SOME REMARKS CONCERNING OUR PAPER ,,ON THE STRUCTURE OF SET-MAPPINGS" . - NON-EXISTENCE OF A TWO-VALUED a-MEASURE FOR THE FIRST UNCOUNTABLE INACCESSIBLE CARDINAL

In accordance with the notations of [4] we say that a cardinal m possesses property P 3 if every two-valued measure μ(X) defined on all subsets of a set S of power m vanishes identically, provided p({x}) =0 for every x E S and p(X) is m-additive . It was well known that tt o fails to possess property P 3 and that every cardinal m < t, possesses property P 3 where t, denotes the first uncountable inaccessible cardinal . Recently A . TARSKI has proved, using a result of P. HANF, that a certain wide class of strongly inaccessible cardinals possesses property P 3 (called strongly incompact cardinals) . H . J . KEISLER gave a purely set-theoretical proof of this result .' After having seen these papers we observed that the special case of this result that t, possesses property P 3 follows almost trivially from some of our theorems proved in (1] . We are going to give this simple proof in § 2 . Our method for the proof is of purely combinatorial character, and although it is certainly weaker than that of A. TARSKI and H . J. KEISLER, we think that it is of interest to formulate how far one can go with these methods at present . Let to , . . ., t,, . . . denote the increasing sequence of the strongly inaccessible cardinals (to = o) and let (O, denote the initial number of t4 . We can prove similarly as in the case of t, that i s possesses property P 3 , provided 0 < ~ 0 4. We only give the outline of this proof. Finally, we are going to formulate some problems . .