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:

woodin-superstrong
Figure 1. The interplay between large cardinals and determinacy axioms.

There might be a lot of links that I’ve missed, but this is at least a first attempt. All arrows in the diagram are the usual consistency implications, where I’ve labeled an arrow with a circle if the implication is strict. The hypotheses on the left-hand side all include \textsf{ZFC} and the ones on the right include \textsf{ZF}. Let me explain what these axioms are, and which results that constitute the arrows.

Starting from the bottom, we have the two well-known theorems due to Woodin that the existence of a Woodin cardinal is equiconsistent to \Delta^1_2-determinacy and the existence of infinitely many Woodins is equiconsistent with \textsf{AD}. Moving one step up,we get to the statement that there is a cardinal \kappa which is a limit of Woodins and there exists a <\kappa-strong cardinal \lambda below \kappa — in Figure 1 I’ve dubbed this a “limit of Woodins with a small strong below”. This is also called the \theta_0<\Theta Hypothesis, or the \Omega>0 Hypothesis. It’s a result due to Woodin and Steel that this hypothesis is in fact equiconsistent to \Omega>0, where \Omega+1 is the length of the Solovay sequence.

Proposition. The existence of a Woodin cardinal \delta and a \delta-strong cardinal \kappa below is consistency-wise stronger than a proper class of Woodins.

Proof. The idea is simply to iterate \kappa using the associated \delta-strong embedding. As the resulting two models agree about V_\delta, \delta is still a Woodin cardinal in the target model, so iterating \kappa out of the universe we leave a proper class of Woodins behind. QED

In particular, as the \Omega>0 Hypothesis is strictly stronger than a Woodin \delta with a \delta-strong below, the former is strictly stronger than a proper class of Woodins. Moving one step up we get the statement that there is a limit of Woodins and small strongs, also called the \textsf{AD}_{\mathbb R} Hypothesis. Woodin and Steel have shown (around 2009-2012) that this hypothesis is in fact equiconsistent to \text{AD}_{\mathbb R} (see [Sarg2013, Theorem 2.14]), giving us an equiconsistency arrow in the diagram.

Also, given a proper class of Woodins and strongs, we can simply construct a sequence of increasing interleaving Woodins and strongs, getting a limit of Woodins and strongs and in particular this limit also satisfies the \textsf{AD}_{\mathbb R} Hypothesis. As there is an inaccessible above this limit we can also prove the consistency of the statement, making this implication strict.

Moving further upwards we encounter the \Theta-regular Hypothesis:

Definition. The \Theta-regular Hypothesis is the statement that there is a cardinal \delta which is an inaccessible limit of Woodin cardinals and <\delta-strong cardinals and whenever \Gamma\subseteq\dot{\Gamma}^\delta_{\text{uB}} is such that \Gamma\models"\Theta\text{ is singular}" then there is some \kappa<\delta such that \kappa coheres \Gamma.

See [Sarg2013, Definition 2.15 and the discussion just before] for a definition of coherence, \dot\Gamma^\delta_{\text{uB}} and \Gamma\models\varphi. It was then shown by Sargsyan and Zhu  [Sarg2013, Theorem 2.18] that this hypothesis is equiconsistent to \textsf{AD}+\Theta\text{ is regular}.

Moving further up on the determinacy side, we get a lot of \Theta-theories, which is proven in [Sarg2013, Lemma 2.6] to have the given (strict) ordering. We then get to \textsf{LSA}:

Definition. The Largest Suslin Axiom (LSA) is the statement that \textsf{AD}^+ holds and for some ordinal \alpha, \Theta=\theta_{\alpha+1} and \theta_\alpha is the largest Suslin cardinal <\Theta.

That \textsf{LSA} is stronger than \textsf{AD}_{\mathbb R}+\Theta\text{ is Mahlo} was shown by Kechris, Klienberg, Moschovakis and Woodin in [KKMW1981]. A very recent result by Sargsyan and Trang [SargTrang2016, Theorem 10.3.1] shows that a Woodin limit of Woodins is stronger than \textsf{LSA}.

With the result of Sargsyan and Trang at hand it is trivial that the top determinacy theory \textsf{AD}_{\mathbb R} + \textsf{HOD}\models\Theta\text{ is a Woodin limit of Woodins} is stronger than \textsf{LSA} as well. Sargsyan in [Sarg2013] also conjectured that this last theory is equiconsistent with a superstrong.

Going back to the large cardinal side we get various “hybrid Woodins”. At the bottom we have an iterable Woodin, or an \omega_1-iterable Woodin, where an iterable cardinal is a notion invented by Gitman in [Git2011, Definition 5.2]:

Definition. A cardinal \kappa is iterable if for every A\subseteq\kappa there is a transitive set \mathcal M of size \kappa, satisfying \mathsf{ZFC}^-, having \kappa,A\in\mathcal M and an \mathcal M-measure \mu on \kappa, which can be iterated through all the ordinals.

Starting off with an iterable Woodin \delta and letting \vec E\subseteq\delta be a (code for an) extender sequence witnessing Woodinness, we can then find a \mathsf{ZFC^-}-model \mathcal M in which \delta is a measurable Woodin. By iterating it out of the universe, we leave (many) Woodin limit of Woodins behind and end up in a model of full \mathsf{ZFC}.

As for the other arrows in the diagram, in [Git2011] it’s shown that Ramseys are iterable and by definition Ramseys are Jónsson. Measurables are also Ramsey by Rowbottom’s theorem. The last steps involve the notion of Hyper-Woodins and Shelahs, where hyper-Woodins were invented by Schimmerling [Schim2002], in which he also showed the given ordering in Figure 1.

One thing to note is the rather peculiar state of a Jónsson Woodin – I’m at least not aware of any upper bound except the trivial Ramsey Woodin one. Even though Jónssons and Ramseys are equiconsistent, a result that is due to Mitchell [Mit1999], Jónssons have a lot lower actual strength than Ramseys, in that they don’t even have to be regular.

As for the current status of inner model theory and descriptive inner model theory, Neeman [Nee2002] has built mice containing a Woodin limit of Woodins using “pure” inner model theoretic methods, which is the best result to date. Using descriptive inner model theoretic methods, Sargsyan and Trang [SargTrang2016] has produced certain hybrid mice satisfying \textsf{LSA}, which is the current best result on the descriptive side. Whether or not \textsf{LSA} is equiconsistent to a Woodin limit of Woodins or if it’s strictly weaker is not known at this moment, as far as I can tell.

EDIT 1: \textsf{LSA} is strictly below a Woodin limit of Woodins, shown by Sargsyan and Trang – I’ve reflected this in the diagram now.

References

  • [Git2011] Gitman, Victoria: “Ramsey-like Cardinals”, The Journal of Symbolic Logic, Volume 76, Number 2, 2011.
  • [Kanamori] Kanamori, Akihiro: “The Higher Infinite”, Second Edition, Springer-Berlin, 2009.
  • [KKMW1981] Kechris, Alexander S. & Kleinberg, Eugene M. & Moschovakis, Yiannis N. & Woodin, W. Hugh: “The axiom of determinacy, strong partition properties and nonsingular measures”. In Cabal Seminar 77-79, volume 839 of Lecture Notes in Math, pages 75-99. Springer-Berlin, 1981.
  • [Mit1999] Mitchell, W.J.: “Jónsson Cardinals, Erdös Cardinals, and the Core Model”, The Journal of Symbolic Logic, Volume 64, Number 3, 1999.
  • [Nee2002] Neeman, Itay: “Inner models in the region of a Woodin limit of Woodin cardinals”, Annals of Pure and Applied Logic, Volume 116, pages 67-155, 2002.
  • [Sarg2013] Sargsyan, Grigor: “Descriptive Inner Model Theory”, Bulletin of Symbolic Logic, Volume 19, Number 1, 2013.
  • [SargTrang2016] Sargsyan, Grigor & Trang, Nam: “The Largest Suslin Axiom”, June 2016, available online at http://www.math.uci.edu/~ntrang/lsa.pdf.
  • [Schim2002] Schimmerling, Ernest: “Woodin Cardinals, Shelah Cardinals and the Mitchell-Steel Core Model”, Proceedings of the American Mathematical Society, Volume 130, Number 11, 2002.