Dan Saattrup Smart

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 $I$ on some $\kappa$, force with the poset of $I$-positive sets and then the generic filter ends up being a $V$-me…

In this day and age we got a massive jungle of forcing notions, each with it's own very specific purpose and technicalities.

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.

In a previous post we proved that whenever a countable mouse M has n Woodins it understands projective sets.

Last time we proved that mice M with finitely many Woodins know about projective sets, using Woodin's genericity iterations and the notion of mice understanding sets of reals.

I've previously covered Woodin's genericity iterations, being a method to "catch" any real using Woodin 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 uncountab…

The last few posts I've been covering a characterisation of pointclasses that admit scales.

So far we've characterised the scaled pointclasses among the projective hierarchy as well as establishing Steel's result that the Sigma_1 pointclass of levels of L which satisfies determinacy.

The last two posts covered the 'classical' theory of scales, meaning the characterisation of the scaled pointclasses in the projective hierarchy.

Last time we got an idea of what scales are and why they're useful.

In classical descriptive set theory a need arose to analyse the analytical and projective hierarchies more abstractly, isolating the key structural properties of the various levels.

Jensen's square principle has proven very useful in measuring the non-compactness of various successor cardinals as well as being an essential tool in finding new lower bounds for forcing axioms like the Proper Forcing Axiom.

I've previously mentioned the open problem of whether inaccessible Jónssons provably are weakly compact.

Inner model theory is all about constructing canonical inner models of set theory that inherits the complexity - i.e. the large cardinals - of the universe.

Weakly compact cardinals have a great variety of properties, which are all interesting enough to warrant independent study.

Previously I've only been talking about large cardinals and determinacy theories as if they were the only consistency hierarchies around.

We currently don't know whether or not there can exist a singular cardinal whose cardinal successor is a Jónsson cardinal.

I was recently playing a (set-theoretic) game and the question of whether it was determined slowly emerged.

In the late 90's it was shown by Steel and Woodin that HOD of L(R) exhibits mouse-like behaviour, and since then there's been a great interest in finding the HODs of other models than L(R).

The proof of Borel determinacy doesn't seem to have the best reputation, as it's both rather long, quite technical and it's really easy to lose track of what's going on.

Games in set theory are usually informally described by describing the rules of the two players and the winning condition.

In the prequel I sketched a proof of how determinacy hypotheses could imply the measurability of both delta^1_1 and delta^2_1 inside L(R).

The _core model_ K is, among other things, great for providing lower bounds for the consistency strength of some interesting theory.

In my previous posts I provided a sketch of how a measurable above a limit of Woodins implies that AD holds in L(R).

The Hahn-Banach Theorem in functional analysis is the theorem saying more or less that normed vector spaces have many bounded linear functionals.

This post is a bit different, as it's somewhat more of a curiosity I recently noticed.

This is the third post in my series on determinacy from Woodins.

This is a continuation of my last post on determinacy, where we began the proof of projective determinacy.

I'm amazed by the history of determinacy.

There are dozens of determinacy axiom around, with Sargsyan's "AD_R + Theta is regular" axiom perhaps being the most well known.

A sentence of type Sigma^2\_1 is a sentence with a second-order existential quantifier as the only second-order quantifier.

In the last post we developed the machinery of generic ultrapowers, which enabled us to go from a uniform normal ideal in the universe to getting an elementary embedding from the universe, assuming th existence of a Woodin cardinal.

The stationary tower is a forcing notion developed by Woodin in the 80's which, assuming there exists a Woodin cardinal, provides us with an elementary embedding into a subset M of an generic extension of the universe which is closed under \

Having introduced the Jónsson cardinals, we now try to investigate how these cardinals relate to the core model K.

Jónsson cardinals are some really strange creatures.

This post finishes the proof of Woodin's genericity iteration theorem, assuming the results related to the extender algebra, introduced in the previous post.

This post introduces Woodin's invention of genericity iterations, which more or less states that any real number is at most an iteration and a forcing away!