Last time we delved into the world of ideals and their associated properties, precipitousness and saturation. We noted that these properties could be viewed as a measure of “how close” a cardinal is to being measurable, and furthermore that all the properties are equiconsistent; i.e. that the existence of a precipitous ideal on some is equiconsistent with the existence of a measurable cardinal. But we can do better.
A long time ago I made a blog post on the fascinating phenomenon of generic ultrapowers, where, roughly speaking, we start off with an ideal on some , force with the poset of -positive sets and then the generic filter ends up being a -measure on . If this sounded like gibberish then I’d recommend reading the aforementioned post first. The cool thing is that we can achieve all this without requiring any large cardinal assumptions! We’re not guaranteed that the generic ultrapower is wellfounded however, but if it happens to be the case then we call precipitous. We have a bunch of other properties these ideals can satisfy however, usually involving the term ‘saturation’. What’s all that about and what’s the connection to precipitousness?
In this day and age we got a massive jungle of forcing notions, each with it’s own very specific purpose and technicalities. For set theorists who aren’t specialists in forcing theory this might seem daunting when stumbling across open questions that cry out for a forcing solution. I’m precisely one of those people, and this is my attempt at providing a brief non-technical toolkit of various forcing notions. I won’t go into how any one of the notions are defined — I’ll purely talk about their properties.
The axiom of choice, by which I mean that every collection of non-empty sets has a choice function, is usually an axiom most working mathematicians accept without further thought. But in set theory we usually get ourselves into situations where we simply cannot have (full) choice — most notably in determinacy scenarios, giving rise to several weakened forms of choice. might seem like an isolated axiom without much direct connection to other axioms, as we usually simply assume choice and get on with our day. But choice is in fact implied by the generalised continuum hypothesis , which can then also be seen as a choice principle, and choice even forces us to work in classical logic.
In a previous post we proved that whenever a countable mouse M has n Woodins it understands sets, implying that whenever A is such a set it holds that . As we mentioned back then, this is not as good as being correct about these sets, which would mean that whenever A of course is non-empty as well. Another way to phrase this is to say that iff for every -sentence. Now, what does it then take for a mouse to be projectively correct?
Last time we proved that mice M with Woodins knows about sets A, meaning , using Woodin’s genericity iterations and the notion of mice understanding sets of reals. But what good is a projectively aware mouse? To give an example of the usefulness of this property, we show that the existence of these projectively aware mice yields determinacy of sets of reals, shown by Neeman (’02).
I’ve previously covered Woodin’s genericity iterations, being a method to “catch” any real using Woodin cardinals. Roughly, given any countable mouse M and a real x, we can iterate M to a model over which x is generic. An application of this is the phenomenon that Woodins present in mice allows them to be more projectively aware.