(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.)
- We define the Side Condition Method.
- We force the P-Ideal Dichotomy.
- A cardinal preserving modification of the side condition is discussed.
- Semi-properness is introduced, and then used to prove a fact about projectively stationary sets.
- The Strong Reflection Principle is introduced.
- Some consequences are discussed.
- A ZFC version of diamond is stated.
Side Condition Method
Side conditions are elements of , that is, -chains of CESMs of some .
Remark 1. For , the condition is -generic in a very strong way:
Fact. , “isn’t needed“) dense open in , such that . i.e.
“They add reals, but they don’t add branches to trees.”
“This works most of the time, so long as 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), . You can step it up to .
“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 and ) then . 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 be a set with a P-Ideal. Assume cannot be written as the countable union , where for all .
To get the PID, we need to force an uncountable such that .
“What is the natural forcing? You need to make some choices.“
Let , with , where and .
“[The condition are] simply functions as before.”
The conditions are maps where:
- ; and
- contains ; and
- ; and
- is separated by .
is really the ideal generated by sets orthogonal to . (???)
- ; and
- we have .
“This is a very natural thing if you are trying to force what we are trying to force.”
So is the intended consequence.
“Go carefully through the copying argument.”
“What if we don’t want to collapse? There is an easy and useful
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 to something isomorphic.”
Let be the collection of all finite conditions , where:
- Each is a finite collection of isomorphic CESMs of ; and
- there is an such that .
“Now let us check the (strong) properness of these. Again, it is the same proof.“
Claim: is (strongly) proper.
proof. Start with , where can be taken to be .
Then is (strongly)--generic.
“This is a very powerful trick of self-reference. A slippery trick.”
“If you are doing something on , this is enough. Sometimes . Not for -PID.“
Fact: is (strongly)--cc.
“Is collapsing that bad?” – Oswaldo
“Yes, it is bad for your health.”
“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 is semi-proper (really, -semi-proper, but that has disappeared over the years.) if for every large enough and CESM containing some and itself, there is an -semi-generic extension .
Definition. With the situation above, is an -semi-generic extension if for every partial function with and there is an and such that .
“This is about -partitions.”
Most of the time , but it doesn’t have to be.
Lemma (for checking semi-proper). Suppose are as above, and suppose there is another CESM such that and . Suppose is -generic, THEN is -semi-generic.
“In some sense, this is all that is known about proving semi-proper.“
You are going to find such that (???), where is an -end-extension.
“Let us examine this notion, by testing them.“
Theorem 1. There is an -stationary set preserving poset of size which collapses .
Theorem 2. If a semi-proper poset collapses then it collapses .
Corollary. If , there is no semi-proper poset which collapses and preserves all other cardinals.
“Semi-proper posets preserves certain sets coded by ordinals in .”
“Let me give you the first example which is interesting even if you aren’t interested in this stuff.“
Real Theorem 1. Suppose that has a stationary subset of size . THEN there is an -preserving poset of size which collapses to . If the stationary subset is projectively stationary, then preserves stationary subsets of .
Remark. For the first assumption, and are the first places where that can fail.
Definition. Suppose that and where . We say is projectively stationary if is stationary in for all stationary .
Exercise. If for some , then there is a projectively stationary subset of cardinality .
proof of Real Theorem 1. “What is the ?”
Fix an injection . Let be the collection of all finite subsets of such that wither or . The ordering is inclusion.
Claim: preserves .
Take a CESM ( around should work), such that and . (This uses the stationarity of ).
Let . Then is -generic.
“Why? By elementarity you can copy.” [End of Proof]
Question. For which is 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 . Let be projectively stationary, THEN you can reflect it on a club. i.e. for every containing , there is a such that contains a club subset of .
Note. 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
Note. Shelah’s Theorem above is easy to show, assuming SRP.
Exercise. If is projectively stationary, then contains a club. “i.e. If you project everywhere, then there is nowhere to escape! So you want .”
Example. Let be projectively stationary, then there is an -stationary-preserving poset which forces , which strongly reflects . “What is the poset? It is the natural one.”
“You know you have to provide this chain of continuous sets:”
- for ;
- for limit ;
How do you code continuity? You say that it can be extended to something continuous.
The conditions. Let be the collection of all partial maps , such that:
- containing ;
- in , ;
- There is a continuous , chain of Elementary Submodels such that . “This reminds me of Baumgartner” – Oswaldo. “Everyone says that. It is not. People should say this reminds them of Cohen.“;
“Now you are doing something. Without this you have nothing.”
Claim. This preserves Stationary subsets of .
“Let us do this.“
Take and a -name for a club in . “Who is this going to be?”
Recall, is stationary. So there is a CESM containing all the objects such that .
Let . (Note , so is uncountable.) Let .
Claim. is an -generic condition, and therefore .
“This is copying as before.” [End of Proof]
Consequences of SRP, left as exercises
- is saturated
- WeakRP, i.e. reflection of stationary subsets of some . “This is the price you pay for the elegance of Jech-Feng SRP.“
- . “You can code reals by ordinals.“
Two Coding Lemmas
“I’ll tell you the real . The which is true in ZFC, but we don’t know if it has nice consequences.“
Lemma. There is a sequence such that:
- ; and
- is projectively stationary.
proof. Look at the oscillation map. Pick a club guessing . Then
And . [End of Sketch]
Lemma (*). For every is projectively stationary.
“All this stuff about coding sets is really this lemma.“