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.

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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.

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An overview of determinacy axioms

I’ve recently started to read up on descriptive inner model theory, and quickly stumbled across a lot of new axioms. Prime among these were Sargsyan’s \textsf{AD}_{\mathbb R} + \Theta\text{ is regular} axiom. After skimming through Sargsyan’s survey paper [Sarg2013] I encountered several variants of these “\Theta-axioms” and also an axiom called \textsf{LSA} (in the paper it’s actually called \textsf{LST}, but the terminology has changed since then). I decided to do some intense Googling and a little bit of thinking, and my current overview of these determinacy axioms and their relation to the large cardinal hierarchy looks like this: Continue reading

Sigma^2_1 absoluteness

A sentence of type \bf\Sigma^2_1 is a sentence of the form \exists X\subseteq\mathbb R\colon\psi(X,r), where r is some fixed real parameter and all the quantifiers occuring in \psi are ranging over the reals or naturals. A particularly famous such sentence is the continuum hypothesis \mathsf{CH}, which is known to be highly non-absolute: we can always both force \mathsf{CH} and force its negation. But it turns out that \mathsf{CH} actually turns out to determine the \bf\Sigma^2_1-truths in models, in that any two forcing extensions in which \mathsf{CH} holds have the same \bf\Sigma^2_1-truths. This is a theorem due to Woodin and Steel independently. My full write-up can be found here, but in this post I’ll just focus on the statement and the key ideas used in the proof(s).

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The Stationary Tower II – the construction

In the last post we developed the machinery of generic ultrapowers, which enabled us to go from a uniform normal ideal in V to getting an elementary embedding j:V\to\mathcal M with \text{crit }j=\omega_1^V lying in a generic extension of Vif we assume that we have a Woodin cardinal. The model \mathcal M furthermore enjoyed the property of being closed under countable sequences in the forcing extension. We now introduce a generalisation of such an embedding, which grants more closure — this is precisely given by the stationary tower.

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