What is K?

The core model K is, among other things, great for providing lower bounds for the consistency strength of some interesting theory. For instance, every Jónsson cardinal is Ramsey in K, showing that the two are equiconsistent. Usually there are a lot of different K’s with different properties, sometimes denoted by things like K^{DJ}, K^{Steel} or something along those lines. This is my attempt at (briefly) explaining what K is, and what the differences between the various versions are. Here’s an overview of the various core models I’ll cover:

Continue reading

From Determinacy to a Woodin I

In my previous posts I provided a sketch of how a measurable above a limit of Woodins implies that \textsf{AD} holds in L(\mathbb R). The “converse”, saying that \textsf{AD} implies that there is a model with infinitely many Woodin cardinals, is a lot more complicated. I will try to simplify a lot of these complications here, to give an idea of what is going on. I will only focus on showing the existence of a single Woodin (for now), where the Woodin in question will be \Theta^{L(\mathbb R)} inside of \text{HOD}^{L(\mathbb R)}. As always, I will be very sketchy in this blog post, but provide more details in my note.

Continue reading

Hahn-Banach sans Zorn

The Hahn-Banach Theorem in functional analysis is the theorem saying more or less that normed vector spaces have many bounded linear functionals. The theorem is usually proven via Zorn’s lemma, giving the impression that Hahn-Banach uses the full power of choice. I’ll here give a proof based on the ultrafilter lemma, that every filter can be extended to an ultrafilter. One of the benefits of this approach, besides relying on a (strictly) weaker assumption, is that we get a little more information about how the functionals are constructed. The idea is that the ultrafilters allow us to “glue” functionals together. A latex’ed version can be found here.

Continue reading

Choiceless non-free algebras

This post is a bit different, as it’s somewhat more of a curiosity I recently noticed. It’s a theorem of \textsf{ZFC} 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

\textsf{ZF}+\textsf{F}:\equiv\textsf{ZF}+"\text{There is an algebraic type which does not admit free algebras}"

has considerable consistency strength. It consistency-wise implies \textsf{AD}^{L(\mathbb R)} and is consistency-wise implied by a proper class of strongly compacts.

Continue reading

Determinacy from Woodins I

I’m amazed by the history of determinacy. As soon as classical descriptive set theorists found the impact the determinacy of games has on regularity properties of sets of reals, a sophisticated program began the goal of which was to characterise the strength of determinacy. The fact that \bf\Delta^1_2-determinacy seemed like an unreachable statement at the time is incredible, until it culminated with Woodin’s 1979 result that \textsf{AD}^{L(\mathbb R)} follows from the incredibly strong hypothesis \bf I_0, after which he isolated the Woodin cardinal as a variant of a Shelah cardinal and proved the well-known equiconsistency result between \textsf{AD} and infinitely many Woodins. For a more detailed historical exposition I can highly recommend [Larson2010].

I’m dedicating a few blog posts to giving an idea of how some of these later results are proven. As some of the proofs are incredibly long and technical, my goal is to give the main ideas and strategies of the proofs, intended to the set theorist who might be interested in what key ideas the determinacy crowd are using. My plan is to accompany most proofs with pdf notes in which I’m writing out the proofs with all the technical details. A tentative plan is to cover:

  1. \textsf{PD} from infinitely many Woodins and a measurable above;
  2. \textsf{AD}^{L(\mathbb R)} from infinitely many Woodins and a measurable above;
  3. The equiconsistency of \textsf{AD} with infinitely many Woodins.

The first result is due to Martin-Steel and the last two are due to Woodin. We’ll start by focusing on the first result.

Continue reading