The equivalence of Axiom (∗)+ and Axiom (∗)++.

Asperó and Schindler have completely solved the Axiom (∗) vs. MM + + problem. They have proved that if MM + + holds then Axiom (∗) holds, with no additional assumptions. The key question now concerns the relationship between MM + + and Axiom (∗) + . This is because the foundational issues raised by the problem of Axiom (∗) vs. MM + + arguably persist in the problem of Axiom (∗) + vs. MM + + . The first of our two main theorems is that Axiom (∗) + is equivalent to Axiom (∗) + + , and as a corollary we show that Axiom (∗) + fails in all the known models of MM + + . This suggests that MM + + actually refutes Axiom (∗) + . Our second main theorem is that the AD + Conjecture holds assuming MM + + (c). This is the strongest partial result known on this conjecture which is one of the central open problems of AD + -theory and Ω -logic. These results identify a fundamental asymmetry between the Continuum Hypothesis and any axiom which is both Σ 2 -expressible and which implies MM + + (c) , on the basis of generic absoluteness for the simplest of the nontrivial sentences of Third-Order Number Theory. These are the Σ 1 2 -sentences with no parameters. Such sentences are those which simply assert the existence of a set A ⊆ ℝ for which some property involving only quantification over ℝ holds. [ABSTRACT FROM AUTHOR]