User jeff - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T21:20:42Z http://mathoverflow.net/feeds/user/642 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/743/what-do-models-where-the-ch-is-false-look-like/1018#1018 Answer by Jeff for What do models where the CH is false look like? Jeff 2009-10-18T09:13:10Z 2009-10-18T09:13:10Z <p>You must start slightly further back. The Axiom of Choice (AC) and the Axiom of Determinacy (AD) are both very desirable, but contradict one another. We usually feel AC has more claim to truth since AC is equivalent to "a cartesian project of non-empty sets is non-empty". So set theorists found weaker versions of AD that don't contradict AC, one being the Axiom of Projective Determinacy.</p> <p>These determinacy axioms basically say that infinite sets behave slightly more like finite sets than one might otherwise imagine. In particular, determinacy axioms have always been found to be equi-consistent with various large cardinal axioms, i.e. axioms that create some cardinal(s) far larger than any that came before. A good example of a large cardinal is just the omega, set of all integers, i.e. the axiom of infinity. You cannot prove that large cardinal axioms are consistent outright, yet they have extremely clear motivation and are unlikely to be inconsistent.</p> <p>CH seems independent of these large cardinal axioms themselves, but Woodin showed that CH contradicts one compelling determinacy axiom.</p> <p>Btw, Freiling's axiom of symmetry (AX) is apparently equivalent to not CH.</p>