On the necessary use of abstract set theory

Abstract

In this paper we present some independence results from the Zermelo-Frankel axioms of set theory with the axiom of choice (ZFC) which differ from earlier such independence results in three major respects. Firstly, these new propositions that are shown to be independent of ZFC (i.e., neither provable nor refutable from ZFC) form mathematically natural assertions about Bore1 functions of several variables from the Hilbert cube I” into the unit interval, or back into the Hilbert cube. As such, they are of a level of abstraction significantly below that of the earlier independence results. Secondly, these propositions are not only independent of ZFC, but also of ZFC together with the axiom of constructibility (V = L). The only earlier examples of intelligible statements independent of ZFC + V= L either express properties of formal systems such as ZFC (e.g., the consistency of ZFC), or assert the existence of very large cardinalities (e.g., inaccessible cardinals). The great bulk of independence results from ZFCLthe ones that involve standard mathematical concepts and constructions-are about sets of limited cardinality (most commonly, that of at most the continuum), and are obtained using the forcing method introduced by Paul J. Cohen (see [2]). It is now known in virtually every such case, that these independence results are eliminated if V= L is added to ZFC. Finally, some of our propositions can be proved in the theory of classes, as formalized by the Morse-Kelley class theory with the axiom of choice for sets (MKC), but not in ZFC. MKC still formalizes only commonly accepted principles of mathematical reasoning. Thus these propositions provide examples of interesting theorems whose proofs necessarily involve the outer limits of what is commonly accepted as valid principles of mathematical reasoning. The starting point for the development of these new propositions was our consideration, in 1976, of certain aspects of Cantor’s basic theorem that the set of all real numbers is not countable. Thus given any sequence of real numbers, there is a real number which is not a term of the sequence. By using nested sequence of closed intervals with rational endpoints, it is easy to