On Some Problems Involving Inaccessible Cardinals

Introduction(') Many problems in set theory and related domains are known which have the following features in common . Each of them can be formulated as the problem of determining all the infinite cardinals which have a given property P. It is known that the property P applies to all infinite cardinals which are not (strongly) inaccessible while it does not apply to the smallest inaccessible cardinal, N 0 . On the other hand, the problem remains open whether P applies to all inaccessible cardinals greater than N O , and in some cases it is not even known whether P applies to any such cardinal . (The meaning of most terms used in this introduction will be explained below .) In the present paper we shall be concerned with four problems of the kind described ; the corresponding properties will be denoted by Pl, P2, P 3 , and P4 . A cardinal 2 is said to have the property P 1 if there is a set A with power A which is simply ordered by a relation 5 in such a way that no subset of A with power A is well ordered by the same relation S or by the converse relation 2! . The property P 2 applies by definition to a cardinal 2 if there is a complete graph on a set of power 2 that can be divided in two subgraphs neither of which includes a complete subgraph on a set of power A. P3 applies to 2 if in the set algebra of all subsets of a set of power A every A-complete prime ideal is principal. Finally, P4 applies to 2 if there is a A-complete and 2-distributive Boolean algebra $ which is not isomorphic to any 1complete set algebra. We do not attempt here to solve fully any of these problems, but we establish some implications among them ; in fact, our main result is that, for every infinite cardinal A, each property P,,, with m = 1, 2,3 implies Pm+1 . The problem remains open whether any of these implications holds in the opposite direction as well . Various equivalent formulations of the properties P3 and P4 are known ; some relevant equivalences will be explicitly formulated and established . (1) This paper was prepared for publication during the period when Tarski was working on a research project in the foundations of mathematics supported by the U.S. National Science Foundation (grants G-6693 and G-14006) .