Some challenges for the philosophy of set theory - Logic at Harvard

IntroductionA theorem of ShelahA theorem of FarahA theorem of Magidor and ShelahConclusion3 / 51Thesis II: Set theory is a branch of mathematics. It happens tobe a branch in which issues of independence and consistencylie close to the surface, and in which the tools of logic haveproven very powerful, but it is not (or not only) about logic. As inany successful field of mathematics, progress in set theorycomes from a dialectic between “internal” developmentselaborating the central ideas, and “external” developments inwhich problems and ideas flow back and forth between settheory and other areas (for example topology, algebra, analysis,the theory of games).

IntroductionA theorem of ShelahA theorem of FarahA theorem of Magidor and ShelahConclusion4 / 51Thesis III: Modern set theory uses a wide range of methodsand discusses a wide range of objects. What is more, from ahistorical perspective there have been quite radicaltransformations in our views about set-theoretic methods andobjects, and I see no reason to believe that this process oftransformation has reached its end. Accordingly we should besuspicious of arguments (especially those based onphilosophical views) that tend to prescribe or circumscribe theobjects or methods which are legitimate. A useful criterion for aphilosophical viewpoint is to ask: “Would this view, if takenseriously, have hampered or helped the development of settheory?”, and to compare it with the view that “Anything goes”.

IntroductionA theorem of ShelahA theorem of FarahA theorem of Magidor and ShelahConclusion5 / 51Thesis IV: Questions about the meaning and truth value of CHhave been mathematically fertile, and have given rise to a newarea in set theory (sometimes called “Set theory of thecontinuum”) with a whole corpus of ideas and techniques.Independence results are present in this area but are by nomeans the whole story. This area has taken on a life of its own,with its own initiatives and insights, and continues to flourish inthe absence of a solution to the Continuum Problem. In fact ifCH were settled positively it would be a blow to “Set theory ofthe continuum”, since a major part of the subject is a richstructure theory inconsistent with CH. In brief: there is more tothe continuum than the Continuum Problem.

IntroductionA theorem of ShelahA theorem of FarahA theorem of Magidor and ShelahConclusion6 / 51Thesis V: It is possible (I would even say quite likely) thatindependence is here to stay, and that questions like CH willnever be settled in a way that commands universal agreement.But we can learn to live with independence. In particularexperience in combinatorial set theory suggests how to reach amodus vivendi with the independence phenomenon.

IntroductionA theorem of ShelahA theorem of FarahA theorem of Magidor and ShelahConclusion7 / 51The rest of the talk consists of a series of case studies. I willdiscuss results by Shelah, Farah and Magidor-Shelah and usethem to defend and elaborate on Theses II-V.

IntroductionA theorem of ShelahA theorem of FarahA theorem of Magidor and ShelahConclusion8 / 51A cardinal fixed point is a cardinal κ such that κ = ℵ κ . Sincethe function κ ↦→ ℵ κ is continuous, there is a closed unboundedclass of such cardinals. The first one is the limit of theincreasing sequenceℵ 0 , ℵ ℵ0 , ℵ ℵℵ0 , . . .The following theorem appears in a paper by Saharon Shelah(“On power of singular cardinals”, Notre Dame Journal ofFormal Logic 27 (1986) 263–299).If ℵ δ is the ω 1 -th cardinal fixed point then ℵ ℵ 1δis less than the(2 2ℵ 1) + -th cardinal fixed point.

IntroductionA theorem of ShelahA theorem of FarahA theorem of Magidor and ShelahConclusion9 / 51The theorem would have made perfect sense to Cantor orHausdorff.Ingredients in the proof includeRamsey cardinals.Core models.The covering lemma.Generic elementary embeddings.Infinite games.

IntroductionA theorem of ShelahA theorem of FarahA theorem of Magidor and ShelahConclusion10 / 51Ramsey cardinals: A Ramsey cardinal is a cardinal κ such thatfor every function F : [κ]

IntroductionA theorem of ShelahA theorem of FarahA theorem of Magidor and ShelahConclusion11 / 51Core models: Gödel’s model L (the constructible universe) isthe least transitive model of ZFC containing all the ordinals, andenjoys many pleasant structural properties: for example theGeneralised Continuum Hypothesis (GCH) holds in L. Themodel L can accommodate small large cardinals, for example ifκ is inaccessible in V then κ is inaccessible in L.On the other hand measurable cardinals (or even Ramseycardinals) can not exist in L. Given a set of ordinals A, one canrelativise the construction of L and obtain L[A], the leasttransitive model which contains all ordinals and has A as amember: L[A] is “L-like above A”, for example above sup(A)GCH holds and there are no Ramsey cardinals.

IntroductionA theorem of ShelahA theorem of FarahA theorem of Magidor and ShelahConclusion12 / 51A major program in set theory (the “inner model program”) hasbeen the construction of L-like models which canaccommodate larger cardinals. An important early step was theconstruction by Dodd and Jensen of K (technically the“Dodd-Jensen core model” or K DJ , but this is the only version ofK needed here). K is (roughly speaking) the largest L-likemodel of “ZFC plus there is no measurable cardinal”. Themodel K shares many of the good properties of L, for exampleit is a model of GCH: but K is more hospitable to largecardinals, for example Mitchell showed that if κ is Ramsey thenκ is Ramsey in K. The construction can be relativised over aset of ordinals A to produce K[A], which has the same relationto K that L[A] has to L.

IntroductionA theorem of ShelahA theorem of FarahA theorem of Magidor and ShelahConclusion13 / 51The covering lemma: Jensen made a very detailed study of the“fine structure” of L and related models like L[A]. The model Lis the union of levels L α , and the fine structure program (for L)involves a careful study of definability over the models of formL α . Although fine structure appears quite formal and syntactic,there is a large mathematical payoff: the theory of core modelslike the model K discussed in the last paragraph, proofs of newstructural facts (squares, diamonds, morasses) which hold in Land K, and (most relevant for us) the covering lemma.

IntroductionA theorem of ShelahA theorem of FarahA theorem of Magidor and ShelahConclusion14 / 51The original form of the covering lemma states that eitherzero-sharp exists or covering holds between V and L, that is tosay for every uncountable set of ordinals X ∈ V there is Y ∈ Lsuch that X ⊆ Y and V |= |X| = |Y|. One can view this as adichotomy theorem, stating that V is either “far from L” or “closeto L”. In the covering case, the combinatorics of V (especiallyat singular cardinals) is quite L-like.We need a generalisation involving a set of ordinals A and astronger principle than zero-sharp: if there is no inner model of“ZFC + there exists a measurable cardinal” containing A as anelement, then covering holds between V and K[A] abovesup(A).

IntroductionA theorem of ShelahA theorem of FarahA theorem of Magidor and ShelahConclusion15 / 51Generic embeddings: Recall that a cardinal κ is measurable ifand only if it is the critical point of an elementary embeddingj : V −→ M ⊆ V, that is to say j ↾ κ = id and j(κ) > κ. A genericembedding is an embedding j defined in a generic extensionV[G] of V, with j : V → M ⊆ V[G]; the critical point of such anembedding can be a small regular uncountable cardinal suchas ω V 1, and the critical point can have reflection propertiesreminiscent of those which follow from measurability; the exactnature of these properties depends on the forcing extensionV[G] and the target model M.The existence of generic embeddings can be expressed interms of objects called precipitous ideals, and the existence ofprecipitous ideals can be formulated in terms of infinite games.

IntroductionA theorem of ShelahA theorem of FarahA theorem of Magidor and ShelahConclusion16 / 51Infinite games: Given a set X with |X| > 1, let ω X be the set ofall ω-sequences from X, and let A ⊆ ω X. We can view A asthe “payoff set” in an infinite game of perfect informationG(X, A) where two players (I and II) alternately play element ofX for ω steps to generate an ω-sequence ⃗x, and then I wins if⃗x ∈ A while II wins if ⃗x A. The notions of “winning strategy”for players I and II are defined in the obvious way, and the set Ais said to be determined if one player has a winning strategy.The axiom of choice implies that there are non-determinedgames, but the payoff sets constructed using choice tend not tohave very simple definitions. In fact it is a theorem of ZFC (dueto Martin) that games with Borel payoff sets are determined,and assuming the existence of large cardinals givesdeterminacy for various kinds of more complex payoff sets.

IntroductionA theorem of ShelahA theorem of FarahA theorem of Magidor and ShelahConclusion17 / 51After this long preamble we can finally outline the proof ofShelah’s result. Let µ = (2 2ℵ 1) + . For every A ⊆ µ we askwhether the following statement φ A holds: there is an innermodel M of “ZFC + there exists a Ramsey cardinal” such thatA ∈ M. Now there are two cases.

IntroductionA theorem of ShelahA theorem of FarahA theorem of Magidor and ShelahConclusion18 / 51Case one: There is A such that φ A fails. In this case we mayappeal to the theory of K[A] to see that covering holds above µ,and GCH holds in K[A] above µ, from which we may obtain thedesired conclusion.Case two: For every A the statement φ A holds. In this case wecan adapt a determinacy result due to Martin to show that thegood player always wins a certain “precipitousness game”,which in turn enables us to construct a certain genericelementary embedding whose existence implies the desiredconclusion.

IntroductionA theorem of ShelahA theorem of FarahA theorem of Magidor and ShelahConclusion19 / 51The proof of the theorem uses methods from a number ofdifferent areas, each with a complex history. To trace itsgenealogy in detail would be to write a good part of the historyof set theory and neighbouring areas in the twentieth century.Here are some very brief historical remarks.

IntroductionA theorem of ShelahA theorem of FarahA theorem of Magidor and ShelahConclusion20 / 51Partition calculus: This originated in work of Ramsey ondecidability problems for finite and countable structures, wasgreatly elaborated (notably by Erdös and his collaborators) inthe setting of uncountable cardinals, and plays a crucial role inthe theory of “small large cardinals” such as weakly compactcardinals and Ramsey cardinals.

IntroductionA theorem of ShelahA theorem of FarahA theorem of Magidor and ShelahConclusion21 / 51Infinite games of perfect information: These were first studiedby Banach and Mazur in connection with problems in analysis,were rediscovered by Gale and Stewart in the setting ofmathematical economics, and then became central objects indescriptive set theory and inner model theory.

IntroductionA theorem of ShelahA theorem of FarahA theorem of Magidor and ShelahConclusion22 / 51Fine structure: Jensen’s work on fine structure builds onGödel’s original work on L, but also on work in computabilitytheory and the study of weak set theories such as admissibleset theory.

IntroductionA theorem of ShelahA theorem of FarahA theorem of Magidor and ShelahConclusion24 / 51Generic elementary embeddings and precipitous ideals:Solovay introduced the study of generic embeddings in hiswork on the theory of saturated ideals. Jech and Prikryintroduced the notion of precipitous ideal and characterisedprecipitousness in terms of games.

IntroductionA theorem of ShelahA theorem of FarahA theorem of Magidor and ShelahConclusion25 / 51Thesis II: Set theory is a branch of mathematics.Shelah’s argument is recognisably the same kind of thing as ahard argument in number theory: Shelah combines partitioncalculus, infinite games, elementary embeddings and so forthto prove a purely combinatorial result in just the same way thata number theorist might combine complex analysis, algebraicgeometry and representation theory in order to prove a purelyarithmetic result.

IntroductionA theorem of ShelahA theorem of FarahA theorem of Magidor and ShelahConclusion26 / 51Thesis III: A useful criterion for a philosophical viewpoint is toask: “Would this view, if taken seriously, have hampered orhelped the development of set theory?”, and to compare it withthe view that “Anything goes”.Shelah’s result is a useful test case here: any viewpoint whichwould discourage interest in large cardinals, the fine structureof L, the theory of infinite games and so forth would apparentlymake it harder to prove certain theorems of ZFC about classicalobjects like cardinals and the continuum function.

IntroductionA theorem of ShelahA theorem of FarahA theorem of Magidor and ShelahConclusion27 / 51We saw here how the entry of new methods and new objects,new views of existing objects, and exchanges of ideas withother areas of mathematics led to progress on a classicalproblem. Returning to Thesis II, this kind of progression istypical of the evolution of a mathematical area. We could drawan analogy with the history of algebraic geometry, where thereformulation of the subject in terms of the highly abstracttheory of schemes led to progress on classical and concreteproblems about curves and surfaces.

IntroductionA theorem of ShelahA theorem of FarahA theorem of Magidor and ShelahConclusion28 / 51The following theorem appears in a paper by by Ilijas Farah(“All automorphisms of the Calkin algebra are inner”, Annals ofMathematics 173 (2011) 619–661)If Todorcevic’s axiom (TA) holds then all automorphisms of theCalkin algebra are inner.After some background I will discuss the ideas that go into thisresult, and then use it as an example of a piece of mathematicswhich is closely related to the continuum but is quite remotefrom the Continuum Problem.

IntroductionA theorem of ShelahA theorem of FarahA theorem of Magidor and ShelahConclusion29 / 51Let H be the complex inner product space l 2 (C) whoseelements are infinite sequences x = (x n ) of complex numberswith ∑ |x n | 2 < ∞ and whose inner product is given byx.y = ∑ n x ny n . Up to isomorphism H is the unique separableinfinite dimensional Hilbert space.B(H) is the space of continuous linear maps from H to H, witha norm ‖T‖ = sup{‖Tx‖ : ‖x‖ 1}. With this norm the spaceB(H) becomes a Banach space. B(H) is closed under thebilinear operation of composition, and with this operation B(H)becomes a Banach algebra.

IntroductionA theorem of ShelahA theorem of FarahA theorem of Magidor and ShelahConclusion30 / 51B(H) has more structure. For any T ∈ B(H) there is a uniqueT ∗ ∈ B(H) such that (Tx).y = x.(T ∗ y). Axiomatising theproperties of B(H) with the “involution” map T ↦→ T ∗ we obtainthe notion of a C ∗ -algebra, that is a Banach algebra equippedwith a map a ↦→ a ∗ satisfying certain identities.A basic theorem (the Gelfand-Naimark-Segal theorem) statesthat every C ∗ -algebra is isomorphic to a norm-closed and∗-closed subalgebra of B(K) for some Hilbert space K. Anothertheorem by Gelfand identifies the unital commutative C ∗ -algebras with spaces C(X) of complex-valued functions oncompact Hausdorff spaces X, and we can view the study ofnoncommutative C ∗ - algebras as a form of “noncommutativegeometry”.

IntroductionA theorem of ShelahA theorem of FarahA theorem of Magidor and ShelahConclusion31 / 51T ∈ B(H) is compact if the image of the unit ball of H under Thas compact closure. We denote the class of compactoperators by K(H); this is a C ∗ –algebra in its own right andforms an ideal in B(H), so we may form a quotient B(H)/K(H)and get the object known as the “Calkin algebra”. It isreasonable to think of the compact operators as “finitary” andso the Calkin algebra consists of “operators modulo finitaryperturbation”.

IntroductionA theorem of ShelahA theorem of FarahA theorem of Magidor and ShelahConclusion32 / 51In any C ∗ algebra with a 1, a unitary element is an element usuch that uu ∗ = u ∗ u = 1. If u is unitary then the map a ↦→ uau ∗is an automorphism, and automorphisms of this type are calledinner automorphisms. A classical question in C ∗ algebrasasked whether all automorphisms of the Calkin algebra areinner.Phillips and Weaver showed that under CH there is an outer(non-inner) automorphism.Farah closed the case by showing that under TA allautomorphisms are inner.

IntroductionA theorem of ShelahA theorem of FarahA theorem of Magidor and ShelahConclusion33 / 51Motivating analogies: the Boolean algebra Pω is analogous tothe Banach algebra B(H), the ideal of finite sets FIN isanalogous to the ideal K(H) of compact operators, and thequotient Pω/FIN is analogous to the Calkin algebraB(H)/K(H).

IntroductionA theorem of ShelahA theorem of FarahA theorem of Magidor and ShelahConclusion34 / 51Say that an automorphism π of Pω is trivial if it has the formA ↦→ f[A] where f is a permutation of ω, andf[A] = {f(n) : n ∈ A}. Since Pω is atomic with atoms thesingleton sets {n}, every automorphism is trivial.Similarly say that an automorphism π of Pω/FIN is trivial if ithas the form [A] FIN ↦→ [f[A]] FIN , where f is a bijection betweencofinite subsets of ω and [B] FIN is the equivalence class of Bmodulo finite: a classical question asks whether allautomorphisms of Pω/FIN are trivial.

IntroductionA theorem of ShelahA theorem of FarahA theorem of Magidor and ShelahConclusion35 / 51Rudin showed that if CH holds there is a non-trivialautomorphism of Pω/FIN. Shelah showed by a hard forcingargument that it is consistent (modulo the consistency of ZFC)that every automorphism of Pω/FIN is trivial, but the argumentis rather specific to Pω/FIN.Shelah and Steprans showed by a different approach that thestrong Proper Forcing Axiom (PFA) implies that everyautomorphism of Pω/FIN is trivial.

IntroductionA theorem of ShelahA theorem of FarahA theorem of Magidor and ShelahConclusion36 / 51Reflecting on the work of Shelah and Steprans, Velickovicshowed that the conjunction of MA and the “Todorcevic Axiom”TA implies that every automorphism of Pω/FIN is trivial.TA is a maximality principle which isolates someRamsey-theoretic consequences of strong forcing axioms; TAfollows from PFA but is known to be consistent relative to theconsistency of ZFC. Farah showed that TA implies that everyautomorphism of the Calkin algebra is inner.

IntroductionA theorem of ShelahA theorem of FarahA theorem of Magidor and ShelahConclusion37 / 51Thesis IV: There is more to the continuum than the ContinuumProblem.Farah’s result is formally an independence result (the existenceof an outer automorphism in the Calkin algebra is independentof ZFC) but this is a misleading frame in which to view it. It isnot proved by clever coding tricks or an elaborate forcingconstruction, but rather by infusing ideas from one field (the settheory of the continuum) into another (the theory of the Calkinalgebra). The proof is fertile for the theory of C ∗ -algebras, forexample it contains ideas which lead to a new and simpler proofof the Phillips-Weaver theorem. It has led to an exchange inwhich ideas and problems flow in both directions between areaswhich have been separated since the era of von Neumann.

IntroductionA theorem of ShelahA theorem of FarahA theorem of Magidor and ShelahConclusion38 / 51The powerful axiom PFA (and the large cardinals and elaborateforcing technology needed to show it is consistent) play no rolein the final result. However PFA was essential in thedevelopment of the circle of ideas which underlie Farah’s proof.

IntroductionA theorem of ShelahA theorem of FarahA theorem of Magidor and ShelahConclusion39 / 51In my last case study I will outline some work by Magidor andShelah (“When does almost free imply free?”, Journal of theAmerican Mathematical Society). I will use this as an exampleof how to make mathematical progress in a field (combinatorialset theory) which is beset by many issues of independence.This work is motivated by problems in group theory involvingfree groups, and is set in a very general framework. I willdescribe a simple case involving the transversal property forfamilies of sets.

IntroductionA theorem of ShelahA theorem of FarahA theorem of Magidor and ShelahConclusion40 / 51A transversal for a family X of non-empty sets is a 1-1 choicefunction on X, that is a function f such that f(A) ∈ A for allA ∈ X and A B =⇒ f(A) f(B). It follows rather easily fromthe compactness theorem in first order logic that if X is a familyof finite sets, and every finite subset of X has a transversal,then X has a transversal. As we will see, the situation forcountable sets is more complicated.

IntroductionA theorem of ShelahA theorem of FarahA theorem of Magidor and ShelahConclusion41 / 51Let S be a stationary subset of ω 1 , choose for each α ∈ S acofinal set x α ⊆ α, and let X = {x α : α ∈ S}. It is not hard to seethat every countable subset of X has a transversal. HoweverFodor’s lemma implies that X itself has no transversal. For aregular κ let R(κ) be the assertion “there is a family X of κmany countable sets such that X has no transversal, whileevery subset with size less than κ does have a transversal”. Weproved R(ℵ 1 ).

IntroductionA theorem of ShelahA theorem of FarahA theorem of Magidor and ShelahConclusion42 / 51R(κ) is a typical “incompactness” or “non-reflection” statement.It asserts that there is a family of sets with a property notshared by any small subset. Such statements are ubiquitous incombinatorial set theory: another example is “there exists aκ-Aronszajn tree”.Incompactness properties tend to follow from V = L. Theirnegations (if consistent at all) tend to have large cardinalstrength, and to be proved consistent by arguments involvinggeneric elementary embeddings.

IntroductionA theorem of ShelahA theorem of FarahA theorem of Magidor and ShelahConclusion43 / 51A stationary subset S of an uncountable cardinal κ is said toreflect if there is an ordinal α < κ such that S ∩ α is stationary inα, and to be non-reflecting if no such α exists. The existence ofa non-reflecting stationary set is a prototypical “incompactness”property of the sort discussed above. For every n < ω, the setℵ n+1 ∩ cof(ℵ n ) is an example of a non-reflecting stationary set.

IntroductionA theorem of ShelahA theorem of FarahA theorem of Magidor and ShelahConclusion44 / 51A general “stepping up” result by Milner and Shelah shows thatif R(κ) holds and there is a non-reflecting stationary set in λconsisting of ordinals of cofinality κ, then R(λ) holds. Using theexamples of non-reflecting sets from the last paragraph, itfollows easily that R(ℵ n ) holds for every n. So now we askwhat about ℵ ω+1 ?A natural first try might be to prove that there are non-reflectingstationary sets in ℵ ω+1 . This is actually true if V = L, or moregenerally as long as there are no inner models with largecardinals. However Magidor showed it to be consistent (modulosome very large cardinals) that every stationary subset ofℵ ω+1 reflects.

IntroductionA theorem of ShelahA theorem of FarahA theorem of Magidor and ShelahConclusion45 / 51Remark: It is a general fact in this area that small cardinals can(sometimes) consistently exhibit the reflection propertiesassociated with large cardinals. In the model built by Magidoreach ℵ n for 0 < n < ω is the critical point of a genericelementary embedding, and these embeddings jointly enforcethe reflection property.

IntroductionA theorem of ShelahA theorem of FarahA theorem of Magidor and ShelahConclusion46 / 51At this point we may begin to speculate that R(ℵ ω+1 ) isindependent of ZFC, but this speculation would be premature.Magidor and Shelah used Shelah’s “PCF theory” to isolate astationary set of points of cofinality ℵ 1 (the “good set”) whichcan be used in place of the non-reflecting set from theMilner-Shelah argument. They used this to show that R(ℵ ω+1 )is true.

IntroductionA theorem of ShelahA theorem of FarahA theorem of Magidor and ShelahConclusion47 / 51Remark: Again there is a general phenomenon to be observed,namely that ZFC is in some ways a surprisingly powerfultheory. In particular ZFC can prove weak forms ofcombinatorial principles (Jensen’s diamond and square) whichfollow from V = L. The stationarity of the good set is a case inpoint. Jensen’s principle square holds if V = L, and implies thatall points are good and non-reflecting stationary sets exist. InZFC alone we can’t prove square, and there may be non-goodpoints, but we can still establish that there are stationarily manygood points and use this to our advantage.

IntroductionA theorem of ShelahA theorem of FarahA theorem of Magidor and ShelahConclusion48 / 51Having reached ℵ ω+1 , we can use similar ideas to proceedinductively and prove R(κ) for all regular κ < ω 2 , at which pointwe again become stuck. PCF theory gives some information:the induction can be pushed to ℵ ω 2 +1 unless a certain subsetof ℵ ω 2 +1 (the “chaotic set”) is stationary. This gives the clueabout how to do a hard forcing construction which shows(modulo large cardinals, by way of generic elementaryembeddings) that R(ℵ ω 2 +1) is consistently false.

IntroductionA theorem of ShelahA theorem of FarahA theorem of Magidor and ShelahConclusion49 / 51Thesis V: We can learn to live with independence.The work I just described gives an example of how to makemathematical progress in an area which is rife withindependence. We can identify some heuristics which wereused in this work, and which have a good track record incombinatorial set theory.

IntroductionA theorem of ShelahA theorem of FarahA theorem of Magidor and ShelahConclusion50 / 51Cast the problem in terms of the tension betweencompactness and incompactness.Settle the problem on the basis of V = L or itsconsequences (GCH, the square principle, the existence ofnon-reflecting stationary sets).Search for refined arguments which rely on theZFC-provable consequences of V = L.When combinatorial arguments fail, mine them for cluesabout how a consistency result would have to look.Search for a consistency result involving genericelementary embeddings.

IntroductionA theorem of ShelahA theorem of FarahA theorem of Magidor and ShelahConclusion51 / 51Challenges for the philosophy of set theory. Say somethingwhichA) Is faithful to set-theoretic practice.B) Does not impose undue methodological restrictions.C) Takes into account the complex history and dynamic natureof the subject.D) Keeps the role of logic and the problem of independence inperspective.