Research Statement Richard O. Ketchersid
My research concerns three closely related areas of set theory: canonical inner models for large cardinals, models of determinacy, and descriptive set theory.
Background Axiomatic Set Theory: Zermelo-Fraenkel (ZF) is the standard theory of sets. Mathematics can be developed in ZF, possibly together with the axiom of choice (ZFC). There are many natural statements that ZFC does not decide. Probably the best known example is Cantor’s Continuum Hypothesis (CH). Gödel built the first canonical inner model, the constructible universe L which contains the ordinals and in which all sets are definable from ordinal parameters. There is a canonical well-ordering of L and CH holds in L. Cohen later introduced the technique of forcing and used it to show that by adding a generic object to a model of set theory, one could arrange the failure of CH. A forcing extension of a model of set theory is akin to a field extension of a field, in it an object satisfying some desired property is added while the background theory is preserved. Most set theorists believe that they are studying the universe of sets rather than various models of ZFC. Thus it is reasonable to say CH is either true or false, but we do not know which at the moment. The universe of sets is denoted V . ZFC presents V as a hierarchy of sets with the ordinals as the spine. For a set x, rk(x) = sup{rk(y) : y ∈ x}, this is always an ordinal. The αth level of V is the set Vα = {x:rk(x) < α}, in particular, Vα+1 = Vα ∪P(Vα ). The first levels of V are: V0 = ∅, V1 = {∅}, V2 = {∅, {∅}}, and so on. ω is the first infinite ordinal (essentially ω = N) and Vω is the set of all hereditarily finite sets. The reals are “contained” in Vω+1 , hence R ∈ Vω+2 . ω1 is the first uncountable ordinal. Large Cardinals and the Consistency Strength Hierarchy: Since the time of Cohen and Gödel the methods of forcing and inner models have developed immensely. We define a partial order on theories, the consistency strength hierarchy, as follows: Let T and S be two theories and let S T mean that given a model of T one can produce a model of S. Say T and S are equiconsistent if T S and S T . Since L has a canonical well-order and given any model M of ZF one can build L inside M , it is clear from the above discussion that ZF is equiconsistent with ZFC + CH as well as ZFC + ¬CH. It is not hard to produce infinite descending chains and incompatible elements in the consistency hierarchy, but it is a surprising empirical fact that all natural extensions of ZFC are well-ordered. Any cardinal whose existence can not be proved in ZFC is considered a large cardinal, for example, κ is inaccessible if and only if Vκ is a model of ZFC and κ is regular, i.e., there is no γ < κ and f : γ → κ such that rng(f ) is cofinal in κ. Clearly, ZFC cannot prove the existence of such a κ. However, most large cardinal assertions indicate some
Richard O. Ketchersid
Research Statement
Background
non-rigidity of the universe of sets. For now call κ a large cardinal if there is an embedding j : V → M which preserves the truth of all statements involving parameters, j(κ) 6= κ and j κ = id κ, and where M is an inner model of V , i.e., M ⊂ V . Any such κ is inaccessible. It is a theorem due to Kunen, that there is no such j : V → V . The more agreement one has between V and M the stronger the large cardinal assertion. A few examples are in order: κ is measurable if Vκ+1 ⊂ M . κ is λ-strong if Vλ ⊂ Mλ . Clearly, κ is measurable if and only if κ is κ + 1-strong. However, if κ is κ + 2-strong, then there are many measurable cardinals below κ and thus being κ + 2-strong is a much stronger assertion than being measurable. A cardinal is strong if it is γ-strong for all γ. Woodin cardinals are key to my research, but are a bit technical to define here. In terms of consistency strength, they are stronger than strong cardinals; however, a Woodin cardinal itself need not even be measurable. A nice example indicating the efficacy of large cardinals in gauging consistency strength is given by: “ZF + All sets are Lebesgue measurable.” and “ZFC + There is an inaccessible cardinal.” are equiconsistent as are “ZFC + There is an extension of Lebesgue measure to all sets of reals.” and “ZFC + There is a measurable cardinal.” It follows that “ZF + All sets are Lebesgue measurable.” is strictly weaker than “ZFC + There is an extension of Lebesgue measure to all reals.” Without the large cardinals to compare these two statements, it would be difficult to see this. Determinacy and Descriptive Set Theory: For technical reasons it is useful to work with Baire space ω ω rather than the Euclidean reals. Since ω ω is homeomorphic to the irrationals, I will, nevertheless, refer to elements of Baire space as reals. For A a set of reals the infinite perfect information game G(A) is defined as follows: Players I and II play elements of ω one at a time with the resulting play x ∈ ω ω . x is a win for player I if x is in A. A strategy is a function σ : ω <ω → ω telling the player how to play. Notice that <ω a strategy is also essentially a real since ω (ω ) is homeomorphic to ω ω . A is determined if one of the players has a winning strategy in the game G(A). Gale and Stewart proved that all open games are determined and later Martin proved that all Borel sets are determined. A pointclass Γ ⊂ P(R) is a collection of sets of reals closed under continuous preimages. For A ⊆ R2 , define the projection of A by x ∈ ∃R A ↔ ∃y A(x, y). For a pointclass Γ, A ∈ ∃R Γ ↔ A = ∃R B for some B ∈ Γ. If Γ is the class of closed sets, then its projection contains all Borel sets and more. Determinacy for pointclasses yields regularity properties for sets in the projections of these pointclasses, i.e., Lebesgue measurability, property of Baire, and so on. Hence, open determinacy already suffices to establish the regularity properties for projections of closed sets. The axiom of determinacy (AD) is the statement that all sets are determined. AD implies that all sets are Lebesgue measurable and hence contradicts the axiom of choice. Thus, it is not a statement studied as an axiom to be adopted, rather one studies inner models of determinacy. In particular inner models of the form L(Γ, R) where Γ ⊆ R. Here for any set X one can build a canonical inner model L(X) where essentially every element is definable -2-
Richard O. Ketchersid
Research Statement
Research Interests
from elements of X and ordinals. I will use ADL(Γ,R) to mean that AD holds in the inner model L(Γ, R). Studying ADL(R) is quite reasonable, since the existence of infinitely many Woodin cardinals and a measurable cardinal above them actually implies ADL(R) . In terms of consistency strength, “ZF + ADL(R) ” is equiconsistent with “ZFC + There are infinitely many Woodin cardinals.” Both of these results are due to Woodin. Certain pointclasses are important to the structure of a model of AD. A tree on X is any T ⊆ X <ω such that s ∈ T → s n ∈ T for all n < lh(s). For T a tree on X, [T ] is the set of all infinite branches through T . Giving X ω the ω-product of the discrete topology on X, a set C ⊆ X ω is closed iff X = [T ] for some tree on X. If T is on X × Y , then ω the projection of T is p[T ] = {f ∈ xω : ∃g ∈ Y ω (f, g) ∈ [T ]}. (This is the same as ∃Y [T ] in the notation introduced previously.) A ⊆ R is κ-Suslin if A = p[T ] for a tree on ω × κ and Sκ is the pointclass of all κ-Suslin sets. So the ω-Suslin sets are just the projections S of the closed subsets of R × R. A cardinal κ is a Suslin cardinal if Sκ \ γ<κ Sγ 6= ∅. The Suslin sets essentially determine the structure of a model of determinacy. The point is that if A ⊆ R × R is Suslin, then A can be uniformized, this is a certain amount of choice and thus is at odds with AD. So the more Suslin sets there are, the stronger the model of determinacy. Inner Model Theory: Inner model theory studies cannonical inner models for large cardinal assertions. These inner models resemble Gödel’s L in many ways, however they may contain many large cardinals and hence resemble V as well. Typically one gets strength from combinatorial principles by proving the existence of a cannonical inner model for some large cardinal assertion. There is a deep connection between the theory of canonical inner models, descriptive set theory (particularly the Suslin sets), and determinacy which there is no room to go into here.
Research Interests My interests primarily center on structural properties of models of determinacy and in particular properties that arise as a result of the sets of reals being definable in various ways from sets of ordinals. For example if A is Suslin, then A = p[T ] where T is a tree on ω × κ for some κ. Many properties can be deduced about A from the fact that it is Suslin. For example, in the absence of an uncountable sequence of reals, A would be measurable, have the property of Baire, etc., moreover, if A ⊆ R × R, then A can be uniformized, so a little bit of choice is reintroduced. One might expect the reintroduction of some choice to strengthen the theory considerably and this intuition is correct. If all sets of reals are Suslin and AD holds, then in fact ADR holds, that is all games where the payoff is in Rω and the players play reals are determined. Recently I have been working primarily with properties that arise from the assumption that all sets are ∞-Borel, which essentially means those sets of reals contained in the smallest class containing the open sets and closed under wellordered unions and intersections as
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Richard O. Ketchersid
Research Statement
Research Interests
well as complementation. In the absence of AC one is required to define first the notion of ∞-Borel code, which is essentially a set of ordinals and then study those sets with codes. After a bit of work it turns out that we can take “A is ∞-Borel ” to mean, there is a set of ordinals, S, and a Σ1 formula of set theory, ϕ, so that A(x) ⇐⇒ L[S, x] |= ϕ(S, x). Clearly all Suslin sets are ∞-Borel, since if T is a tree witnessing A is Suslin, then A(x) ⇐⇒ L[T, x] |= Tx is illfounded. where Tx = {s : (x|i, s) ∈ T }. There are ∞-Borel sets that are not Suslin in models of AD, unless all sets are Suslin and hence ADR holds. Woodin came up with a method for converting ∞-Borel codes to Suslin representations, when possible. I have recently written a survey of these and related results [Ket09]. Andrés Caicedo and I have used the structure given by all sets being ∞-Borel together with additional assumptions, for example AD, AD+ , or simply the existence of certain measures on Pω1 (R) to study the cardinal structure in models various cannonical inner models containing the reals. In [CK] we show that under either of the assumptions (1) AD+ + V = L(P(R)) or (2) the existence of a fine measure on Pω1 (R) and V = L(S, R) for a set S ⊆ OR, for any pre-partial order (X, ≤) exactly one of the following hold: (1) X is the wellordered union of pre-chains. (2) There is a perfect set of ≤-incompatible classes and the quotient of X/E by the induced equivalence relation, E, embeds into (2α , ≤) for some α, (3) There is an embedding of R/E0 into X/E whose range is pairwise incompatible classes. In [CK10a] we extend the G0 dichotomy [KST99] to arbitrary sets in canonical models of AD+ . We show here that under AD+ + V = L(P(R)) for every graph G, either (1) G is colorable by ordinals, or (2) there is a homomorphism of G0 into G. This is an ongoing project which is fuelled by results from Ben Miller, in which we attempt to generalize Miller’s results to the AD+ setting using technology developed to work with ∞-Borel sets. In [CK10b] we use these same techniques to investigate the cardinal successors of R in canonical models of AD+ . -4-
Richard O. Ketchersid
Research Statement
Research Interests
Another area in which I work is concerned with using directed systems of countable canonical inner models, which I will call mice from here on. My investigations grew out of work in my dissertation. In [Ket00], motivated by work of Woodin, such systems are used to calculate HODM for an inner model, M , of AD+ + V = L(P(R)) containing R. This presents HODM as a mouse in a generic collapse of R to ω, the so called next mouse. This is a key step in using core model techniques to get large cardinal strength from combinatorial principles on small cardinals. The main result from [Ket00] is: Suppose NS S is ω1 -dense for a dense set of stationary S + CH +Ψ where Ψ is a, possibly strong, assumption on the generic embeddings. Then there is a pointclass Γ ⊆ P(R) such that L(Γ, R) |= “ AD+ + Θ < Θ0 ”. In terms of inner models, Woodin’s “Mouse Set Theorem” then yields mice for roughly a strong cardinal overlapping a Woodin cardinal, the exact conclusion is the existence of nontame mice. AD+ is a (potential) strengthening of AD, on L(R) there is no difference and it is an open question whether AD and AD+ are the same. This work is currently written up in a book on the core model induction [SS07]. Directed systems of mice were first used by Steel and Woodin to calculate HOD of various models of AD+ + V = L(P(R)) and thereby extend many theorems of classic descriptive set theory. This area of research has potential for explaining and extending the already known, but surprising, phenomena of determinacy. A canonical example is the fact, due to Steel, that for every regular κ, the ω-club filter is a κ-complete ultrafilter, a truly surprising fact. Steel’s proof both extends what was known before and explains why it is true in the sense that the measures are coming from measures in the models of the directed systems. At a high level, to each Suslin cardinal there corresponds a directed system that computes the determinacy model up to that Suslin cardinal. This is already a good reason to expect the study of these systems to be quite useful in the analysis of models of determinacy. The fact that there has already been a good deal of success with this technique strengthens this expectation. I am working on applying directed systems of mice to recast and extend work of Jackson on the structure of cardinals in L(R). For example, the Coding Lemma has as a corollary that every subset of ω1 is constructible from a real. Jackson’s work extends this corollary to show that every subset of ωn is constructible from a real together with the first n − 1 uniform indiscernibles. This extension does not follow directly from the coding lemma and seems to need some deeper analysis of the structure of ordinals below ωω . One project is to prove Jackson’s local coding for subsets of ωn using directed systems of mice. An issue that comes up immediately is how to code sets of ordinals in the directed systems. On ω1 the coding is straightforward. Jackson and I applied the technique to show that the functions f ∈ ω1ω1 living in nice inner models, those arising as limits of directed systems, form an initial segment under
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Richard O. Ketchersid
Research Statement
Research Interests
to give an analysis of the supercompactness measures in L(R) on Pω1 (δ). By combining systems for various Suslin cardinals, the supercompactness measures on Pκ (λ) may also be dealt with in this manner. Neeman has already taken initial steps in this direction [Nee]. I am also working on a variety of other set theoretic issues. There are many combinatorial statements whose consistency strength should be past the existence of non-tame mice, but for which no proof is known yet, for example, the failure of square at a singular strong limit. Similarly, there are combinatorial facts holding in L(R) under AD from which it is not known whether AD (consistently) holds in L(R). For example, does ΘL(R) being a regular limit cardinal in L(R) give ADL(R) ? These are all problems whose solution, most likely, will require a core model induction argument. In [KZ06], we showed that if Woodin’s iteration to make a real generic for the free extender algebra Bδ succeeds, then consistently there is a Woodin cardinal. It would be nice to improve this to show directly that if Bδ is δ-cc, then δ is a Woodin cardinal. In [KLZ07, KLZ10], we showed that from a stationary limit of stationary limits of Woodin cardinals, there is a homogeneous forcing B giving an embedding jG : V → M ⊂ V [G] with crit(jG ) = ω1 and M is ω-closed in V [G]. From the existence of such a B Steel has used the core model induction to get the consistency of “many” Woodin cardinals, via determinacy in L(R), L(R] ), etc. It would be nice to tighten up this gap in consistency strength. In [FKLM08], we showed that if a tree on R in a model M = L(Γ, R) of AD+ of height ω1 fails to have ω1 -branches, then there is a kind of absolute impediment (in M ) to the tree having ω1 -branches. Here Γ ⊆ P(R). This is then used in several applications involving the “∃uncountable ” quantifier. I have an ongoing project with Larson [KLZ09] looking into what effect adding a single ultrafilter on ω (a “minimal amount of choice”) to L(R) has on the (non-well-ordered) cardinal structure. This, of course, is part of a bigger project to understand what happens to the cardinal structure of L(R) when some choice is added in by forcing. Finally, as alluded earlier, I have ongoing work with Jackson involving applications of directed systems of mice in studying the “fine structure” of AD+ models.
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Richard O. Ketchersid
Research Statement
Research Interests
References [CK09]
Andrés Caicedo and Richard Ketchersid, A trichotomy theorem in natural models of AD+ , http://unixgen.muohio.edu/~ketchero/preprints, 2009.
[CK10a]
, A G0 -dichotomy in natural models of AD+ , http://unixgen.muohio. edu/~ketchero/preprints, 2010.
[CK10b]
, Successors of R, http://unixgen.muohio.edu/~ketchero/preprints, 2010.
[FKLM08] Ilijas Farah, Richard Ketchersid, Paul Larson, and Menachem Magidor, Absoluteness for universally Baire sets and the uncountable II, computational Prospects of Infinity, vol. 15, 2008, pp. 163 – 192. [JK07]
Steve Jackson and Richard Ketchersid, An initial segment property of inner models, http://math.boisestate.edu/~ketchers/ (in preparation), 2007.
[Ket00]
Richard Ketchersid, Toward ADR from the Continuum Hypothesis and an ω1 dense ideal, Ph.D. thesis, U. C. Berkeley, 2000.
[Ket09]
Richard Ketchersid, More structural properties of AD and AD+ , http:// unixgen.muohio.edu/~ketchero/preprints, 2009.
[KLZ07]
Richard Ketchersid, Paul Larson, and Jindřich Zapletal, Increasing δ21 and Namba-style forcing, J. Symbolic Logic 72 (2007), no. 4, 1372–1378. MR MR2371211 (2008i:03058)
[KLZ09]
Richard Ketchersid, Paul Larson, and Jindrich Zapletal, Borel cardinals and ramsey ultrafilters, in progress, 2009.
[KLZ10]
, Σ22 -maximality, homogeneity, and the stationary tower, in print, 2010.
[KST99]
A. S. Kechris, S. Solecki, and S. Todorcevic, Borel chromatic numbers, Adv. Math. 141 (1999), no. 1, 1–44. MR MR1667145 (2000e:03132)
[KZ06]
Richard Ketchersid and Stuart Zoble, On the extender algebra being complete, MLQ Math. Log. Q. 52 (2006), no. 6, 531–533. MR MR2282396 (2007k:03134)
[Nee]
Itay Neeman, Inner models and ultrafilters in L(R), http://www.math.ucla. edu/~ineeman/luminy1.pdf/.
[SS07]
Ralf Schindler and John Steel, The core model induction, http://www.math. berkeley.edu/~steel/publications/, 2007.
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