Logic across Mathematics

Transcription

1 Logic across Mathematics Three Examples in Algebra Gabriele Gullà University of Roma-Tor Vergata

2 No one can forbid to violate rules of logic. And, in fact, they are respected very rarely. But, in order to obtain truth, you have to respect them. A. Zinov ev

3 Attack Plan In what follows I will present 3 examples: 1) In first example we will see how to use (very) high Large Cardinals Axioms to solve problems about Laver Tables, which are objects apparently closely related to Braids and Knots theory 2) Second one is about the study of relations between set-theoretic axioms and Operator Algebras, in particular C -Algebras problems 3) Last one concerns the proof by Saharon Shelah (1974) of the independence of Whitehead Problem (a group theory problem from the 50s) from ZFC (the usual Zermelo-Fraenkel set theory with the Axiom of Choice)

4 We will work here : 1) x y[ z(z x z y) x = y] 2) x[ a(a x) y(y x z(z y z x))] 3) a 1,..., a n A B x(x B [x A φ(x, a 1,..., a n )]) 4) x y z(x z y z) 5) A B Y x[(x Y Y A) x B] 6) A a 1 a 2... a n [ x(x A!yψ) B x(x A y(y B ψ))] 7) X [ X y(y X y {y} X )] 8) x y z[z x z y] 9) X R(R well-orders X )

5 C est dire... 1)Extensionality: two sets are equal if they have the same elements. 2)Foundation: Every non-empty set x contains a member y such that x and y are disjoint sets. 3)Comprehension: for any set A, there is a set B such that, given any set x, x is a member of B if and only if x is a member of A and φ holds for x. 4)Pair: If x and y are sets, then there exists a set z which contains x and y as elements. 5)Union: for any set of sets A there is a set B containing every element that is a member of some member of A. 6)Replacement: the image of a set under any definable function is contained inside a set. 7)Infinity: there exists at least an infinite set. 8)Power: for any set x, there is a set y that contains every subset of x. 9)Choice: every set can be well-ordered.

6 I: Laver Tables Let be a binary operation on a set; we define the Left Self-Distributive Law of as Classical examples are: x (y z) = (x y) (x z) 1) X a module and x y := (1 λ)x + λy 2) G a group and g h := ghg 1 In all this cases there is idempotence ( x x x = x), which means substructures mono-generated are trivial...

7 R.Laver, in 1995, proved the following theorem: for every n 1 there is a unique operation on {1, 2,..., 2 n }, such that for all x, y and x 1 = x + 1 mod N x (y 1) = (x y) (x 1) Figure : Richard Laver,

8 We call this system a Laver table and we write A n for the table with 2 n elements. Examples: A 0 : A 1 :

9 A 2 : It is simple to see that, for n 1, in A n there is no idempotence (1 1 = 2 1)! That s why (theorem): A n is generated by 1 via the representation 1 1 [2 n ] = 1 where x [k] = (...((x x) x)...) x, k times

10 Theorem 1 For every m 2 n, there is a number π n (m) (which is a power of 2) such that the mth row of A n is the periodic repetition of π n (m) values increasing from m + 1 mod 2 n to 2 n. For example, in A 2 (see last slide) we have π 2 (1) = 2, π 2 (2) = 2, π 2 (3) = 1, π 2 (4) = 4

11 Now let see what happens to the periods of 1 and 2... n π n (1) π n (2) ) First spontaneous (maybe naive) question is: will π n ( ) reach 32? 1-2) And then: does π n (2) π n (1) always hold? Does π n (1) tend to with n? To solve this problems we need set theory, so far...

12 First we need sets : let fix the following (transfinitely inductive) Von Neumann Hierarchy: V 0 := V α+1 := P(V α ) V λ := α<λ V α if λ is limit Every V α of this hierarchy is called a rank. Remark: If α is limit, f : V α V α a function, then f V 2 β belongs to V α for every β < α. Proof: Left as an exercise!

13 Now we need maps: let M and N be two structures (whatever it meant...); now let j be an injective map from M into N. We say j is an elementary embedding iff it satisfies the following schema: for any formula of the language we use (in ZFC case just ) φ and x 1,..., x n M M = φ[x 1,..., x n ] N = φ[j(x 1 ),..., j(x n )] We write crit(j) for the first ordinal moved by j. N.B. I do not want to go deep inside the model theory behind this slide...

14 Now we can state the crucial hypothesis I3: For some cardinal λ there is a non-trivial elementary embedding j from V λ into itself. This axiom (which is not provable nor refutable in ZFC) has a incredible consistency strenght, in fact Kunen Theorem (1971): ZFC For any λ there are no non-trivial elementary embeddings from V λ+2 into itself. Thats why for several years I3 (with his brothers I2, I1 and I0...) was considered Inconsistent!

15 Now, If i and j are self-embedding of V λ, we apply i to j by defining i[j] := i(j Vα) 2 α<λ Now, being a self-embedding and being the image of are -definable statements, so: 1) i[j] is a self-embedding (N.B. Application is NOT composition!) 2) l = j[k] i[l] = i[j][i[k]], i.e. i[j[k]] = i[j][i(k]] Last one is a left-distributive law!!

16 So, Proposition 2 I3 the set Iter(j) of iterates of j form a self-distributive structure. Now, let k, k Iter(j); we write that kρ n k k and k coincide till crit(j [2 n ]) level where j [m] = j[j][j]..., m times. Then Proposition 3 ρ n is a congruence (i.e. compatible equivalence relation) on Iter(j) with 2 n classes j, j [2],..., j [2 n ] = id. Non-trivial proof...

17 Corollary 4 The quotient Iter(j)/ ρn is isomorphic to A n. Proof (the laziest one...): Iter(j)/ ρn is a self distributive structure with 2 n elements which obey, for all m, j [m] j = j [m+1 mod 2 n ] Thesis follows from the unicity of A n.

18 Proposition 5 - The Translator For m n and p 2 n, the period of p jumps from 2 m to 2 m+1 between A n and A n+1 j [p] maps crit m (j) to crit n (j), where crit n (j) := (n + 1) th ordinal in {crit(i) i Iter(j)} Not difficult proof, but...

19 Lemma 6 If j is a self-embedding, then j[j](α) j(α) for every ordinal α. Proof: Of course there is a β such that j(β) > α; this means there is a smallest β such that j( β) > α and γ < β (j(γ) α). We apply j to the last formula: ( ) γ < j( β) j[j](γ) j(α) So, if we choose γ = α in the last formula, we obtain the thesis.

20 Proposition 7 For all n π n (2) π n (1) Proof: Write π n (1) = 2 m+1 and let n be the maximal one such that n < n and π n (1) 2 m. This means, by construction, that period of 1 jumps from 2 m to 2 m+1 between A n and A n+1. By the Translator j maps crit m (j) to crit n (j). Hence, by Lemma 6, j[j] maps crit m (j) to crit n (j). This means there exists an n n such that j[j](crit m (j)) = crit n (j). Again, from the Translator, the period of 2 jumps from 2 m to 2 m+1 between A n and A n +1. Hence in A n period of 2 is at least 2 m+1.

21 Lemma 8 If j is an elementary embedding of V λ, then Proof is difficult... Proposition 9 π n (1) tends to with n. λ = sup n {crit n (j)} Proof: Assume π n (1) = 2 m. We want to show that there is a n n such that π n (1) = 2 m and π n+1 (1) = 2 m+1. So for the Translator, we want to show that j maps crit m (j) to crit n (j).

22 Now, crit m (j) = crit(j [2 m ]), and j maps it to crit(j[j [2 m ]]). As j[j [2 m ]] belongs to Iter(j), by Lemma 8, we have crit(j[j [2 m ]]) = crit n (j) for some n.

23 Why Laver tables could be useful? (Very concise answer...) Left-distributive structures are well known in low-dimensional topology, so the idea of a lots of mathematicians (Dehornoy, Lebed, Drapal, Fromentin...) is to use Laver tables to find invariants about isotopy of links, knots and braids by colouring strands with a finite sets of colours; this set, with respect precise operations, turns to be a left-distibutive structure. Moreover someone is investigating the possible relation with solutions of Yang-Baxter equations...

24 II: Operator Algebras I know you are experts, but... following is for me!! Definition Let H be a Hilbert space; the set B(H) of all bounded operators is a Banach space (normed and complete with respect the norm metric) and a Banach algebra with respect the product (composition of operators) and the involution (the function which maps an operator to its adjoint). A C -algebra is a closed subset of B(H) which is also closed under the adjoint operation. A Von Neumann algebra is a C -algebra which contains the identity and is closed with respect the topology which makes the function A Av, w continuous for all v, w H.

25 First we see something about M[0, 1], i.e. the space of complex Borel measures on [0, 1]. Theorem (Mauldin) ZFC + CH For every bounded linear functional T on M[0, 1] there is a bounded function ψ : P([0, 1]) Borel C such that for all µ M[0, 1]. T (µ) = ψdµ

26 The integral here means the limit over the directed set of Borel partitions {B 1,..., B n } of [0, 1] of the quantity ψ(b i )µ(b i ). ψ can be chosen such that the limit always exists. The proof of the theorem begins by chosing a maximal family of mutually singular Borel measure on [0, 1]; then (using CH...) this family is indexed by countable ordinals and ψ is defined then by transfinite recursion.

27 Sometimes CH is too much. It is possible to weak hipothesys by using Martin s Axiom: Definition (MA(κ)) X compact Hausdorff topological space. It satisfies the CAC i.e. every antichains in X is countable. Then X is not the union of κ nowhere dense subsets (sets whose closure has empty interior, or, equivalently, sets which are not dense in any open non-empty set). If this is true for all κ < c, then Martin s Axiom holds. N.B. While MA(c) fails in ZFC (check [0, 1]...), MA(ℵ 0 ) is a theorem of ZFC (the Rasiowa-Sikorski lemma), so CH MA!

28 Theorem (Blass-Weiss 1978) ZFC + MA Every proper ideal of B(l 2 ) that properly contains the finite rank operators is the join of two smaller ideals. Theorem (Blass-Shelah 1987/1989) The failure of K(l 2 ) to be the join of two smaller ideals is equivalent to the assertion that any two non-principal ultrafilters over ω are related in both directions by a finite-to-one map from ω to itself. Last condition is consistent with ZFC.

29 Theorem (Ackemann-Anderson-Pedersen 1986) ZFC + MA Any sequence of pure states on a separably acting Von Neumann algebra that is supported by a sequence of mutually orthogonal, positive, norm-one elements has at least one limit point which is a pure state. What is needed in this proof is the nonexistence of an inextendible κ-tower (which is a particular sequence of complete ultrafilters) in P(ω)/Fin for any κ < c I do not want to define perfect C -algebras and I will say just that a diffuse sequence (whatever it is...) is trivial if A n 0. Theorem 8 (Ackemann-Anderson-Pedersen 1986) A separable C -algebra is perfect iff it has only trivial diffuse sequences.

30 Theorem (Ackemann-Anderson-Pedersen) ZFC + CH Every separably acting Von Neumann algebra has only trivial diffuse sequences. So: Conjecture: Every Von Neumann algebra is perfect. The proof of last theorem is based in showing first that the existence of a nontrivial diffuse sequence implies the existence of a mutually orthogonal, positive, norm-one diffuse sequence and then authors invoke their first theorem to reach a contradiction.

31 Masas and Pure States on Calkin algebra Definition A masa of a C -algebra is a maximal abelian C -subalgebra. If A is a masa in B(l 2 ) then π(a) is a masa in the Calkin algebra C(l 2 ) (π is the quotient map...) Theorem (Anderson) ZFC + CH There are other masas of C(l 2 ) besides those of the form π(a). And...

32 Theorem (Anderson) ZFC + CH For every countable set of states on C(l 2 ) there is a masa A such that the restriction of each state to A is pure. Moreover Theorem (Anderson) ZFC + CH There is a masa A of C(l 2 ) with the property that every pure state on A has only one (necessarily pure) state extension to C(l 2 ). This relates to the open problem of classifying pure states on C(l 2 ), which correspond precisely to non-trivial pure states on B(l 2 ). Here there is a well known analogy between B(l 2 ), C(l 2 ) and K(l 2 ) on one side and βω, ω and βω \ ω on the other...

33 III: The Whitehead s Problem The problem was presented by J.H.C. Whitehead (nephew of Alfred N. Whitehead...) in Warsaw in May 1952 We need some definition... Figure : J.H.C. Whitehead

34 A, B two abelian groups; π : B A a surjective homomorphism. We say that π splits if there is a homomorphism ρ : A B such that πρ = 1 A. An abelian group A is called W -group if satisfies the following: for all surjective homomorphisms π : B A, if the kernel of π is isomorphic to Z (the group of integers), then π splits. That free abelian groups are W -group is a theorem (cause it is possible to show that an abelian group is free iff every onto-homomorphism splits); Whitehead s Problem asks if the converse holds N.B. The problem was defined in homological terms, and could be stated in terms of topological groups, but I choose what I think is the simplest way...

35 Theorem (Stein, 1951) Every countable W -group is free. The proof of this theorem is based on two facts: a sub-group of a W -group is a W -group, and every W -group is torsion-free (i.e. only the identity has finite order) But what about uncountable group??

36 Now, we need the definition of Gödel s constructible set hierarchy. I will present an intuitive definition: L 0 := L α+1 := G(L α ) L λ := α<λ L α if λ is limit L := α Ord where G( ) is the Gödel operation which constructs sets from other sets, via first-order definability. L α

37 If you remember Von Neumann hyerarchy, it is quite natural to ask which is the relation between V and L. Well, the interesting fact is that V = L is a proper axiom with respect to ZF, and in fact Theorem (Gödel, 1940) Con(ZF ) Con(ZF + V = L) Moreover Theorem (Gödel, 1940) ZF + V = L AC + CH V = L is called Constructibility Axiom.

38 It follows from last two theorems that Con(ZF ) Con(ZFC). Not only, Theorem (Solovay-Tennenbaum, 1971) Con(ZFC) Con(ZFC + MA + CH)

39 Figure : On the left, Shelah awarded with Hausdorff Medal during ESTC conference, Budapest 2017 Using Gödel s theorem and Solovay-Tennenbaum theorem, Shelah proved the following:

40 Theorem (Shelah, 1974) 1) ZFC + V = L Every W -group of cardinality ℵ 1 is free 2) ZFC + MA + CH There is a W -group of cardinality ℵ 1 which is not free. This implies that either the affirmative or negative answer to Whitehead s Problem is consistent with ZFC, and so the question is undecidable with respect to ZFC In 1975 Shelah proved something more: part (2) can be generalized to every uncountable cardinality κ by taking the direct sum of κ copies of the non-free W -group of cardiality ℵ 1. Also part (1) generalizes to every W -group. But moreover, a negative answer to WP for groups of cardinality ℵ 1 is consistent with ZFC + CH, which means WP is undecidable also with respect to ZFC + CH.

41 Some idea about Shelah s proof: For the first part of his theorem, Shelah used two facts, under V = L. 1) Let A be a W -group of cardinality ℵ 1 ; then A satisfies the following Chase s Condition: A is a ℵ 1 -free group, i.e. every countable subgroup is free, and this countable sub-groups are contained in a countable subgroup B of A such that A/B is ℵ 1 -free. 2) It is used a characterization of free groups in terms of stationary subsets of ω 1. A stationary set is, in a very very very intuitive way, a big set (cause it intersect a lot of other sets)

42 For the second part, Shelah shows that MA + CH Any group of cardinality ℵ 1 which satisfies Chase s condition is a W -group. This implies the big result via the following: There is a group of cardinality ℵ 1 which satisfies the Chase s condition but is not free.

43 So... don t be scared by Large Cardinals!

44 ...Maybe

45 That s all, folks! Thanks!!

46 References I example Dehornoy, P.: Laver s result and low dimensional topology ; Dehornoy, P.: Braid Groups and Left Distributive Operations ; Laver, R.: The left-distributive law and the freeness of an algebra of elementary embeddings, Advances in Mathematics; Laver, R.: On the algebra of elementary embeddings of a rank into itself, Advances in Mathematics; Laver, R.: Braid group actions on left distributive structures, and well orderings in the braid groups, Journal of Pure and Applied Algebra;

47 References II example Farah, I.: Logic and operator algebras ; Farah, I.-Wofsey, E.: Set Theory and Operator Algebras ; Weaver, N,: Set Theory and C -Algebras

48 References III example Shelah, S.: Infinite Abelian groups, Whitehead problem and some constructions, Israel Journal of Mathematics, 18 (3); Shelah, S.: Whitehead groups may not be free, even assuming CH. I, Israel Journal of Mathematics, 28 (3); Shelah, S.: Whitehead groups may not be free, even assuming CH. II, Israel Journal of Mathematics, 35 (4);