Some challenges for the philosophy of set theory

theory of schemes led to progress on classical and concrete problems about curves and surfaces. 3. A theorem of Farah The following theorem appears in a paper by by Ilijas Farah (“All automorphisms of the Calkin algebra are inner”, Annals of Mathematics 173 (2011) 619–661) If Todorcevic’s axiom (TA) holds then all automorphisms of the Calkin algebra are inner. Since the focus of this lecture series is the Continuum Hypothesis (CH), I have chosen for my second case study a theorem which involves CH and the mathematics around it in an essential way. My central point,on which I will elaborate after discussing Farah’s work, is that considerations of whether CH has a truth value or scenarios for settling that truth value are perhaps not the main issue in this very impressive piece of CH-related mathematics. The problem which Farah solved is fairly easy to state, but we need to recall a little background in functional analysis and the theory of C∗-algebras. A (complex) Hilbert space is a complex inner product space H which is complete with respect to the metric induced by the inner product. We will assume that the space H is infinite dimensional and separable: there is just one such space up to isomorphism, the space l(C) whose elements are infinite sequences x = (xn) of complex numbers with ∑ |xn| <∞ and whose inner product is given by x.y = ∑ n xnȳn. B(H) is the space of continuous linear maps from H to H; if T is such a map then the norm ‖T‖ of T is sup{|Tx‖ : ‖x‖ ≤ 1}. With this norm the space B(H) becomes a complete normed space (a Banach space); B(H) is closed under the bilinear operation of composition and |ST |≤ ‖S|‖T |, that is to say B(H) is a Banach algebra. For any T ∈ B(H) there is a unique T ∗ ∈ B(H) such that (Tx).y = x.(T ∗y). We have the identities T ∗∗ = T , (TS)∗ = S∗T ∗, ‖T‖ = ‖T ∗‖, |T ∗T‖ = ‖T‖. A Banach algebra with an operation a 7→ a∗ satisfying the identities just listed is known as a C∗-algebra. A basic theorem (the GelfandNaimark-Segal theorem) states that every C∗-algebra is isomorphic to a norm-closed and ∗-closed subalgebra of B(K) for some Hilbert space K. Another theorem by Gelfand identifies the unital commutative C∗algebras with spaces C(X) of complex-valued functions on compact Hausdorff spaces X, and we can view the study of noncommutative C∗algebras as a form of “noncommutative geometry”. SOME CHALLENGES FOR THE PHILOSOPHY OF SET THEORY 9 An operator T ∈ B(H) is compact if the image of the unit ball of H under T has compact closure. We denote the class of compact operators by K(H); this is a C∗–algebra in its own right and forms 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 is reasonable to think of the compact operators as “finitary” and so the Calkin algebra consists of “operators modulo finitary perturbation”. In any C∗ algebra with a 1 a unitary element is an element u such that uu∗ = u∗u = 1. If u is unitary then the map a 7→ uau∗ is an automorphism, and automorphisms of this type are called inner automorphisms. A classical question in C∗ algebras asked whether all automorphisms of the Calkin algebra are inner: Phillips and Weaver showed that under CH there is an outer (non-inner) automorphism, and Farah closed the case by showing that under other hypotheses all automorphisms are inner. The starting point for Farah’s work is a series of analogies with wellstudied objects in classical set theory, namely Pω and Pω/FIN where FIN is the ideal of finite sets: the Boolean algebra Pω is analogous to the Banach algebra B(H), the ideal of finite sets FIN is analogous to the ideal K(H) of compact operators, and the quotient Pω/FIN is analogous to the Calkin algebra B(H)/K(H). Say that an automorphism π of Pω is trivial if it has the form A 7→ f [A] where f is a permutation of ω, and f [A] = {f(n) : n ∈ A}. Since Pω is atomic with atoms the singleton sets {n}, every automorphism is trivial. Similarly say that an automorphism π of Pω/FIN is trivial if it has the form [A]FIN 7→ [f [A]]FIN , where f is a bijection between cofinite subsets of ω and [B]FIN is the equivalence class of B modulo finite: a classical questions asks whether all automorphisms of Pω/FIN are trivial. Walter Rudin showed that if CH holds there is a non-trivial automorphism of Pω/FIN ; the point is that under CH Pω/FIN is a “saturated” (roughly speaking, homogeneous) Boolean algebra of size א1 and so has 2א1 automorphisms, while there are only 2א0 candidates for the bijection f . Shelah showed by a hard forcing argument that it is consistent (modulo the consistency of ZFC) that every automorphism of Pω/FIN is trivial, but the argument is rather specific to Pω/FIN . The Proper Forcing Axiom (PFA) is a “maximality principle” which goes along the same lines as Martin’s Axiom, and which states (roughly) that objects of size א1 which can be added to V by proper forcing already exist in V . It was shown by Baumgartner that PFA is consistent modulo the existence of a supercompact cardinal, and PFA is a very