Stevo’s Forcing Class Fall 2012 – Class 10

(This is the tenth lecture in Stevo Todorcevic’s Forcing class, held in the fall of 2012. You can find the ninth lecture here. Quotes by Stevo are in dark blue; some are deep, some are funny, some are paraphrased so use your judgement. As always I appreciate any type of feedback, including reporting typos, in the comments below. I am sure that this one has a ton of typos. There are also some omissions which I will try to patch up.)

Side Condition Method

Side conditions are elements of \epsilon\textrm{-Col}(\kappa) , that is, \in -chains of CESMs of some H_\theta .

Remark 1. For p, \epsilon\textrm{-Col}(\kappa) \in M \prec H_{{2^\kappa}^+} , the condition p \cup \{M \cap H_\kappa\} is (M, \epsilon\textrm{-Col}(\kappa)) -generic in a very strong way:

Fact. \forall r \leq p \cup \{M \cap H_\kappa\} , \forall \mathcal{D} (\in M isn’t needed“) dense open in \epsilon\textrm{-Col}(\kappa) \cap M , \exists \overline{r} \in \mathcal{D} \cap M such that r \not \perp \overline{r} . i.e.
\displaystyle  p \cup \{M \cap H_\kappa\} \Vdash \dot G \cap \epsilon\textrm{-Col}(\kappa) \cap M \textrm{ is } (M, \epsilon\textrm{-Col}(\kappa))\textrm{-generic}

They add reals, but they don’t add branches to trees.

This works most of the time, so long as p = (H_p, \mathcal{N}_p) are on good terms.

I strongly suggest you go through the Fubini argument. Most of the errors and fantastic results happen here.

For the ideal (XXX), \mathcal{H} := \mathcal{P} \setminus I . You can step it up to \mathcal{H}^{n+1} := \{W \subseteq S^{n+1} : \{x \in S :\{\vec y : x ^\frown \vec y \in W\} \in \mathcal{H}^n\}\in \mathcal{H}\} .

This tells you you have lots of chance for continuing.”

This is the usual Fubini definition. You can switch first and last, but you cannot go to the middle and split.

(Why we do this side condition thing. When \mathcal{H}_p \in W \in \min \mathcal{N}_p and W \in S^{\vert \mathcal{H}_p \vert} ) then W \in \mathcal{H}^{\vert \mathcal{H}_p \vert} . i.e. It is large.

The OCA proof is actually more simple than the general case.”

The first step is an easy argument (world of squares). After that you are in the world of rectangles. There is no way you can avoid this. There is no clever move.

Forcing PID – Example of a construction which uses the rectangular copying procedure

Let S be a set with \mathcal{I} \subseteq [S]^{\leq \aleph_0} a P-Ideal. Assume S cannot be written as the countable union \bigcup_{n < \omega} X_n , where X_n \perp \mathcal{I} for all n .

To get the PID, we need to force an uncountable Y \subseteq S such that [Y]^{\aleph_0} \subseteq \mathcal{I} .

What is the natural forcing? You need to make some choices.

Let S,\mathcal{I} \in H_\kappa , with N \prec H_\kappa \mapsto a_N \subseteq N \cap S , where a_N \in \mathcal{I} and \forall x \in \mathcal{I}\cap N, x \subseteq^* a_N .

[The condition are] simply functions as before.”

The conditions are maps p : \mathcal{N}_p \rightarrow S where:

  • \mathcal{N}_p \in \epsilon\textrm{-Col}(\kappa^+) ; and
  • \min \mathcal{N}_p contains S, \mathcal{I} ; and
  • N \mapsto a_N ; and
  • H_p = \textrm{dom} (p) is \mathcal{I}- separated by \mathcal{N}_p .

\mathcal{I} is really the \sigma- ideal generated by sets orthogonal to \mathcal{I} . (???)

Extension. p \leq q iff

  • p \supseteq q ; and
  • \forall M \in \mathcal{N}_q, \forall N \in \mathcal{N}_p \cap M we have p(N) \in a_M .

This is a very natural thing if you are trying to force what we are trying to force.”

So \forall M \in \mathcal{N}_p, r \Vdash \dot y_G \cap M \subseteq^* a_M is the intended consequence.

Go carefully through the copying argument.

What if we don’t want to collapse? There is an easy and useful adaptation modification.”

A Cardinal Preserving Modification of the Method of Side Conditions

For this we have to assume something about the ground model. But for now we pretend.

This is much like before, except we are allowed to blow up \mathcal{N}_k to something isomorphic.”

Let \PP_\kappa be the collection of all finite conditions p = \{\mathcal{N}_0, \mathcal{N}_1, ..., \mathcal{N}_k\} , where:

  • Each \mathcal{N}_i is a finite collection of isomorphic CESMs of H_\theta ; and
  • \forall i \leq j \leq k, \forall M \in \mathcal{N}_i there is an N \in \mathcal{N}_j such that M \in N .

Now let us check the (strong) properness of these. Again, it is the same proof.

: \PP_\kappa is (strongly) proper.

. Start with p, \PP_\kappa \in M \prec (H_\lambda, \in) , where \lambda can be taken to be (2^\kappa)^+ .

Then q := p \cup \{M \cap H_\kappa\} is (strongly)-(\PP_\kappa, M) -generic.

This is a very powerful trick of self-reference. A slippery trick.

If you are doing something on \aleph_1 , this is enough. Sometimes \aleph_2 . Not for \aleph_2 -PID.

: \PP_\kappa is (strongly)-\aleph_2 -cc.

Is collapsing that bad?” – Oswaldo

Yes, it is bad for your health.”

Semi-Proper Posets

Now we go to some uses, but we need a less restrictive class of posets. Semi-proper is important because that is how MM was discovered.

Definition. A poset \mathbb{P} is semi-proper (really, \aleph_1 -semi-proper, but that has disappeared over the years.) if for every large enough H_\theta and CESM M \prec H_\kappa containing some p \in \mathbb{P} and \mathbb{P} itself, there is an (M, \mathbb{P}) -semi-generic extension q \leq p .

Definition. With the situation above, q is an (M, \mathbb{P}) -semi-generic extension if for every partial function f : \mathbb{P} \rightarrow \omega_1 with f\in M, r \leq q and r \in \textrm{dom} (f) there is an \alpha < M \cap \omega_1 and \overline{r} \in f^{-1} (\alpha) such that r \not \perp \overline{r} .

This is about \aleph_1 -partitions.”

Most of the time f^{-1}(\alpha) \in M , but it doesn’t have to be.

Lemma (for checking semi-proper). Suppose M, p, \mathbb{P} are as above, and suppose there is another CESM \overline{M} \prec H_\kappa such that M \subseteq \overline{M} and M \cap \omega_1 = \overline{M} \cap \omega_1 . Suppose q \leq p is (\overline{M}, \mathbb{P}) -generic, THEN q is (M, \mathbb{P}) -semi-generic.

In some sense, this is all that is known about proving semi-proper.

You are going to find \overline{r} \in \textrm{dom} (f) such that (???), where \overline{r} is an \omega_1 -end-extension.

Let us examine this notion, by testing them.

Theorem 1. There is an (\omega_1) -stationary set preserving poset of size \aleph_2 which collapses \aleph_2 .

Theorem 2
. If a semi-proper poset collapses \aleph_2 then it collapses \mathfrak{c} .

. If \mathfrak{c} > \aleph_2 , there is no semi-proper poset which collapses \aleph_2 and preserves all other cardinals.

Semi-proper posets preserves certain sets coded by ordinals in \aleph_2 .”

Let me give you the first example which is interesting even if you aren’t interested in this stuff.

Real Theorem 1
. Suppose that [\kappa]^{\aleph_0} has a stationary subset of size \kappa . THEN there is an \omega_1 -preserving poset \mathbb{P}(S) of size \kappa which collapses \kappa to \aleph_1 . If the stationary subset S is projectively stationary, then \mathbb{P}(S) preserves stationary subsets of \omega_1 .

. For the first assumption, \aleph_\omega and \aleph_{\omega+1} are the first places where that can fail.

. Suppose that E \subseteq \omega_1 and S \subseteq [\kappa]^{\aleph_0} where S(E) := \{X \in S : X \cap \omega_1 \in E\} . We say S is projectively stationary if S(E) is stationary in [\kappa]^{\aleph_0} for all stationary E \subseteq \omega_1 .

. If \kappa = \omega_n for some n \in \omega , then there is a projectively stationary subset S \subseteq [\kappa]^{\aleph_0} of cardinality >\kappa .

proof of Real Theorem 1
. “What is the \mathbb{P}(S) ?

Fix an injection i : S \rightarrow \kappa . Let \mathbb{P}(S) be the collection of all finite subsets p of S such that \forall x,y \in \mathbb{P} wither X \cup \{i(X)\} \subseteq Y or Y \cup \{i(Y)\} \subseteq X . The ordering is inclusion.

: \mathbb{P}(S) preserves \omega_1 .

Take a CESM M \prec H_\lambda (\lambda around {2^{2^\kappa}}^+ should work), such that p,\PP(S) \in M and M \cap \kappa \in S . (This uses the stationarity of S ).

Let q := p \cup \{M \cap \kappa\} . Then q is (M, \mathbb{P}(S)) -generic.

Why? By elementarity you can copy.” [End of Proof]

Question. For which S \subseteq [H_\kappa]^{\aleph_0} is \mathbb{P}(S) semi-proper?

Strong Reflection Principle (SRP)

So now we go to do something.”

There two versions of the strong reflection theorem “that are of course equivalent.” (Feng-Jech did one version, Stevo did another.)

Feng-Jech, SRP
for stationary subsets of some [\kappa]^{\aleph_0} . Let S \subseteq [\kappa]^{\aleph_0} be projectively stationary, THEN you can reflect it on a club. i.e. for every X \in [\kappa]^{\aleph_1} containing \omega_1 , there is a X \subseteq Y \in [\kappa]^{\aleph_1} such that S \cap [Y]^{\aleph_0} contains a club subset of [\kappa]^{\aleph_0} .

. Stevo’s version of SRP has to do with the amount of reflection required.

Corollary [MM]
. SRP holds.

Theorem (Shelah)
. MM is equivalent to SPFA

.  Shelah’s Theorem above is easy to show, assuming SRP.

. If S \subseteq [\omega_1]^{\aleph_0} is projectively stationary, then S contains a club. “i.e. If you project everywhere, then there is nowhere to escape! So you want \kappa \geq \omega_2 .”

. Let S [\kappa]^{\aleph_0} be projectively stationary, then there is an \omega_1 -stationary-preserving poset \mathbb{P}(S) which forces Y , which strongly reflects S . “What is the poset? It is the natural one.

You know you have to provide this chain of continuous sets:

  1. Y := \bigcup_{\alpha < \omega_1} Y_\alpha ;
  2. Y_\alpha \subseteq Y_\beta for \alpha < \beta ;
  3. Y_\lambda := \bigcup_{\alpha < \lambda} Y_\alpha for limit \lambda ;
  4. \{Y_\alpha : \alpha < \omega_1\} \subseteq S .

How do you code continuity? You say that it can be extended to something continuous.

The conditions. Let \mathbb{P}(S) be the collection of all partial maps p : \omega_1 \rightarrow [H_{\kappa^+}]^{\aleph_0} , such that:

  1. \forall \alpha \in \textrm{dom} (p), p(\alpha) \prec H_{\kappa^+} containing S, X ;
  2. \forall \alpha < \beta in \textrm{dom} (p) , p(\alpha) \in p(\beta) ;
  3. There is a continuous f: \omega_1 \rightarrow [H_\kappa]^{\aleph_0} , \in- chain of Elementary Submodels such that f \supseteq p . “This reminds me of Baumgartner” – Oswaldo. “Everyone says that. It is not. People should say this reminds them of Cohen.“;
  4. \forall \alpha \in \textrm{dom} (p), p(\alpha) \cap \kappa \in S .

Now you are doing something. Without this you have nothing.”

Claim. This preserves Stationary subsets of \omega_1 .

Let us do this.

Take E \in \textrm{stat}(\omega_1), p \in \mathbb{P}(S) and a \mathbb{P}(S) -name \dot C for a club in \omega_1 . “Who is this going to be?

Recall, S(E) := \{A \in [\kappa]^{\aleph_0} : A \cap \omega_1 \in E\} is stationary. So there is a CESM M \prec H_{\textrm{large}} containing all the objects such that M \cap \kappa \in S(E) .

Let \delta := M \cap \omega_1 . (Note \delta \in E , so E is uncountable.) Let q := p \cup \{<\delta, M \cap H_{\kappa^+}>\} .

Claim. q is an (M, \mathbb{P}(S)) -generic condition, and therefore q \Vdash \delta \in \dot C \cap E .

This is copying as before.” [End of Proof]

Consequences of SRP, left as exercises

  1. \diamond_{\omega_2}^{\textrm{cof}(\omega_1)}
  2. NS_{\omega_1} is saturated
  3. WeakRP, i.e. reflection of stationary subsets of some [\kappa]^{\aleph_0} . “This is the price you pay for the elegance of Jech-Feng SRP.
  4. 2^{\aleph_0} = 2^{\aleph_1} . “You can code reals by ordinals.

Two Coding Lemmas

I’ll tell you the real \diamond . The \diamond which is true in ZFC, but we don’t know if it has nice consequences.

Lemma. There is a sequence \{S_x : x \in [\omega_2]^{\aleph_0}\} such that:

  1. S_x \subseteq x ; and
  2. \forall W \subseteq \omega_2, S_W := \{ x \in [\aleph_2]^{\aleph_0} : s_x = W\cap x\} is projectively stationary.

proof. Look at the oscillation map. Pick a club guessing \{C_\delta : \delta < \omega_2, \textrm{cof}(\delta) = \omega_\} . Then
\displaystyle  S_x \cong \textrm{Osc}(x) = \{ n : x \cap [C_{\textrm{sup}(x)}(n), C_{\textrm{sup}(x)}(n+1)) \neq \emptyset\}
And C_\delta = \{C_\delta (n) : n < \omega\} \uparrow \delta . [End of Sketch]

Lemma (*). For every r \in [\omega]^{\omega}, \mathcal{S}_r := \{X \in [\omega_2]^{\aleph_0} : \textrm{Osc}(X) = r\} is projectively stationary.

All this stuff about coding sets is really this lemma.

11 thoughts on “Stevo’s Forcing Class Fall 2012 – Class 10”

    1. אם היית פה, יהיו פה שלשה “אסף”ים ביחד. אולי היינו מגיעים למסקנה שכל מתמטיקאי ישראלי נקרא אסף (אלא שיש פה גם אוהד).‏

      פעם הייתי בועידה שממנה היה אפשר לחשוב שכל מתמטיקאי נקרא “מנחם”. היו שם שנימנחמים: מגידור וקוג’מן.‏

      תפילות לבטחון לכולכם בארץ ישראל.‏


      1. Sounds like a sitcom….

        One Assaf sets up the other to do all his office work while he goes out on another Assaf’s hot date.


  1. Some more quotes:

    “You remember the Fubini argument from kindergarten.”

    “Lots of fantastic results come from mistaken uses of Fubini.”

    “I don’t know whether I ever published this.” (Sorry I don’t remember what the “this” refers to, but it was somewhere after “It is bad for your health” and before “Now you are doing something.”)

    “I’ll give you a break for five minutes in case you want to escape.”


  2. In the “Fact” after Lemma 1 (which is really the proof sketch of Lemma 1), the “isn’t needed” wasn’t meant to refer to \in M . I think it was referring to r \in D , which you didn’t write. The point is that in the definition of (M, \mathcal P) -generic, you require either D is dense or r \in D (it is an interesting exercise using elementarity of M to show that the two definitions are equivalent), but you don’t need both, because if D is dense then even if r \notin D you can extend it to a condition in D .


    1. Although I haven’t quite understood what he meant by “in a very strong way” (I think he said “strongly (M, \mathcal P) -generic) so maybe that has something to do with this.


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