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

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