This post is a bit different, as it’s somewhat more of a curiosity I recently noticed. It’s a theorem of that every type of of algebraic structure has a free algebra, similar to the notion of a free group, free module and so on. How about without choice? It turns out that the theory
has considerable consistency strength. It consistency-wise implies and is consistency-wise implied by a proper class of strongly compacts.
This is the third post in my series on determinacy from Woodins. In the last post we showed Martin-Steel’s result that follows from the existence of infinitely many Woodins and a measurable above. We’ll now give the main ideas of Woodin’s incredible strengthening of this result, showing from the same assumption that holds.
This is a continuation of my last post on determinacy, where we began the proof of projective determinacy. We’ve reduced the statement to showing that every projective set is homogeneously Suslin, which will be shown here, modulo a key lemma from [MS89].