# An exceptional review about inner model theory

Today I’m making two blog posts about exceptional reviews: one review of a book and one of a paper.  This post is about Grigor Sargsyan‘s exceptional review of a paper:  John Steel‘s chapter, An outline of inner model theory, in the Handbook of Set Theory edited by Matthew Foreman and Akihiro Kanamori [MR2768678].  The other post is about Harald Helfgott‘s review of Terry Tao‘s book Expansion in finite simple groups of Lie type

MR2768698
Steel, John R.(1-CA)
An outline of inner model theory. Handbook of set theory. Vols. 1, 2, 3, 1595–1684, Springer, Dordrecht, 2010.
03E45 (03E15 03E35 03E55 03E60)

It is a well-documented phenomenon that the set-theoretic universe becomes more complex as one assumes stronger and stronger large cardinal axioms. For instance, the existence of a measurable cardinal implies that the universe is far away from the constructible universe $L$, and this can be measured via objective means. If there is a measurable cardinal then the constructible universe computes no successor cardinal correctly and, in particular, the set of reals of the constructible universe is countable.

There are other ways in which the universe becomes complex under the assumption that it contains large cardinals. For example, the existence of infinitely many Woodin cardinals implies that there is no projective well-ordering of the reals. Or the existence of a supercompact cardinal implies the failure of the square principle above it.

On the other hand, it is known that many large cardinals can exist in universes that are well understood and permit detailed analysis, universes that after being analyzed and thoroughly studied are not so complex anymore. For example, it is a well-known fact that, assuming the existence of a measurable cardinal, there is a minimal transitive model that contains all ordinals and has a measurable cardinal. It is important to note that by minimal we do not mean “minimal under inclusion” but rather that it generates all the others, or is embedded into all the others. This minimal model has the form $L[\mu]$ where $L[\mu]\models “\mu$ is a normal $\kappa$-complete ultrafilter on $\kappa$ for some $\kappa$”. It was shown by K. Kunen that any other model of this form is obtained from the minimal one by iterating the ultrapower construction. The modern representation of $L[\mu]$ as a mouse is somewhat more involved but has the additional advantage that it permits a complete analysis similar to the fine structural analysis of $L$ carried out by R. B. Jensen in his seminal paper “The fine structure of the constructible hierarchy” [Ann. Math. Logic 4 (1972), 229–308; erratum, ibid. 4 (1972), 443; MR0309729].

Similar facts are known for large cardinals up to the level of a Woodin cardinal that is itself a limit of Woodin cardinals and somewhat past it. Showing that such minimal and canonical models exist for all large cardinals is the goal of inner model theory.

Any substantial theory needs test problems and unexplained phenomena that lead the way to further developments. Resolving these test problems and explaining the unresolved phenomena is what keeps us, the practitioners and builders of the theory, honest in our pursuit. Otherwise, in the case of inner model theory, why not just declare $V$ as canonical and be done with it? Inner model theory has both test questions that it tries to resolve and unexplained phenomena that it tries to elucidate.

The inner model-theoretic approach to building minimal and canonical models for large cardinals has been used to calibrate lower bounds of set-theoretic statements. For instance, it is the most successful method for deriving strength from the Proper Forcing Axiom, which has been conjectured to have large cardinal strength—that of a supercompact cardinal. The problem of whether the consistency of the Proper Forcing Axiom implies the consistency of ${\sf{ZFC}}\,+\,$“there is a supercompact cardinal” is one of the main test questions of inner model theory.
The existence of canonical inner models for large cardinals explains our experience with determinacy. For instance, projective determinacy is true because for each $n$ there is a canonical inner model, a mouse, with $n$ Woodin cardinals. In fact the two theories, projective determinacy and the theory ${\sf{ZFC}}\,+\,$”  for each $n$ there is a mouse with $n$ Woodin cardinals”, are equiconsistent. Similar results hold for more complex forms of determinacy and more complex canonical inner models. In a sense, the goal of inner model theory is to show that all determinacy is a consequence of the existence of mice with large cardinals and vice versa. That this is true is the content of the Mouse Set Conjecture.

In short, inner model theory is simply great. However, anyone who has any experience with it can say that learning it is tough. The chapter under review (in what follows, the Outline), written by John R. Steel, one of the most experienced and accomplished inner model theorists, makes the experience of learning it significantly easier. I don’t know how students who came before me, before the time of the Outline, learned inner model theory. For me, learning this subject simply means developing a thorough understanding of every section of the Outline. It is not so clear to me what the meaning of “learning inner model theory” would be in a world that is not blessed with this chapter.
The goal of the author is clear; he states it in the second paragraph. He is setting out to “present the theory in the greatest possible generality known at the time”. This goal is too grand, yet the author accomplishes it with as few shortcomings as is possible. It is the best exposition of inner model theory available, and I highly recommend it to both students and researchers.

Since we have been glorifying inner model theory and the Outline, it is fair that we start with some of the Outline’s shortcomings. There is simply no quick fix, no magic potion that we take and after that we know inner model theory. Anyone who aims to present the theory in the “greatest possible generality” must make some concessions.

The first concession made is the amount of background material that is assumed. There is a lack of standard graduate textbooks in set theory, and so it is hard to say what the minimum required level is for reading the Outline with a relative ease. To read the later sections, one certainly has to be familiar with forcing. While notions such as extenders, elementary embeddings and Woodin cardinals are introduced in the chapter, I do think one has to have a prior knowledge of these topics. The introduction to extenders is too quick and there are no exercises, so while theoretically it is possible to just learn extenders from the chapter and ambitiously continue, I think those who have worked out some exercises on extenders and have read the relevant chapters of the books by T. J. Jech [Set theory, the third millennium edition, revised and expanded, Springer Monogr. Math., Springer, Berlin, 2003; MR1940513] and A. Kanamori [The higher infinite, second edition, Springer Monogr. Math., Springer, Berlin, 2003; MR1994835] will have a better understanding of the material.

The next basic shortcoming of the chapter is the lack of a historical introduction. In the modern world of mathematical publishing it is taboo to write papers without an appropriate introduction. The author does so and he gets away with it perfectly, as what he presents in the chapter has a far greater value than conforming to certain unjustified taboos. Nevertheless, he pays a relatively high cost for this.

The chapter is simply not accessible to those who don’t already know what inner model theory is about. The author, however, knows that his intended audience has this knowledge. After all, what follows a one-page Introduction is a section that introduces premice, the main object studied by inner model theorists. The reasons behind our endless search for canonical models for large cardinals among the models of the form $L[\vec{E}]$ are not given. Fine structural notions are not motivated, and the motivation behind the definition of fine extender sequence is not given (Definition 2.4). The non-sophisticated reader, one who has not acquired the proper background for reading the chapter, might have the feeling that one sunny day these notions just fell down from the sky.

If our ambitious reader, the one who ignored the warning to become more sophisticated elsewhere, made it past the initial hurdles, it is hard to see how they could get past Definition 2.7. Here three types of premice are introduced without any prior warning. The author makes an attempt to encourage our ambitious reader to venture on by saying that these distinctions are not important in what follows and only appear in technical details suppressed by the Outline. This is indeed true, so if you are the ambitious reader with no or limited prior knowledge of inner model theory don’t give up yet; however, there will be more of this. For example, wait until you reach the point where you are suddenly asked to go and read about the $r\Sigma_n$ hierarchy somewhere else (prior to Definition 2.20). Let me stop this. I simply do not recommend the Outline to anyone who wants to acquire practical knowledge of inner model theory and wants to use the Outline as their first source. In all fairness to the author, it is indeed the case that if you work out the courage to ignore some of the technicalities, then you can work through the Outline. But there is a better, more productive way to use the chapter.

The following is a reasonable way of approaching the Outline. First one should master forcing and basic large cardinal theory. The excellent books by Kunen [Set theory, reprint of the 1980 original, Stud. Logic Found. Math., 102, North-Holland, Amsterdam, 1983; MR0756630; Set theory, Stud. Log. (Lond.), 34, Coll. Publ., London, 2011; MR2905394], Jech [op. cit.] and Kanamori [op. cit.] can be used for this goal. One also needs to have an understanding of the theory of $0^\sharp$ and the theory of $L[\mu]$. Again, the texts of Jech and Kanamori can be used for this. A reader of the Outline who expects to understand the material should in addition to the aforementioned topics also work through Jensen’s “Fine structure of the constructible universe” [op. cit.]. One also needs to be motivated. The Introduction of the Outline contains excellent references that deal with history and motivations. I highly recommend Jensen’s “Inner models and large cardinals” [Bull. Symbolic Logic 1 (1995), no. 4, 393–407; MR1369169]. At this point one can start reading the Outline with relative ease.

I would also recommend fragments of “Fine structure and iteration trees” by W. J. Mitchell and Steel [Lecture Notes Logic, 3, Springer, Berlin, 1994; MR1300637] and “A proof of projective determinacy” [J. Amer. Math. Soc. 2 (1989), no. 1, 71–125; MR0955605] and “Iteration trees” [J. Amer. Math. Soc. 7 (1994), no. 1, 1–73;MR1224594] by D. A. Martin and Steel. One can also read more modern expositions, for instance “Fine structure” by R.-D. Schindler and M. Zeman [in Handbook of set theory. Vols. 1, 2, 3, 605–656, Springer, Dordrecht, 2010; MR2768688] or “Determinacy in $L({\Bbb R})$” by I. Neeman [in Handbook of set theory. Vols. 1, 2, 3, 1877–1950, Springer, Dordrecht, 2010; MR2768701]. See also “A recommended roadmap into inner models”, a question by Asaf Karagila at MathOverflow (http://mathoverflow.net/questions/73075/). Then everything in the Outline will make perfect and clear sense.

Sections 1–6 of the Outline are taken from various papers. Nevertheless, the Outline offers a short and succinct presentation of these topics, and often it communicates ideas that cannot be found elsewhere. For instance, the Outline contains a sketch of the proof of Condensation that has all the main ideas clearly presented. In Section 6, the proof of the Branch Existence Theorem is left out and instead it is explained how it leads to iterability via unique branches. This is also where the all too useful zipper argument is presented (Theorem 6.10). Also, the idea of $\mathcal{Q}$-structures is introduced and it is shown how it leads to uniqueness of branches (Corollary 6.14). I am unaware of any other publication prior to the Outline that makes a strong case for the idea that the good branches must be identified via good $\mathcal{Q}$-structures. This is one of the ideas that invites descriptive set theory into inner model theory. One starts looking for such canonical $\mathcal{Q}$-structures inside the largest countable set operators associated with nice pointclasses.

Section 2 introduces premice and fine structure. Section 3 introduces iteration trees and proves the Comparison Lemma (Theorem 3.11). Section 4 contains technical yet important lemmas that are used in showing that the various constructions of mice converge. Here one can learn about the Dodd-Jensen lemma and the weak Dodd-Jensen lemma. These are important results used in the proofs of various fine structural facts such as universality and solidity, which are the topic of Section 5. Section 6 deals with the constructions producing mice, the $K^c$ constructions. Here everything introduced up to this point comes together. The convergence of the $K^c$constructions is reduced to the iterability conjecture (Conjecture 6.5), which says that the countable substructures of the models appearing in a $K^c$ construction are iterable.

One shortcoming is that there is no discussion of the different background conditions that can be used in $K^c$ constructions. Varying such conditions produces different models, and in many applications the countably certified $K^c$ defined in the Outline is not good enough. Another shortcoming is that there is no review of core model theory (for a quick introduction to this topic, the reader may consult the paper by B. Löwe and Steel [in Sets and proofs (Leeds, 1997), 103–157, London Math. Soc. Lecture Note Ser., 258, Cambridge Univ. Press, Cambridge, 1999; MR1720574]). Perhaps the biggest disappointment is that there are no proofs of iterability anywhere in these sections. However, the Outline should be viewed as a section of the Handbook of set theory edited by M. Foreman and Kanamori. Here it is published alongside Neeman’s “Determinacy in $L({\Bbb R})$” [op. cit.], E. Schimmerling’s “A core model toolbox and guide” [in Handbook of set theory. Vols. 1, 2, 3, 1685–1751, Springer, Dordrecht, 2010; MR2768699], Schindler and Zeman’s “Fine structure” [op. cit.] , P. D. Welch’s “$\Sigma^*$ fine structure” [in Handbook of set theory. Vols. 1, 2, 3, 657–736, Springer, Dordrecht, 2010; MR2768689], and W. J. Mitchell’s “Beginning inner model theory” [in Handbook of set theory. Vols. 1, 2, 3, 1449–1495, Springer, Dordrecht, 2010; MR2768696] and “The covering lemma” [in Handbook of set theory. Vols. 1, 2, 3, 1497–1594, Springer, Dordrecht, 2010;MR2768697]. These works together contain most of what is left out by the Outline. In particular, Schimmerling’s chapter contains a guide to core model theory and Neeman’s chapter contains a proof of iterability.

The highlight of the Outline is its last two sections, sections 7 and 8, which can be viewed as an introduction to descriptive inner model theory [see G. Sargsyan, Bull. Symbolic Logic 19 (2013), no. 1, 1–55; MR3087400]. This is a subject that lies in the crossroads of descriptive set theory and inner model theory. The two subjects meet in several ways. Often descriptive set-theoretic methods are used to show that the $\omega_1$-iteration strategies that we build are universally Baire, and hence, generically absolute. The aforementioned $\mathcal{Q}$-structure idea is one way of showing that $\omega_1$-iteration strategies are universally Baire. For example, if $V$ is closed under the sharp function, $x\mapsto x^\sharp$, and an iteration strategy is guided by $\mathcal{Q}$-structures whose descriptive set-theoretic complexity is below the $\sharp$ operator, then the strategy in question is universally Baire.

Often inner model theory is used to analyze the structure of the universe under the assumption of ${\sf{AD}}$ or ${\sf{AD}}^+$. A typical example is the computation of the inner model ${\rm HOD}$ of models of determinacy. It is a long-standing open problem whether the ${\rm HOD}$ of models of ${\sf{AD}}^++V=L({\wp}({\Bbb R}))$ satisfies ${\sf{GCH}}$. It is known that it satisfies ${\sf{CH}}$. There has been a great deal of progress on this question using methods from inner model theory.

The goal of the last two sections of the Outline is to introduce two important topics of descriptive inner model theory. The first one is mouse capturing. This is the statement that under ${\sf{AD}}^++V=L({\wp}({\Bbb R}))\,+\,$ “there is no mouse with a superstrong cardinal”, for $x, y\in {\Bbb R}$, $x$ is ordinal definable from $y$ if and only if $x$ is in some mouse over $y$.

The Outline only deals with mouse capturing in $L({\Bbb R})$ under the additional assumption that $\mathcal{M}_\omega$ exists. This is not the optimal hypothesis. W. H. Woodin has shown that mouse capturing in $L({\Bbb R})$ is a consequence of ${\sf{AD}}^{L({\Bbb R})}$, and also that the set-theoretic strength of ${\sf{AD}}^{L({\Bbb R})}$ is weaker than that of the existence of $\mathcal{M}_\omega$. The proof of this result can be found, for instance, in “A theorem of Woodin on mouse sets” by Steel [in Ordinal definability and recursion theory: The Cabal Seminar, Volume III, 243–256, Cambridge Univ. Press, 2016].

The first part of Section 7 contains the proof of the first half of mouse capturing in $L({\Bbb R})$. In general, this portion of mouse capturing is just a consequence of comparison, but the author wants to prove more, namely that the ordinal definable reals are exactly those that are in $\mathcal{M}_\omega$. For this one needs techniques that track fragments of the iteration strategy of $\mathcal{M}_\omega$ inside $L({\Bbb R})$. Definition 7.7 introduces weak iterability, and after some discussion a conclusion is drawn that weak iterability is absolute between $V$ and $L({\Bbb R})$ (Theorem 7.8). It is then shown that under $V=L({\Bbb R})+{\sf{AD}}$, weak iterability is equivalent to iterability. This then naturally leads to the fact that every initial segment of $\mathcal{M}_\omega|\omega_1^{\mathcal{M}_\omega}$ is iterable in $L({\Bbb R})$.

The second part of Section 7 contains the proof of the second part of mouse capturing, namely that the ordinal definable reals are only those that are in $\mathcal{M}_\omega$ (Corollary 7.18). This part of the proof uses a great deal of sophisticated machinery. First, Woodin’s extender algebra and genericity iterations are introduced (Theorem 7.14). Genericity iterations are one of the most important techniques of descriptive inner model theory. Given an iterable transitive model $M$with a Woodin cardinal $\delta$ and a real $x$, it is possible to iterate $M$ to make $x$ generic over the iterate.

Such iterations can be dovetailed together to obtain a representation of $L({\Bbb R})$ as a symmetric extension of some iterate of $\mathcal{M}_\omega$, a very useful fact (Theorem 7.15). This fact, coupled with homogeneity of the Lévy collapse, easily implies that every ordinal definable real is in $\mathcal{M}_\omega$. At this point it would have been nice to explain Woodin’s derived model theorem but the author refers to his paper “A stationary-tower-free proof of the derived model theorem” [in Advances in logic, 1–8, Contemp. Math., 425, Amer. Math. Soc., Providence, RI, 2007; MR2322359] for a complete proof of this result that only uses inner model-theoretic techniques.

Section 7 ends with the Mouse Set Conjecture (see the discussion after Definition 7.23), one of the modern driving forces of inner model theory. The Mouse Set Conjecture states that mouse capturing holds. The last sentence of Section 7 is rather dramatic. It expresses the author’s belief that it is impossible to build mice with superstrong cardinals without first proving the Mouse Set Conjecture. This is a belief that has greatly influenced a generation of inner model theorists that grew up reading the Outline.

The second important topic is the computation of ${\rm HOD}$ of models of ${\sf{AD}}^++V=L({\wp}({\Bbb R}))$. Section 8 is devoted to it. Assume $\mathcal{M}_\omega$ exists. Let $\mathcal{H}={\rm HOD}^{L({\Bbb R})}$. It was known to the Cabal group that ${\sf{CH}}$ holds in $\mathcal{H}$, and whether ${\sf{GCH}}$ holds in $\mathcal{H}$ was left open. Motivated by the result that the reals of $\mathcal{M}_\omega$ are exactly those that are in $\mathcal{H}$, Steel and Woodin started investigating the structure of $\mathcal{H}$. Soon Steel showed that, assuming $\mathcal{M}_\omega$ exists, $V_\Theta^{\mathcal{H}}$ is a mouse, and Woodin showed that it is in fact an initial segment of an iterate of $\mathcal{M}_\omega$. Here $\Theta$ is the successor of the continuum. Woodin also computed the full $\mathcal{H}$ by showing that it has the form $L(V_\Theta^{\mathcal{H}}, \Sigma)$, where $\Sigma$ is some fragment of the iteration strategy of $V_\Theta^{\mathcal{H}}$. It is then easy to conclude that $\mathcal{H}\models {\sf{GCH}}$.

Let $\delta$ be the least cardinal such that $L_\delta({\Bbb R})$ is a $\Sigma_1$-elementary substructure of $L({\Bbb R})$. Section 8 contains a proof of the fact that $\mathcal{H}\cap V_{\delta}^{L({\Bbb R})}$ is a mouse (Theorem 8.20). A discussion of other results mentioned above follows but the proofs of these results are not presented. Recently Steel and Woodin wrote a long paper [in Ordinal definability and recursion theory: The Cabal Seminar, Volume III, 257–346, Cambridge Univ. Press, 2016] on ${\rm HOD}$ of models of ${\sf{AD}}^+ + V=L({\wp}({\Bbb R}))$ that contains the proofs of all these facts. Section 8 ends with the result due to the author that in $L({\Bbb R})$, every regular cardinal below $\Theta$ is a measurable cardinal (Theorem 8.27).

There are several important techniques and notions that are used in modern inner model theory that are not covered by the Outline. For example, the theory of homogenously Suslin sets and universally Baire sets is not present, and there is no in-depth discussion of the derived model theorem [see J. R. Steel, in Logic Colloquium 2006, 280–327, Lect. Notes Log., 32, Assoc. Symbol. Logic, Chicago, IL, 2009; MR2562557]. Certainly these are topics that a student of inner model theory must master. Topics that were still under development at the time the Outline was written are also, naturally, not in there. For instance, the reader will not find much on the core model induction or hod mice in the Outline. However, all of these topics have received a fair treatment in other publications [see R.-D. Schindler and J. R. Steel, “The core model induction”, unpublished manuscript, available at wwwmath.uni-muenster.de/u/rds; J. R. Steel, J. Symbolic Logic 70 (2005), no. 4, 1255–1296; MR2194247; G. Sargsyan, Mem. Amer. Math. Soc. 236 (2015), no. 1111, viii+172 pp.; MR3362806], and the Outline does its job at informing the reader of the existence of a world beyond what it covers.

At the beginning of this review, I made the claim that learning inner model theory means mastering the topics covered by the Outline. Then it seemed that I self-contradicted by saying that many details or several topics are omitted. In the Outline what the reader has is exactly what the title promises, an outline of a subject and its main components up to the time it was written. Undoubtedly the reader will have to consult other sources to develop a deeper understanding of inner model-theoretic techniques and notions, but it is absolutely invaluable information to know what exactly to learn, and this is the job of the Outline. A subject that is as spread out through various publications as inner model theory would be close to impossible to learn if we were not blessed with this chapter. The Outline is a torch that guides our way through the labyrinth whose walls are made of extenders, iteration trees, universally Baire representations, mice, ${\rm HOD}$, Woodin cardinals and other unworldly creatures. The prize we get for going through the labyrinth is the knowledge of topics needed to do research in one of the most beautiful and deep areas of set theory.
{For the collection containing this paper see MR2768678.}

Reviewed by Grigor Sargsyan