Generalising the weak compactness of ω

Transcription

1 Generalising the weak compactness of ω Andrew Brooke-Taylor Generalised Baire Spaces Masterclass Royal Netherlands Academy of Arts and Sciences 22 August 2018 Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

2 Overview One obstacle to generalising results about cardinal characteristics of the continuum is if the arguments use compactness properties of ω. Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

3 Overview One obstacle to generalising results about cardinal characteristics of the continuum is if the arguments use compactness properties of ω. In these cases, assuming your cardinal is weakly compact will often allow the argument to generalise. Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

4 Overview One obstacle to generalising results about cardinal characteristics of the continuum is if the arguments use compactness properties of ω. In these cases, assuming your cardinal is weakly compact will often allow the argument to generalise. In this tutorial I want to give a couple of examples of this, digging into the necessary preliminaries on the way. Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

5 Weak compactness There are many equivalent formulations of weak compactness; we will use a couple of different ones. Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

6 Weak compactness There are many equivalent formulations of weak compactness; we will use a couple of different ones. Notice in each case that, if we didn t simply decree that weakly compact cardinals (and inaccessible cardinals) must be uncountable, then ω would fit the definition. Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

7 Weak compactness There are many equivalent formulations of weak compactness; we will use a couple of different ones. Notice in each case that, if we didn t simply decree that weakly compact cardinals (and inaccessible cardinals) must be uncountable, then ω would fit the definition. Recommended Reference: The Exercises for Section 4.2 of Chang & Keisler s Model Theory (but note that their definition of weakly compact needs to have inaccessibility added to it). Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

8 Infinitary languages For any vocabulary Σ (i.e. set of function and relation symbols) and for any cardinal κ the language L κ,κ consists of formulas built via the usual construction rules along with: Conjunctions and disjunctions of less than κ many formulas: if δ < κ and ϕ γ is a formula for every γ < δ, then and are formulas. γ<δ ϕ γ Less than κ-fold quantifications: if x = (x γ : γ < δ) is an δ-tuple of variables for some δ < κ and ϕ is a formula, then are formulas. γ<δ xϕ and xϕ Satisfaction of these formulas is defined as you would expect. ϕ γ Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

9 Infinitary languages E.g. Being well-ordered can be expressed by a sentence of L ω1,ω 1 : ( ) (x i : i ω) x i > x i+1 i ω Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

10 Infinitary languages E.g. Being well-ordered can be expressed by a sentence of L ω1,ω 1 : ( ) (x i : i ω) x i > x i+1 Weak compactness, 1st formulation An uncountable cardinal κ is weakly compact if and only if, for every set of T of L κ,κ sentences over a vocabulary Σ of cardinality at most κ, if every subset of T of cardinality less than κ has a model then T itself has a model. i ω Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

11 Infinitary languages E.g. Being well-ordered can be expressed by a sentence of L ω1,ω 1 : ( ) (x i : i ω) x i > x i+1 Weak compactness, 1st formulation An uncountable cardinal κ is weakly compact if and only if, for every set of T of L κ,κ sentences over a vocabulary Σ of cardinality at most κ, if every subset of T of cardinality less than κ has a model then T itself has a model. i ω Note that if the constraint on Σ is dropped, then this defines strongly compact cardinals. Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

12 Elementary embeddings Weak compactness, 2nd formulation An uncountable cardinal κ is weakly compact if and only if, for any structure M of size κ for a vocabulary Σ of cardinality κ, there is a Σ-structure N such that M is a proper elementary substructure of N in the L κ,κ sense. Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

13 Elementary embeddings Weak compactness, 2nd formulation An uncountable cardinal κ is weakly compact if and only if, for any structure M of size κ for a vocabulary Σ of cardinality κ, there is a Σ-structure N such that M is a proper elementary substructure of N in the L κ,κ sense. Proof sketch that formulation 1 formulation 2 Add to the vocabulary a constant c m for each element m of M, and consider the the complete L κ,κ theory T of M for this language, with c m interpreted as m. Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

14 Elementary embeddings Weak compactness, 2nd formulation An uncountable cardinal κ is weakly compact if and only if, for any structure M of size κ for a vocabulary Σ of cardinality κ, there is a Σ-structure N such that M is a proper elementary substructure of N in the L κ,κ sense. Proof sketch that formulation 1 formulation 2 Add to the vocabulary a constant c m for each element m of M, and consider the the complete L κ,κ theory T of M for this language, with c m interpreted as m. Now add another constant c to the vocabulary and add to the theory all of the sentences c c m. Every subset A of this extended theory T with cardinality < κ has a model (M with a suitable choice of c), so by formulation 1, T has a model; this will be N. Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

15 Elementary embeddings Weak compactness, 2nd formulation An uncountable cardinal κ is weakly compact if and only if, for any structure M of size κ for a vocabulary Σ of cardinality κ, there is a Σ-structure N such that M is a proper elementary substructure of N in the L κ,κ sense. Proof sketch that formulation 1 formulation 2 Add to the vocabulary a constant c m for each element m of M, and consider the the complete L κ,κ theory T of M for this language, with c m interpreted as m. Now add another constant c to the vocabulary and add to the theory all of the sentences c c m. Every subset A of this extended theory T with cardinality < κ has a model (M with a suitable choice of c), so by formulation 1, T has a model; this will be N. Remember that well-foundedness is definable in L κ,κ for κ > ω, so these embeddings can be nice from a set theory point of view. Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

16 Trees A κ-tree is a tree T of height κ such for every α < κ there are fewer than κ many nodes of T of height α. Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

17 Trees A κ-tree is a tree T of height κ such for every α < κ there are fewer than κ many nodes of T of height α. A cardinal κ has the tree property if and only if every κ-tree has a cofinal branch (i.e. a branch of height κ). Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

18 Trees A κ-tree is a tree T of height κ such for every α < κ there are fewer than κ many nodes of T of height α. A cardinal κ has the tree property if and only if every κ-tree has a cofinal branch (i.e. a branch of height κ). Weak compactness, 3rd formulation A cardinal is weakly compact if and only if it is inaccessible and has the tree property. Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

19 Trees A κ-tree is a tree T of height κ such for every α < κ there are fewer than κ many nodes of T of height α. A cardinal κ has the tree property if and only if every κ-tree has a cofinal branch (i.e. a branch of height κ). Weak compactness, 3rd formulation A cardinal is weakly compact if and only if it is inaccessible and has the tree property. Proof sketch that formulation 2 formulation 3 Code your κ-tree T as a subset of V κ, and take M = V κ,, T, enough extra stuff to make V κ rigid. κ is in the model N = X, E, T,... given by formulation 2, and T below level κ is just T. Choosing any node t of T at level κ, the set of nodes below t is then a cofinal branch through T. Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

20 Warm-up example: s κ For A, B κ of cardinality κ, say that A splits B if B A = B A = κ. Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

21 Warm-up example: s κ For A, B κ of cardinality κ, say that A splits B if B A = B A = κ. A family A [κ] κ is a splitting family if for every B κ with B = κ, there is an A A which splits B. Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

22 Warm-up example: s κ For A, B κ of cardinality κ, say that A splits B if B A = B A = κ. A family A [κ] κ is a splitting family if for every B κ with B = κ, there is an A A which splits B. The splitting number s κ is the least cardinality of a splitting family. Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

23 Proposition s ω ω 1 Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

24 Proposition s ω ω 1 Proof. Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

25 Proposition s ω ω 1 Proof. Diagonalise! Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

26 Proposition s ω ω 1 Proof. Diagonalise! Consider any A = {A i : i ω} [ω] ω. We will inductively define a sequence of infinite subsets B i of ω and a sequence of elements c i of ω such that no A in A splits C = {c i : i ω}. Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

27 Proposition s ω ω 1 Proof. Diagonalise! Consider any A = {A i : i ω} [ω] ω. We will inductively define a sequence of infinite subsets B i of ω and a sequence of elements c i of ω such that no A in A splits C = {c i : i ω}. For the base case, let B 0 = ω and c 0 = 0. Having defined B i, at least one of B i A i and B i A i is infinite, so pick one that is infinite, and take that to be B i+1. Then let c i+1 be the least element of B i+1 that is greater than c i. Note that for each i ω, {c j : j > i} B i+1, and so is disjoint from or contained in A i. So no A i splits C. Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

28 Generalising to higher κ Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

29 Generalising to higher κ Potential problems To generalise this inductive argument to higher κ, we have to be able to deal with limit stages. At limit stages α along the way, the natural choice is to take B α = γ<α B γ. What if this intersection is empty, or even just of size < κ? Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

30 Generalising to higher κ Potential problems To generalise this inductive argument to higher κ, we have to be able to deal with limit stages. At limit stages α along the way, the natural choice is to take B α = γ<α B γ. What if this intersection is empty, or even just of size < κ? The tree of possible choices we could have made for Bγ is a binary tree. If κ is inaccessible, then at any stage α, since there are only 2 α < κ possible nodes, and they between them partition κ, at least one of them must have cardinality κ. Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

31 Generalising to higher κ Potential problems To generalise this inductive argument to higher κ, we have to be able to deal with limit stages. At limit stages α along the way, the natural choice is to take B α = γ<α B γ. What if this intersection is empty, or even just of size < κ? The tree of possible choices we could have made for Bγ is a binary tree. If κ is inaccessible, then at any stage α, since there are only 2 α < κ possible nodes, and they between them partition κ, at least one of them must have cardinality κ. Does this tree of possibilities have a cofinal branch, allowing us to define C? Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

32 Generalising to higher κ Potential problems To generalise this inductive argument to higher κ, we have to be able to deal with limit stages. At limit stages α along the way, the natural choice is to take B α = γ<α B γ. What if this intersection is empty, or even just of size < κ? The tree of possible choices we could have made for Bγ is a binary tree. If κ is inaccessible, then at any stage α, since there are only 2 α < κ possible nodes, and they between them partition κ, at least one of them must have cardinality κ. Does this tree of possibilities have a cofinal branch, allowing us to define C? If κ also has the tree property, then yes! Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

33 So: Proposition (Kamo; see Zapletal [4]) If κ is weakly compact, then s κ > κ. Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

34 So: Proposition (Kamo; see Zapletal [4]) If κ is weakly compact, then s κ > κ. Actually, this is if and only if. Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

35 So: Proposition (Kamo; see Zapletal [4]) If κ is weakly compact, then s κ > κ. Actually, this is if and only if. To get s κ > κ + one requires even more large cardinal strength see Zapletal [4]. Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

36 More involved example: e κ Evasion and prediction were introduced by Blass in a paper motivated by group theory [1]. Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

37 More involved example: e κ Evasion and prediction were introduced by Blass in a paper motivated by group theory [1]. Definition A predictor is a a function π such that dom(π) κ and dom(π) = κ, and for each α dom(π), π(α) is a function from κ α to κ. Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

38 More involved example: e κ Evasion and prediction were introduced by Blass in a paper motivated by group theory [1]. Definition A predictor is a a function π such that dom(π) κ and dom(π) = κ, and for each α dom(π), π(α) is a function from κ α to κ. Definition Given a predictor π and a function f : κ κ, we say π predicts f if there is some α < κ such that for all β > α, π(β)(f β) = f (β). Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

39 Definition The evasion number e κ is the bounding number for prediction : e κ = min{ F : F κ κ predictor π f F(π predicts f )}. Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

40 Definition The evasion number e κ is the bounding number for prediction : e κ = min{ F : F κ κ predictor π f F(π predicts f )}. In the framework of relations introduced in the first talk today, e κ is the norm of the dual relation to (κ κ, predictors, is predicted by). Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

41 e vs b e κ = min{ F : F κ κ predictor π f F(π predicts f )}. b κ = min{ F : F κ κ g κ κ f F(g f )}. (Recall g f means there is some α < κ such that for all β α, g(β) f (β).) Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

42 e vs b e κ = min{ F : F κ κ predictor π f F(π predicts f )}. b κ = min{ F : F κ κ g κ κ f F(g f )}. (Recall g f means there is some α < κ such that for all β α, g(β) f (β).) e ω is independent of b ω : there is a model in which e ω > b ω, a model in which e ω < b ω, and a model in which e ω = b ω. Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

43 e vs b e κ = min{ F : F κ κ predictor π f F(π predicts f )}. b κ = min{ F : F κ κ g κ κ f F(g f )}. (Recall g f means there is some α < κ such that for all β α, g(β) f (β).) e ω is independent of b ω : there is a model in which e ω > b ω, a model in which e ω < b ω, and a model in which e ω = b ω. I will focus on Brendle and Shelah s construction of a model with e ω > b ω [2]. Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

44 Getting e > b How to get a model of e ω > b ω? The natural approach: start with a model of CH, where e ω = b ω = ω 1, Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

45 Getting e > b How to get a model of e ω > b ω? The natural approach: start with a model of CH, where e ω = b ω = ω 1, force to make e ω large, whilst keeping b ω small. Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

46 Getting e > b How to get a model of e ω > b ω? The natural approach: start with a model of CH, where e ω = b ω = ω 1, force to make e ω large, whilst keeping b ω small. A natural approach to this latter: do the most obvious forcing to make e ω large, and hope for the best. Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

47 Getting e > b How to get a model of e ω > b ω? The natural approach: start with a model of CH, where e ω = b ω = ω 1, force to make e ω large, whilst keeping b ω small. A natural approach to this latter: do the most obvious forcing to make e ω large, and use a clever argument to show that b ω remains small. Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

48 Making e large There is a standard way to make a bounding number -type cardinal characteristic large: force to kill all ground model unbounded sets: add a new bound for them all, Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

49 Making e large There is a standard way to make a bounding number -type cardinal characteristic large: force to kill all ground model unbounded sets: add a new bound for them all, do a long (however large you want the cardinal) iteration of these forcings with finite support. Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

50 Making e large There is a standard way to make a bounding number -type cardinal characteristic large: force to kill all ground model unbounded sets: add a new bound for them all, do a long (however large you want the cardinal) iteration of these forcings with finite support. Any small potentially unbounded set appears after an initial fragment of the iteration, and then is forced to be bounded at the next step. Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

51 Making e large There is a standard way to make a bounding number -type cardinal characteristic large: force to kill all ground model unbounded sets: add a new bound for them all, do a long (however large you want the cardinal) iteration of these forcings with finite support. Any small potentially unbounded set appears after an initial fragment of the iteration, and then is forced to be bounded at the next step. In particular, to make e ω equal to some regular cardinal λ > ω 1, we: force to add a new predictor that predicts all ground model functions, Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

52 Making e large There is a standard way to make a bounding number -type cardinal characteristic large: force to kill all ground model unbounded sets: add a new bound for them all, do a long (however large you want the cardinal) iteration of these forcings with finite support. Any small potentially unbounded set appears after an initial fragment of the iteration, and then is forced to be bounded at the next step. In particular, to make e ω equal to some regular cardinal λ > ω 1, we: force to add a new predictor that predicts all ground model functions, do a length λ iteration with finite supports of this forcing. Any set F ω ω in the extension of cardinality less than λ must have appeared by some initial stage of the forcing, and then the predictor added at the next step predicts it. Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

53 Adding a predictor How do we force to add a predictor that predicts all ground model functions? Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

54 Adding a predictor How do we force to add a predictor that predicts all ground model functions? By finite approximations! Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

55 Adding a predictor Definition (Brendle-Shelah [2]) We define the single step predictor forcing P. Conditions are triples d, π, F such that Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

56 Adding a predictor Definition (Brendle-Shelah [2]) We define the single step predictor forcing P. Conditions are triples d, π, F such that 1 d 2 <ω Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

57 Adding a predictor Definition (Brendle-Shelah [2]) We define the single step predictor forcing P. Conditions are triples d, π, F such that 1 d 2 <ω (a finite approximation to the characteristic function of the domain of the predictor), Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

58 Adding a predictor Definition (Brendle-Shelah [2]) We define the single step predictor forcing P. Conditions are triples d, π, F such that 1 d 2 <ω (a finite approximation to the characteristic function of the domain of the predictor), 2 π is a function with domain d 1 {1} such that n d 1 {1}, π(n) is a finite partial function from ω n to ω Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

59 Adding a predictor Definition (Brendle-Shelah [2]) We define the single step predictor forcing P. Conditions are triples d, π, F such that 1 d 2 <ω (a finite approximation to the characteristic function of the domain of the predictor), 2 π is a function with domain d 1 {1} such that n d 1 {1}, π(n) is a finite partial function from ω n to ω (a finite approximation to the predictor), Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

60 Adding a predictor Definition (Brendle-Shelah [2]) We define the single step predictor forcing P. Conditions are triples d, π, F such that 1 d 2 <ω (a finite approximation to the characteristic function of the domain of the predictor), 2 π is a function with domain d 1 {1} such that n d 1 {1}, π(n) is a finite partial function from ω n to ω (a finite approximation to the predictor), 3 F ω ω is finite and for f g F, max({n : f n = g n}) < dom(d) Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

61 Adding a predictor Definition (Brendle-Shelah [2]) We define the single step predictor forcing P. Conditions are triples d, π, F such that 1 d 2 <ω (a finite approximation to the characteristic function of the domain of the predictor), 2 π is a function with domain d 1 {1} such that n d 1 {1}, π(n) is a finite partial function from ω n to ω (a finite approximation to the predictor), 3 F ω ω is finite and for f g F, max({n : f n = g n}) < dom(d) (a promise to predict the functions in F from now on). Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

62 Adding a predictor Definition (Brendle-Shelah [2]) We define the single step predictor forcing P. Conditions are triples d, π, F such that 1 d 2 <ω (a finite approximation to the characteristic function of the domain of the predictor), 2 π is a function with domain d 1 {1} such that n d 1 {1}, π(n) is a finite partial function from ω n to ω (a finite approximation to the predictor), 3 F ω ω is finite and for f g F, max({n : f n = g n}) < dom(d) (a promise to predict the functions in F from now on). We say d, π, F d, π, F if and only if d d, π π, and F F, and Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

63 Adding a predictor Definition (Brendle-Shelah [2]) We define the single step predictor forcing P. Conditions are triples d, π, F such that 1 d 2 <ω (a finite approximation to the characteristic function of the domain of the predictor), 2 π is a function with domain d 1 {1} such that n d 1 {1}, π(n) is a finite partial function from ω n to ω (a finite approximation to the predictor), 3 F ω ω is finite and for f g F, max({n : f n = g n}) < dom(d) (a promise to predict the functions in F from now on). We say d, π, F d, π, F if and only if d d, π π, and F F, and for all f F and n (d ) 1 {1} d 1 {1}, π (n)(f n) = f (n) (and in particular is defined). Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

64 Centred-ness Let P be a partial order. A subset X of P is (1, < ω)-centred if any finitely many conditions in X have a common extension in P. A subset Y of P is (λ, < ω)-centred if Y may be decomposed as Y = γ<λ Y γ where each Y γ is (1, < ω)-centred in P. P is said to be σ-centred if it is (ω, < ω)-centred (as a subset of itself). Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

65 Centred-ness Let P be a partial order. A subset X of P is (1, < ω)-centred if any finitely many conditions in X have a common extension in P. A subset Y of P is (λ, < ω)-centred if Y may be decomposed as Y = γ<λ Y γ where each Y γ is (1, < ω)-centred in P. P is said to be σ-centred if it is (ω, < ω)-centred (as a subset of itself). Clearly any σ-centred forcing is ccc. Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

66 Centred-ness Let P be a partial order. A subset X of P is (1, < ω)-centred if any finitely many conditions in X have a common extension in P. A subset Y of P is (λ, < ω)-centred if Y may be decomposed as Y = γ<λ Y γ where each Y γ is (1, < ω)-centred in P. P is said to be σ-centred if it is (ω, < ω)-centred (as a subset of itself). Clearly any σ-centred forcing is ccc. Note that the predictor forcing P is σ-centred: any set of conditions with the same d and π components are compatible: take the union of the F components, and extend d to take the value 0 for long enough to satisfy requirement 3 on conditions. Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

67 So P is ccc, and hence preserves cardinals and cofinalities. Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

68 So P is ccc, and hence preserves cardinals and cofinalities. Given a generic filter G for P, the union of the π components of the conditions in G is a predictor, with domain the union of the d components, which predicts every ground model function from ω to ω. Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

69 So P is ccc, and hence preserves cardinals and cofinalities. Given a generic filter G for P, the union of the π components of the conditions in G is a predictor, with domain the union of the d components, which predicts every ground model function from ω to ω. Iterating P with finite support for length λ a regular cardinal makes e ω = λ in the generic extension, as outlined above. Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

70 So P is ccc, and hence preserves cardinals and cofinalities. Given a generic filter G for P, the union of the π components of the conditions in G is a predictor, with domain the union of the d components, which predicts every ground model function from ω to ω. Iterating P with finite support for length λ a regular cardinal makes e ω = λ in the generic extension, as outlined above. To show that b ω = ω 1 in the extension, we use the following lemma: Lemma If F is an unbounded (with respect to ) family of functions from ω to ω, then P ˇF is unbounded. Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

71 To prove the Lemma, we use the following definitions. For a condition p = d, π, F P define I p = {f dom(d) : f F }. For ḣ a P-name for a function in ωω define h d,π,i (ω + 1) ω by h d,π,i (n) = min{m ω : there is no p P of the form p = d, π, F with I p = I such that p ḣ(n) > m}. Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

72 To prove the Lemma, we use the following definitions. For a condition p = d, π, F P define I p = {f dom(d) : f F }. For ḣ a P-name for a function in ωω define h d,π,i (ω + 1) ω by h d,π,i (n) = min{m ω : there is no p P of the form p = d, π, F with I p = I such that p ḣ(n) > m}. Main Claim Actually, h d,π,i ω ω. Proof. Compactness of ω. Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

73 Part II Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

74 Recap π predicts f if for all large enough n in dom(π), π(n)(f n) = f (n). e ω = min{ F : F ω ω predictor π f F(π does not predict f )} Want to force e ω > b ω with a long iteration. Conditions in the individual forcing step: d, π, F finite approximations to the generic predictor and its domain, and a promise to predict the functions in F. Lemma: F unbounded before the forcing F unbounded after. Towards proof of the Lemma, let I p = {f dom(d) : f F }, and given a P-name ḣ for a funtion ω ω, define h d,π,i (n) = min{m ω : there is no p P of the form p = d, π, F Main Claim: h d,π,i (n) is finite for all n. with I p = I such that p ḣ(n) > m}. Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

75 How the Main Claim proves the Lemma Given ḣ, there are only countably many functions h d,π,i (as d, π and I vary), so there is some h : ω ω such that h d,π,i h. Since F is unbounded there is some f F such that f (n) > h (n) for infinitely many n. Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

76 How the Main Claim proves the Lemma Given ḣ, there are only countably many functions h d,π,i (as d, π and I vary), so there is some h : ω ω such that h d,π,i h. Since F is unbounded there is some f F such that f (n) > h (n) for infinitely many n. Then also P ˇf (n) > ḣ(n) for infinitely many n. ( ) Otherwise, for some m and some p we would have p P n ˇm(ˇf (n) ḣ(n)). But taking d, π and I corresponding to p and n m such that f (n) > h (n) h d,π,i (n), we d have p P ȟ d,π,i (ň) < ˇf (ň) ḣ(ň), contradicting the definition of h d,π,i. So ( ) holds, and so F is not bounded by the function named by ḣ, which was arbitrary; so F remains unbounded. Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

77 Proving the Main Claim Main Claim: h d,π,i (n) is finite for all n, where h d,π,i (n) = min{m ω : there is no p P of the form p = d, π, F with I p = I such that p ḣ(n) > m}. Suppose not. Then there are d, π, I and p i for i ω with p i = d, π, F i and I pi = I such that p i P ḣ(ň) > ǐ. Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

78 Proving the Main Claim Main Claim: h d,π,i (n) is finite for all n, where h d,π,i (n) = min{m ω : there is no p P of the form p = d, π, F with I p = I such that p ḣ(n) > m}. Suppose not. Then there are d, π, I and p i for i ω with p i = d, π, F i and I pi = I such that p i P ḣ(ň) > ǐ. Let s index I as I = { f l : l < I } and each F i as F i = {fl i : l < I } such that fl i dom(d) = f l. We may thin out the sequence of p i so that for each l < I, either g l ω ω i(f i l i = g l i), or i l ω ĝ l ω i l i(f i l i l = ĝ l f i l (i l) > i). Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

79 Proving the Main Claim Main Claim: h d,π,i (n) is finite for all n, where h d,π,i (n) = min{m ω : there is no p P of the form p = d, π, F with I p = I such that p ḣ(n) > m}. Suppose not. Then there are d, π, I and p i for i ω with p i = d, π, F i and I pi = I such that p i P ḣ(ň) > ǐ. Let s index I as I = { f l : l < I } and each F i as F i = {fl i : l < I } such that fl i dom(d) = f l. We may thin out the sequence of p i so that for each l < I, either g l ω ω i(f i l i = g l i), or i l ω ĝ l ω i l i(f i l i l = ĝ l f i l (i l) > i). Compactness of ω has been used here! Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

80 With the p i thinned out to get this dichotomy, we have enough hands-on control to build a contradiction. Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

81 Generalising to κ (Joint work with Jörg Brendle) To carry over the proof for the single step of the iteration, the main obstacle is to generalise this thinning out. Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

82 Generalising to κ (Joint work with Jörg Brendle) To carry over the proof for the single step of the iteration, the main obstacle is to generalise this thinning out. There is a natural tree to consider the set of all restrictions fl i k. But it would still be helpful to think of climbing up through the tree rather than just using a branch that the tree property hands down to use. Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

83 Generalising to κ (Joint work with Jörg Brendle) To carry over the proof for the single step of the iteration, the main obstacle is to generalise this thinning out. There is a natural tree to consider the set of all restrictions fl i k. But it would still be helpful to think of climbing up through the tree rather than just using a branch that the tree property hands down to use. To do this we can use the embedding form of weak compactness to give us an ultrafilter to follow. Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

84 Proposition Let κ be a weakly compact cardinal. Then for suitable structures M of size κ for a vocabulary of size at most κ, there is an M-normal ultrafilter: a set U such that M,, U = U is a κ-complete normal ultrafilter on κ. In particular, if M <κ M, then U really is closed under < κ-fold intersections. Proof sketch. Use the embedding formulation of weak compactness, define for X κ in M, X U j(x ) κ. Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

85 So working in a κ-sized model containing everything needed (such as the tree), we can use this ultrafilter to guide our way up the tree, and at the end, use normality to get the final thinned out sequence. We get: Lemma Suppose κ is a weakly compact cardinal, γ is a cardinal less than κ, and for each β γ, f δ β : δ κ is a sequence of functions in κκ. Then there is a strictly increasing sequence of ordinals less than κ, δ η : η κ, such that for every β γ, either (a) β : g β κ κ η < κ(f δη β η = g β η) or (b) β : ι β < κ ĝ β κ ι β η < κ(f δη β ι β = ĝ β f δη β (ι β) η). Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

86 Finishing the generalisation The above Lemma is enough to generalise the single forcing step. How about the rest of the proof? Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

87 Finishing the generalisation The above Lemma is enough to generalise the single forcing step. How about the rest of the proof? More problems: We want κ to remain weakly compact for later stages of the forcing. Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

88 Finishing the generalisation The above Lemma is enough to generalise the single forcing step. How about the rest of the proof? More problems: We want κ to remain weakly compact for later stages of the forcing. Solution: Johnstone [3] showed that strongly unfoldable cardinals, which are somewhat stronger than weakly compact cardinals but still far below supercompact, can be made indestructible to a class of forcings including these. So we just assume this stronger large cardinals. Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

89 Finishing the generalisation The above Lemma is enough to generalise the single forcing step. How about the rest of the proof? More problems: We want κ to remain weakly compact for later stages of the forcing. Solution: Johnstone [3] showed that strongly unfoldable cardinals, which are somewhat stronger than weakly compact cardinals but still far below supercompact, can be made indestructible to a class of forcings including these. So we just assume this stronger large cardinals. The iteration theorems that deal with limit stages of the iteration in the ω case don t carry over for small cofinality limit stages. Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

90 Finishing the generalisation The above Lemma is enough to generalise the single forcing step. How about the rest of the proof? More problems: We want κ to remain weakly compact for later stages of the forcing. Solution: Johnstone [3] showed that strongly unfoldable cardinals, which are somewhat stronger than weakly compact cardinals but still far below supercompact, can be made indestructible to a class of forcings including these. So we just assume this stronger large cardinals. The iteration theorems that deal with limit stages of the iteration in the ω case don t carry over for small cofinality limit stages. Solution: Work with the fact that P is κ centred with canonical lower bounds (caution: not written down yet... ). Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

91 Questions What large cardinal assumption is really needed for e κ > b κ? Strong unfoldability? Weak compactness? None? Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

92 Questions What large cardinal assumption is really needed for e κ > b κ? Strong unfoldability? Weak compactness? None? What other cardinal characteristics of the continuum results use compactness & need weak compactness to generalise? Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

93 Questions What large cardinal assumption is really needed for e κ > b κ? Strong unfoldability? Weak compactness? None? What other cardinal characteristics of the continuum results use compactness & need weak compactness to generalise? This is saying something about the necessity of compactness for these arguments from the ω case. Is there an interesting way to view this from a reverse mathematics perspective? Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32

94 Andreas Blass. Cardinal characteristics and the product of countably many infinite cyclic groups. Journal of Algebra, 169: , Jörg Brendle and Saharon Shelah. Evasion and prediction II. Journal of the London Mathematical Society, 53(1):19 27, Thomas Johnstone. Strongly unfoldable cardinals made indestructible. Journal of Symbolic Logic, 73(4): , December Jindřich Zapletal. Splitting number at uncountable cardinals. Journal of Symbolic Logic, 62(1):35 42, Andrew Brooke-Taylor Generalising the weak compactness of ω Generalised Baire / 32