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 axiom. After skimming through Sargsyan’s survey paper [Sarg2013] I encountered several variants of these “-axioms” and also an axiom called (in the paper it’s actually called , 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:
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 and the ones on the right include . 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 -determinacy and the existence of infinitely many Woodins is equiconsistent with . Moving one step up,we get to the statement that there is a cardinal which is a limit of Woodins and there exists a -strong cardinal below — in Figure 1 I’ve dubbed this a “limit of Woodins with a small strong below”. This is also called the Hypothesis, or the Hypothesis. It’s a result due to Woodin and Steel that this hypothesis is in fact equiconsistent to , where is the length of the Solovay sequence.
Proposition. The existence of a Woodin cardinal and a -strong cardinal below is consistency-wise stronger than a proper class of Woodins.
Proof. The idea is simply to iterate using the associated -strong embedding. As the resulting two models agree about , is still a Woodin cardinal in the target model, so iterating out of the universe we leave a proper class of Woodins behind. QED
In particular, as the Hypothesis is strictly stronger than a Woodin with a -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 Hypothesis. Woodin and Steel have shown (around 2009-2012) that this hypothesis is in fact equiconsistent to (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 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 -regular Hypothesis:
Definition. The -regular Hypothesis is the statement that there is a cardinal which is an inaccessible limit of Woodin cardinals and -strong cardinals and whenever is such that then there is some such that coheres .
See [Sarg2013, Definition 2.15 and the discussion just before] for a definition of coherence, and . It was then shown by Sargsyan and Zhu [Sarg2013, Theorem 2.18] that this hypothesis is equiconsistent to .
Moving further up on the determinacy side, we get a lot of -theories, which is proven in [Sarg2013, Lemma 2.6] to have the given (strict) ordering. We then get to :
Definition. The Largest Suslin Axiom (LSA) is the statement that holds and for some ordinal , and is the largest Suslin cardinal .
That is stronger than 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 .
With the result of Sargsyan and Trang at hand it is trivial that the top determinacy theory is stronger than 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 -iterable Woodin, where an iterable cardinal is a notion invented by Gitman in [Git2011, Definition 5.2]:
Definition. A cardinal is iterable if for every there is a transitive set of size , satisfying , having and an -measure on , which can be iterated through all the ordinals.
Starting off with an iterable Woodin and letting be a (code for an) extender sequence witnessing Woodinness, we can then find a -model in which 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 .
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 , which is the current best result on the descriptive side. Whether or not 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: is strictly below a Woodin limit of Woodins, shown by Sargsyan and Trang – I’ve reflected this in the diagram now.
- [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.