Object generators, sets, and a foundation for mathematics

Abstract: Beginning with two primitives, four axioms, and weak non-binary logic, we develop what seems to be a complete foundation for mainstream deductive mathematics. In more detail, the primitives are object generators'', essentially theobject'' primitive of category theory; and morphisms of these, essentially functions. The main assumption is the axiom of choice. The logic does not include a global equality operator, quantification, or excluded middle. Sets are defined inside this context, and most of the standard properties of sets are proved rather than assumed as axioms. The main technical result is that, in this context there is an implementation of the Zermillo-Fraenkel-Choice axioms, except that our use of function'' as a primitive instead ofmembership'' avoids the traditional need for first-order logic. We abbreviate the axioms with this omission as ZFC-1. This model is maximal in the sense that any other model of ZFC+/-1 is (uniquely) a transitive submodel of this one. Finally, we formulate the ``coherent limit axiom'' that characterizes the maximal model among all ZFC-1 models. In mainstream mathematics, the coherent limit axiom is regarded as self-evident and used extensively. Therefore, mainstream deductive work implicitly takes place in the set theory described here. This is even true of elementary calculus. For set theory, one implication is that much of the axiomatic set theory of the last century is irrelevant to mainstream mathematics. However, questions such as the Continuum Hypothesis are well-posed for the maximal model, and some of the vast literature on ZFC might be sharpened in this context.