Cardinal addition and the axiom of choice

Abstract

The quenstions which are discussed in this paper originate with Traski and concern the decision problem for the calss of theorems on the addition of cardinal numbers which are provable without the axiom of choice. Our first result is in the negative direction and takes the following form. We postulate the existence of a sequence of sets which satisfies a number of special conditons. Then we show, by the methods of Tarski-Mostowski-Robinson's “Undecidable Theoris”, that whenever a system of set theory is compatible with this additional postulate, then the class of theorems in the elementary theory of cardinal addition, which are provable within the system, is undecidable. Cohen's method of consistency proof is used to show that the postulate is compatible with “natural” systems of set theory such as the Zermelo-Fraenkel and Bernays-Gödel systems excluding the axiom of choice, provided that these systems are themselves consistent. Therefore a positive solution of the decision problem is possible only for a restricted class of theorems. Most of the theorems on cardinal addition which are have been proved without the axiom of choice take a form in which it is assumed that cardinals a 0,… a 1, satisfy a cerntain system equations and inequalities and then shown that there are cardinals b 0, … b m which together with a 0, … a n satisfy another system of equations and inequalities. The following theorem illustrates the type of results we have in mind.