Stevo’s Forcing Class Fall 2012 – Class 8

(This is the eighth lecture in Stevo Todorcevic’s Forcing class, held in the fall of 2012. You can find the seventh lecture here. Quotes by Stevo are in dark blue; some are deep, some are funny, some are paraphrased so use your judgement. I didn’t take these notes, so there are few Stevo quotes. Thanks to Dana Bartasova for letting me reproduce her notes here. As always I appreciate any type of feedback, including reporting typos, in the comments below.)

Warmup Lemma

Lemma: CC implies \theta_2 = \aleph_2

proof. Given a function f: \omega_2 \times \omega_2 \rightarrow \omega we need to find infinite sets A,B \subseteq \omega_2 such that f \upharpoonright [A \times B] is constant.

By CC, find an elementary submodel M \prec H_{\aleph_3} such that f \in M , \alpha := M \cap \omega_1 is countable, but M \cap \omega_2 is uncountable. Find an uncountable x \subseteq M \cap \omega_2 and n < \omega such that f(\alpha, \beta) = n for all \beta \in X .

By elementarity of M , for all finite F \subseteq X the set \{\xi \neq \alpha : \forall \beta \in F, f(\alpha,\beta = n)\} is unbounded in \alpha .

We are done by the previous agrument: Construct \{\alpha_i : i < \omega\} \subseteq \alpha , \{\beta_i : i < \omega\} \subseteq X such that f(\alpha_i, \beta_j) = n for all i,j \in \omega . [QED]

The CC Theorem

Theorem. CC failing is equivalent to the following:

There is an e:[\omega_2]^2 \rightarrow \omega_1 such that:

  1. e(\alpha, \gamma) \leq \max_{\alpha \leq \beta \leq \gamma < \omega_2} \{e(\alpha, \beta), e(\beta, \gamma)\} ,
  2. e(\alpha, \beta) \leq \max_{\alpha \leq \beta \leq \gamma < \omega_2} \{e(\alpha, \gamma), e(\beta, \gamma)\} ,
  3. For any uncountable family \mathcal{A} of pairwise disjoint finite subsets of \omega_2 and \nu < \omega_1 there is an uncountable \mathcal{B} \subseteq \mathcal{A} such that (\forall a \neq b \in \mathcal{B})(\forall \alpha \in a)(\forall \beta \in b)[e(\alpha, \beta) > \nu]

Note 1: For all \beta , e(\dot, \beta): \beta \rightarrow \omega_1 is countable-to-1.

Otherwise, we get an uncountable \mathcal{A} \subseteq \beta and \nu < \omega_1 such that e(\alpha, \beta) \leq \nu for all \alpha \in \mathcal{A} . By (b) e(\alpha, \alpha^\prime) \leq \nu for all \alpha, \alpha^\prime \in \mathcal{A} .

Note 2: If (C_\alpha : \alpha < \omega_2) is a square sequence (\square_{\omega_1}) and \rho = \rho(C_\alpha : \alpha <\omega_2) , then \rho has these properties. Consequently, \square_{\omega_1} implies \neg CC. (Use \square_{\omega_1} to construct a ccc poset that adds a kurepa tree, but that would be killed by CC.)

Proof of Theorem.

[\neg 1 \Rightarrow \neg 2 ] Fix e as above. We need to force a function g: \omega_2 \times \omega_2 \rightarrow \omega which is not constant on the product of any two infinite subsets of \omega_2 .

Let \mathbb{P} be the collection of all finite maps p : [\mathcal{D}_p]^2 \rightarrow \omega such that
\displaystyle   \textbf{(*)}\forall \xi<\alpha<\beta \in \mathcal{D}_p, e(\xi, \alpha) > e(\alpha, \beta) \Rightarrow p(\xi, \alpha) \neq p(\xi, \beta)

For p,q \in \mathbb{P} , set p \leq q if p \supseteq q and
\displaystyle   \textbf{(**)} \forall \alpha < \beta \in \mathcal{D}_q, \forall \xi \in \mathcal{D}_p \setminus \mathcal{D}_q, p(\xi, \alpha) \neq p(\xi, \beta)

Claim: \mathbb{P} is ccc (even property K!)

proof. Let A \subseteq \mathbb{P} be uncountable. We may assume that

  • \{\mathcal{D}_p : p \in A\} forms a Delta system with root D
  • F_p := e^{\prime\prime} [\mathcal{D}_p]^2 forms a Delta system on \omega_1 with root F , increasing
  • The p are isomorphic as finite models (+ identity on \mathcal{D} ??)

By Lemma 3(c) find an uncountable B \subseteq A such that
\displaystyle  \textbf{(***)} \forall p \neq q \in N \forall \alpha \in \mathcal{D}_p \setminus D \forall \beta \in \mathcal{D}_q \setminus D e(\alpha, \beta) > \max (F)

Claim: B is a 2-linked subset of \mathbb{P} .

Take p \neq q \in B . Define r : [\mathcal{D}_p \cup \mathcal{D}_q]^2 \rightarrow \omega to extend p,q and to be 1-1 on new pairs, avoiding odd values.

Check (*) on r.

Let \xi < \alpha <\beta be given in \mathcal{D}_r such that e(\xi, \alpha) > e(\alpha, \beta) . We need to show that r(\xi, \alpha) \neq r(\xi, \beta) .

We may assume \{\xi, \alpha\}, \{\xi, \beta\} \in [\mathcal{D}_p]^2 \cup [\mathcal{D}_q]^2 . So we may assume \xi \in D , \alpha \in \mathcal{D}_p \setminus \mathcal{D}_q and \beta \in \mathcal{D}_q \setminus \mathcal{D}_p .

By properties (a), (b) of e we know that:
e(\xi, \alpha) > e(\alpha, \beta) implies e(\xi, \alpha) = e(\xi, \beta)
implies e(\xi, \alpha) = e(\xi, \beta) \in F (A contradiction. This line might be superfluous.)
e (\xi, \beta) \leq \max \{e(\xi, \alpha), e(\alpha, \beta)\} = e(\xi, \alpha)
e (\xi, \alpha) \leq \max \{e(\xi, \beta), e(\alpha, \beta)\} = e(\xi, \beta)

Check that r \leq p,q . (We check only r \leq p ).

Choose \alpha <\beta in \mathcal{D}_p and q \in \mathcal{D}_q \setminus \mathcal{D}_p such that \xi < \alpha <\beta . We need to show that r(\xi, \alpha) = r(\xi, \beta) .

This is automatic if one of the pairs is new. So WLOG, \{\xi, \alpha\}, \{\xi, \beta\} \in [\mathcal{D}_p]^2 \cup [\mathcal{D}_q]^2 . But then r(\xi, \alpha) = q (\xi, \alpha) and r(\xi, \beta) = q (\xi, \beta) .

Now we look at (*) for q, (i.e. to see that e(\xi, \alpha) > e(\alpha, \beta) .)

Otherwise, e(\xi, \alpha) \in F since \alpha, \beta \in D .

By the isomorphism condition we get that \forall s \in B, e(\xi(s), \alpha) \in F , where \xi(s) is the image of \xi relative to the isomorphism between s and q .

So X = \{ \xi(s) : s \in B\} is an uncountable subset of \alpha on which e( \dot, \alpha) is constant. (A contradiction with note 1).

Forcing with \mathbb{P} gives f: [\omega_2]^2 \rightarrow \omega such that \{\xi < \alpha : f(\xi, \alpha) = f(\xi, \beta)\} for all \alpha, \beta .

Define g: \omega_2 \times \omega_2 \rightarrow \omega by g(\alpha, \beta) =

  • 2 f(\alpha, \beta) + 1 if \alpha <\beta
  • 0 if \alpha = \beta
  • 2 f(\alpha, \beta) + 2 if \alpha > \beta

Note g is not constant on any product of infinite sets.

Definable CH

Recall: CH^2 is equivalent to “Every compact ccc space of weight \leq \mathfrak{c} ” has a Luzin set.

Note: CH^2 implies CH^{<\omega}

CH^{<\omega} says “Every ccc poset \mathbb{P} of size \leq \mathfrak{c} has an uncountable collection \mathcal{F} of centred subsets of \mathbb{P} such that for all dense open D \subseteq \mathbb{P} the set \{F \in \mathcal{F} : F \cap D \neq \emptyset\} is countable.”

Note: If we replace \mathcal{F} by filters, then this is equivalent to CH.

\mathbb{P} := (P, \leq_\mathbb{P}, \textrm{Comp}_\mathbb{P}^n)_{n<\omega} , where \textrm{Comp}_\mathbb{P}^n (p_0, ..., p_{n-1}) iff \exists q \in \mathbb{P} such that q \leq p_i for all i < n .

\mathbb{P} is definable if \mathbb{P} \subseteq \mathbb{R} and there exists \leq, \textrm{Comp}^n (n < \omega) in L(\R) such that \leq_\mathbb{P} = <\upharpoonright P and \textrm{Comp}_\mathbb{P}^n = \textrm{Comp}^n \upharpoonright P for all n < \omega .

Exercise: Every \sigma -centred poset in definable.

All posets we have constructed so far (except for the one in CH^3 implies \theta_2 = \aleph_2 ) are definable.

Definable CH” is the statement “CH^2 for definable posets”.

Note: Definable CH^2 implies that all sets of reals are \aleph_1 -Borel.

Theorem: Definable CH^2 is not equivalent to CH. (Large cardinals, \mathfrak{c} = \aleph_{\omega_1} )

Question: Is CH^2 the same as CH?

4 thoughts on “Stevo’s Forcing Class Fall 2012 – Class 8”

  1. Here are some quotes I recorded:

    “What about Namba in \omega_1 ? I like this better. We will analyze Namba forcing on \omega_1 , using \mathfrak m > \omega_1 . (We really only need a weaker hypothesis, such as either \mathfrak p or \mathfrak b .)”

    “This proof has several interesting useful things in it.” (referring, I think, to the theorem that CC is equivalent to the statement “Every ccc poset forces \theta_2 = \omega_2 “)

    “What is the combinatorial argument we always use?”

    “You can’t control infinite sets. Infinite sets show up all over the place.”

    “Whenever you construct a partial ordering of size \aleph_2 , you have to ask: ‘What is the influence of Chang’s Conjecture on it?’ ”

    (Regarding the Claim that P is ccc:) “Actually, I’m going to show it has Property K. It probably also has precalibre \aleph_1 , but I haven’t tried to show that. It would be a good exercise to show that it has precalibre \aleph_1 .”

    \Delta -system on \omega_2 is very tricky, not like on \aleph_1 . On \omega_1 , \Delta -systems are beautiful: root, tail, tail. On \omega_2 , you have: root, tail, tail, pieces of root, tail, tail, etc.”

    “If you look at the analogue between set theory and geometry (and there are many such analogues), we have: \omega_2 has dimension 3, \omega_3 has dimension 4, etc. We cannot visualize 4 dimensions, so it is harder to understand \omega_3 .”

    “So far I haven’t seen an interesting ccc partial ordering on \omega_3 . Maybe there will be one.”


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