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:

large_cardinals — 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 $\textsf{LSA}$ . In [Sarg2013], Sargsyan conjectures that these two theories are in fact equiconsistent.

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.

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.