##
**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 P_{1}, ... ,P_{4},Q,R be the following properties: P_{1}(\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 . P_{2}(\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. P_{3}(\lambda): Every \lambda-complete prime ideal in the set algebra formed by all subsets of \lambda is principal. P_{4}(\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 S_{1} {^{D} ––>} S_{2} we mean that for every infinite cardinal in D property S_{1} implies S_{2}. And let C, AC, SL, IC be the class of all infinite, accessible, singular strong limit and inaccessible cardinals, respectively. We write S_{1} ––> S_{2} instead of S_{1} {^{C} ––>} S_{2}.

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

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag