55 Large cardinals
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