We show that the existence of a non-Lebesgue measurable set cannot be proved in Zermelo-Frankel set theory (ZF) if use of the axiom of choice is disallowed. In fact, even adjoining an axiom DC to ZF,… Expand

Let ZFM be the set theory ZF together with an axiom which asserts the existence of a measurable cardinal. It is shown that if ZFM is consistent then ZFM is consistent with every sentence φ whose… Expand

We consider interpretations of modal logic in Peano arithmetic (P) determined by an assignment of a sentencev* ofP to each propositional variablev. We put (⊥)*=“0 = 1”, (χ → ψ)* = “χ* → ψ*” and let… Expand

This is the expository paper on strong axioms of infinity and elementary embeddings origi;,ally to have been authored by Reinhardt and Solovay. It has been owed for some time and already cited with… Expand

A uniform distribution a from a uniform distribution on the set 1, 2, 3, 4, 5 is a random number and if a and n are relatively prime, compute the residue varepsilon.Expand

We investigate relativized versions of the open question of whether every language accepted nondeterministically in polynomial time can be recognized deterministically in polynomial time. For any set… Expand

We can characterize the real line, up to order isomorphism, by the following list of properties: R is order complete, order dense, has no first or last elements, and contains a countable dense… Expand