# A brief overview of forcing notions

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.

# Choice principles

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. $\textsf{AC}$ 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 $\textsf{GCH}$, which can then also be seen as a choice principle, and choice even forces us to work in classical logic.

# Projectively correct mice

In a previous post we proved that whenever a countable mouse M has n Woodins it understands $\bf\Pi^1_{n+1}$ sets, implying that whenever A is such a set it holds that $A\cap M\in M$. As we mentioned back then, this is not as good as being correct about these sets, which would mean that $A\cap M\neq\emptyset$ whenever A of course is non-empty as well. Another way to phrase this is to say that $V\models\sigma$ iff $M\models\sigma$ for every $\bf\Pi^1_{n+1}$-sentence. Now, what does it then take for a mouse to be projectively correct?

# From mice to determinacy

Last time we proved that mice M with $n<\omega$ Woodins knows about $\bf\Sigma^1_{n+1}$ sets A, meaning $A\cap M\in M$, 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).

# Projective understanding via Woodins

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.

# Categorification of large cardinals?

When working with most of modern set theory we tend to transcend ZFC, always working with some strong background hypothesis, whether it being the existence of some elementary embedding, a colouring for some partition property, a generic for some uncountable poset or something completely different. When it comes to using these strong hypotheses in mainstream mathematics it seems that we hit a brick wall, as most of our strong hypotheses don’t easily translate to the language of everyday mathematics.

# Pcf scales and squares

The last few posts I’ve been covering a characterisation of pointclasses that admit scales. To make scale theory even more confusing there’s a completely different notion of scale, which really has nothing to do with our previous one — this one being of a more combinatorial nature. To avoid unnecessary confusion I’ll call these new objects pfc scales (but usually they’re simply called scales as well, however).