(This is the seventh lecture in Stevo Todorcevic’s Forcing class, held in the fall of 2012. You can find the sixth 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.)
 Talk about the CH “hierarchy”.
 Point out the MA analogue.
 Show that it is possible to force a function to be continuous on a large set.
 Show that Chang’s Conjecture is preserved by ccc forcing.
The Combinatorial CH Statements
Recall the combinatorial versions of CH:
 CH: , i.e. every ccc poset of size is centred.
 CH: Every ccc poset of size is nlinked.
 CH: Every ccc poset of size is 2linked.
“They are linearly ordered, but maybe they collapse.”
Axiom  Consequences 

CH  Everything, right? 
CH  Density of Lebesgue measure algebra is 
CH  
CH 

CH  ; 
Question 1: Which of these tells you that ?
Question 2: Which one tells you that every subset of the reals is borel?
One of These Consequences
Theorem: CH implies that is the union of many continuous subfunctions.
What about the union of monotone functions? “Oh, no, I do not think that is possible. Monotone functions are nice. They are always continuous, modulo something small. In general, it is a two dimensional object.”
Corollary: CH implies .
proof. “The proof is important, but easy by today’s standards. Look at Sierpinski’s function that is not continuous on any set of size .”
If , then there is a function such that for every partial Borel function there is an such that for all . [QED]
“This trick is important. When you do this for all finite dimensions, you enter the world of Baumgartner’s theorem. All subsets of the reals are isomorphic.“
proof of Theorem. We may assume that . Define iff is finite and for all distinct we have implies . Call this the metric property.
“You really don’t want this property. You really want to say that preserves the splitting structure of the tree. Unfortunately, this is probably not ccc. This is the short blanket phenomenon.“
Note: If satisfies the metric property, then is continuous.
“Now can you show ccc? You have to make it linked… almost. It is highly not 3linked. (Plug in the Sierpinski function). As always, the countable invariants are freezing the metric structure below some height.“
“Now comes the payoff. Checking the metric property for the union. Draw the picture.” [QED]
“What is CH telling you?” Suppose is 3linked in for some . By CH, , where each is 3linked. In fact, , with each satisfying the metric property.
The MA Analogue for CH
Whereas the CH were “global” properties, we describe “local” properties
Axiom  Definition  Consequences 

MA  Every ccc poset has an generic filter.  Everything else, right? 
Every ccc poset has an uncountable linked family.  
Every ccc poset has an uncountable linked family.  embeds into all ultrapowers  
Every ccc poset has an uncountable linked family. 
“The interesting thing is connecting these to things that aren’t related to the continuum.“
Chang’s Conjecture and the Countable Chain Condition, i.e. cc&ccc
“Let us do something different. Now for something completely different.”
Theorem: TFAE
 Chang’s Conjecture;
 Every ccc poset forces .
There are also strengthenings of Chang’s Conjecture denoted .
Recall: Chang’s conjecture states – For every structure of the form there is an elementary substructure of such that and .
Useful Note: Chang’s Conjecture is equivalent to the statement that for all sufficiently large regular cardinals and there is , with , and . “You first find a code for the closure of in …(?)” It seems you want to use a wellordering of something.
Lemma: Chang’s Conjecture is preserved by any ccc forcing.
proof. “Nice exercise to see if you really understand forcing.”
“I’ll do it unless anyone complains. You should complain if you want. Complaining shows that someone is alive.“
Let be ccc and for be names for functions from to .
Let be such a with for all . By the note, choose , with and .
Claim: Liberally applying checks,
“Closed under functions is easy as everything is countable. Elementary submodel takes work.“
proof. For , again with checks,
“Now ccc has to show up. All the possibilities, by elementarity, have to be in .” [QED]
“Most posets destroy Chang’s Conjecture. For example, adding an analogue of a Hechler real kills CC.”
“I did not do anything I planned. This is the definition of a good lecture. You just talk.”
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