Combinatorial dichotomies and cardinal invariants

Transcription

1 Dilip Raghavan (Joint with Stevo Todorcevic) National University of Singapore ASL North American annual meeting, University of Waterloo May 9, 2013

2 Outline 1 The Project 2 3 4

3 Calibrating some consequences of PFA Cardinal invariants can be used to calibrate certain mathematical statements in the presence of some combinatorial dichotomies. These statements are consequences of PFA that contradict CH.

4 Calibrating some consequences of PFA Cardinal invariants can be used to calibrate certain mathematical statements in the presence of some combinatorial dichotomies. These statements are consequences of PFA that contradict CH. Prototypical Theorem Assume ZFC + CD. Then the following are equivalent: 1 x > ω 1. 2 φ. Here CD is a combinatorial dichotomy, x is some cardinal invariant, and φ is some mathematical statement.

5 Two recent examples The statement φ can be varied: can come from combinatorics, topology, Banach space theory...

6 Two recent examples The statement φ can be varied: can come from combinatorics, topology, Banach space theory... Theorem (Todorcevic and Torres-Perez) Assume ZFC + Rado s Conjecture (RC). Then the following are equivalent: 1 c > ω 1 2 There are no special ω 2 -Aronszajn trees.

7 Two recent examples Theorem (Brech and Todorcevic) Assume ZFC + (PID). Then the following are equivalent: 1 b > ω 1. 2 Every non-separable Asplund space has an uncountable bi-orthogonal system.

8 Two recent examples Theorem (Brech and Todorcevic) Assume ZFC + (PID). Then the following are equivalent: 1 b > ω 1. 2 Every non-separable Asplund space has an uncountable bi-orthogonal system. In these examples PID and RC are the combinatorial dichotomies and c and b are the cardinal invariants. This talk will mostly focus on PID.

9 P ideals Let X be an uncountable set. An ideal I [X] ω is called a P-ideal if for every countable collection {a n : n ω} I, there is a I such that n ω [a n a]. Here a b means a \ b is finite. All ideals are assumed to be non-principal, meaning that [X] <ω I.

10 P ideals Let X be an uncountable set. An ideal I [X] ω is called a P-ideal if for every countable collection {a n : n ω} I, there is a I such that n ω [a n a]. Here a b means a \ b is finite. All ideals are assumed to be non-principal, meaning that [X] <ω I. Example Let X be an uncountable set and let Y be an uncountable subset of X. I = [X] <ω [Y] ω.

11 Example Let X be an uncountable set. Let {X n : n ω} be a collection of uncountable subset of X. Put I = { a [X] ω : n ω [ a X n < ω] }

12 The P-ideal dichotomy (PID) is the following statement: For any P-ideal I on an uncountable set X either (1) There is an uncountable set Y X such that [Y] ω I (so I contains a copy of the first example) or (2) There exist {X n : n ω} such that the X n are pairwise disjoint, X = n ωx n, and n ω [[X n ] ω I = 0] (so I is contained in a copy of the second example).

13 PID is a consequence of PFA However PID is consistent with CH.

14 PID is a consequence of PFA However PID is consistent with CH. Some consequences of PID: Suslin s Hypothesis (Todorcevic). Singular Cardinals Hypothesis (Viale). κ,ω fails for all uncountable cardinals κ (R.).

15 PID is a consequence of PFA However PID is consistent with CH. Some consequences of PID: Suslin s Hypothesis (Todorcevic). Singular Cardinals Hypothesis (Viale). κ,ω fails for all uncountable cardinals κ (R.). Question How does PID influence statements that contradict CH?

16 PID is a consequence of PFA However PID is consistent with CH. Some consequences of PID: Suslin s Hypothesis (Todorcevic). Singular Cardinals Hypothesis (Viale). κ,ω fails for all uncountable cardinals κ (R.). Question How does PID influence statements that contradict CH? Answer: It tends to push many such statements down to combinatorial questions about sets of reals.

17 Cardinal Invariants For functions f, g ω ω, f < g means n ω [f(n) < g(n)]. A set F ω ω is said to be unbounded if there is no g ω ω such that f F [f < g]. b = min { F : F ω ω F is unbounded}

18 Cardinal Invariants For functions f, g ω ω, f < g means n ω [f(n) < g(n)]. A set F ω ω is said to be unbounded if there is no g ω ω such that f F [f < g]. b = min { F : F ω ω F is unbounded} A family F [ω] ω is said to have the finite intersection property (FIP) if for any a 0,..., a k F, a 0 a k is infinite. p = min { F : F [ω] ω F has the FIP b [ω] ω a F [b a] }

19 Cardinal Invariants cov(m) is the least κ such that R can be covered by κ many meager sets. c = 2 ω.

20 Cardinal Invariants cov(m) is the least κ such that R can be covered by κ many meager sets. c = 2 ω. It is easy to show ω 1 p b c. Also p cov(m) c. b and cov(m) are independent.

21 Cardinal Invariants cov(m) is the least κ such that R can be covered by κ many meager sets. c = 2 ω. It is easy to show ω 1 p b c. Also p cov(m) c. b and cov(m) are independent. PID + p > ω 1 implies many of the consequences of PFA that contradict CH.

22 An example A regular hereditarily separable space that is not Lindelöf is called an S-Space. Theorem (Todorcevic) Assume PID + p > ω 1. Then there are no S Spaces. Moreover, if b = ω 1, then there is a (first countable) S Space.

23 An example A regular hereditarily separable space that is not Lindelöf is called an S-Space. Theorem (Todorcevic) Assume PID + p > ω 1. Then there are no S Spaces. Moreover, if b = ω 1, then there is a (first countable) S Space. There is a gap between b and p. There are other examples like this. We will see 2 in a bit.

24 The project General Problem 1 Given a statement φ which is a consequence of PID + MA ℵ1, find a cardinal invariant x such that φ is equivalent to x > ω 1. A slightly less ambitious project is General Problem 2 Given a statement φ which is a consequence of PID + p > ω 1, investigate whether φ is equivalent to p > ω 1

25 Why do this? Allows one to calibrate the relative strength of various consequences of PFA over ZFC + PID.

26 Why do this? Allows one to calibrate the relative strength of various consequences of PFA over ZFC + PID. One often needs to find sharper proofs. Decomposes the influence of PFA on φ into two parts: A part that is consistent with CH + the essential combinatorial bit contradicting CH captured by the cardinal invariant.

27 A model for General Problem 2 Let S be a coherent Suslin tree. PFA(S) is the following statement. If P is a poset which is proper and preserves S and {D α : α < ω 1 } is a collection of dense subsets of P, then there is a filter G on P such that α < ω 1 [G D α 0].

28 A model for General Problem 2 Let S be a coherent Suslin tree. PFA(S) is the following statement. If P is a poset which is proper and preserves S and {D α : α < ω 1 } is a collection of dense subsets of P, then there is a filter G on P such that α < ω 1 [G D α 0]. PFA(S) says that the maximal amount of PFA compatible with the existence of S holds. After forcing with S over a model of PFA(S) many consequences of PFA hold.

29 A model for General Problem 2 Let S be a coherent Suslin tree. PFA(S) is the following statement. If P is a poset which is proper and preserves S and {D α : α < ω 1 } is a collection of dense subsets of P, then there is a filter G on P such that α < ω 1 [G D α 0]. PFA(S) says that the maximal amount of PFA compatible with the existence of S holds. After forcing with S over a model of PFA(S) many consequences of PFA hold. In particular, after forcing with S, PID holds. Also after forcing with S, most cardinal invariants are greater than ω 1. For example, b > ω 1 and cov(m) > ω 1.

30 A model for General Problem 2 However p = ω 1. So after forcing with S over a model of PFA(S), one gets almost all of the consequences of PFA that are consistent with PID + p = ω 1. So if φ some consequence of PFA which you suspect is not equivalent to p > ω 1 over ZFC + PID, then this model is a good place to look.

31 5 cofinal types Given two (upward) directed posets P and Q, we say that a map f : P Q is a Tukey map if it maps (upward) unbounded sets in P to unbounded sets in Q. We say that a map g : Q P is a convergent map if the image of every (upward) cofinal subset of Q is cofinal in P.

32 5 cofinal types Given two (upward) directed posets P and Q, we say that a map f : P Q is a Tukey map if it maps (upward) unbounded sets in P to unbounded sets in Q. We say that a map g : Q P is a convergent map if the image of every (upward) cofinal subset of Q is cofinal in P. Fact: There is a Tukey map f : P Q iff there is a convergent g : Q P We say P is Tukey reducible to Q and we write P T Q if there is a a Tukey map f : P Q.

33 5 cofinal types We have a natural equivalence P T Q iff P T Q and Q T P. Then we say P and Q are Tukey equivalent or have the same Tukey type.

34 5 cofinal types We have a natural equivalence P T Q iff P T Q and Q T P. Then we say P and Q are Tukey equivalent or have the same Tukey type. Fact: P T Q iff both P and Q embed a cofinal subsets of another directed set R.

35 5 cofinal types We have a natural equivalence P T Q iff P T Q and Q T P. Then we say P and Q are Tukey equivalent or have the same Tukey type. Fact: P T Q iff both P and Q embed a cofinal subsets of another directed set R. Theorem (Todorcevic[2]) Under PFA there are only 5 Tukey types of size at most ℵ 1 : 1, ω, ω 1, ω ω 1, [ω 1 ] <ω. In fact, PID + p > ω 1 implies this.

36 5 cofinal types cof(f σ ) is the least κ such that there exists a tall F σ ideal I on ω and a directed cofinal X I (i.e. X is cofinal in I, ) such that X = κ.

37 5 cofinal types cof(f σ ) is the least κ such that there exists a tall F σ ideal I on ω and a directed cofinal X I (i.e. X is cofinal in I, ) such that X = κ. cof(f σ ) has been investigated by Hrušák and Zapletal [1]. It is not hard to see that cov(m) cof(f σ ).

38 5 cofinal types Theorem (R. and Todorcevic) Assume PID. The following are equivalent. 1 min { b, cof(f σ ) } > ω , ω, ω 1, ω ω 1, and [ω 1 ] <ω are the only cofinal types of directed sets Remark of size at most ℵ 1. b and cof(f σ ) are independent even over ZFC + PID. So one cannot really simplify (1) above.

39 A strong version of Dushnik-Miller Theorem For an ordinal α, ω 1 (ω 1, α) 2 means that for any c : [ω 1 ] 2 2 either there exists X [ω 1 ] ω 1 such that c [X] 2 = {0}, or there exists X ω 1 with otp(x) = α and c [X] 2 = {1}.

40 A strong version of Dushnik-Miller Theorem For an ordinal α, ω 1 (ω 1, α) 2 means that for any c : [ω 1 ] 2 2 either there exists X [ω 1 ] ω 1 such that c [X] 2 = {0}, or there exists X ω 1 with otp(x) = α and c [X] 2 = {1}. ω 1 (ω 1, ω + 1) 2 is a theorem of ZFC (Dushnik-Miller) ω 1 (ω 1, ω 1 ) 2 is false (Sierpinski).

41 A strong version of Dushnik-Miller Theorem For an ordinal α, ω 1 (ω 1, α) 2 means that for any c : [ω 1 ] 2 2 either there exists X [ω 1 ] ω 1 such that c [X] 2 = {0}, or there exists X ω 1 with otp(x) = α and c [X] 2 = {1}. ω 1 (ω 1, ω + 1) 2 is a theorem of ZFC (Dushnik-Miller) ω 1 (ω 1, ω 1 ) 2 is false (Sierpinski). Theorem (Todorcevic) PFA implies that ω 1 (ω 1, α) 2, for every α < ω 1. In fact, this follows from PID + p > ω 1. Moreover, if b = ω 1, then ω 1 (ω 1, ω + 2) 2.

42 A strong version of Dushnik-Miller Theorem Theorem (R. and Todorcevic) PFA(S) implies that the coherent Suslin tree S forces ω 1 hold for all α < ω 1. (ω 1, α) 2 to

43 A strong version of Dushnik-Miller Theorem For any A S, A [2] = {{a, b} : a, b A and a < b}. For t S, pred(t) denotes the set of predecessors of t, that is {s S : s t}. For a set X and t S, pred X (t) = pred(t) X.

44 A strong version of Dushnik-Miller Theorem For any A S, A [2] = {{a, b} : a, b A and a < b}. For t S, pred(t) denotes the set of predecessors of t, that is {s S : s t}. For a set X and t S, pred X (t) = pred(t) X. We also consider the following variation of A [2] : Let Y S and g : Y S. Then Y [2] g denotes { } {a, b} : a, b Y and a < b and g(a) b.

45 A strong version of Dushnik-Miller Theorem If S S and c : S [2] 2 is a coloring, then K i = { {s, t} S [2] : c ({s, t}) = i }, for any i {0, 1}

46 A strong version of Dushnik-Miller Theorem Theorem Assume PFA(S). Let S [S] ω 1 and c : S [2] 2. Then either there exist Y [S] ω 1 and g : Y S such that y Y [g(y) y] and Y [2] g K 0 or for each α < ω 1, there exists s S and B pred S (s) such that otp(b) = α and B [2] K 1.

47 A strong version of Dushnik-Miller Theorem Theorem Assume PFA(S). Let S [S] ω 1 and c : S [2] 2. Then either there exist Y [S] ω 1 and g : Y S such that y Y [g(y) y] and Y [2] g K 0 or for each α < ω 1, there exists s S and B pred S (s) such that otp(b) = α and B [2] K 1. Theorem Let S be a (coherent) Suslin tree. There is c : S [2] 2 such that 1 There is no X [S] ω 1 such that X [2] K 0. 2 There is no s S and B pred S (s) such that otp(b) = ω 2 and B [2] K 1.

48 Question Can we find a cardinal invariant x such that x > ω 1 is equivalent over ZFC + PID to ω 1 (ω 1, α) 2 for all α < ω 1?

49 Question Can we find a cardinal invariant x such that x > ω 1 is equivalent over ZFC + PID to ω 1 (ω 1, α) 2 for all α < ω 1? Question Can we find a cardinal invariant x such that x > ω 1 is equivalent over ZFC + PID to the statement that there are no S spaces? Question Is there an S space after forcing with the coherent Suslin tree S over a model of PFA(S)?

50 M. Hrušák and J. Zapletal, Cofinalities of borel ideals, preprint (2012). S. Todorčević, Directed sets and cofinal types, Trans. Amer. Math. Soc. 290 (1985), no. 2,