Stevo’s Forcing Class Fall 2012 – Class 3

 (This is the third lecture in Stevo Todorcevic’s Forcing class, held in the fall of 2012. You can find the second lecture here. Quotes by Stevo are in dark blue; some are deep, some are funny, some are paraphrased so use your judgement.)

Summary

  • We look at 4 examples of posets that collapse \omega_1 to \omega .
  • A problem of Foreman is mentioned that asks about collapsing versus adding.
  • Talking about anti-Suslin trees.

Tying up loose ends from last class is the proof that \mathfrak{m}_{SH} \leq \mathfrak{m} . Thanks to Oswaldo for explaining it here (pdf).

On a bureaucratic note, Stevo intends this course to be a supplemental course to Alan Dow’s course in Forcing at the Fields Institute. As such the classes might just end up being Stevo answering students’ questions about Alan Dow’s course. “I am Alan Dow’s teaching assistant.” Today is one of those days, and we chat about a variety of ways of collapsing cardinals.

Collapsing \omega_1

First we recall the usual poset that collapses \omega_1 . This is \text{Col}(\omega, \omega_1) which is finite partial functions from \omega to \omega_1 . You can see how a generic object will be a function from \omega onto the ground model \omega_1 .

Theorem (Kripke). For \mathbb{P} = \text{Col}(\omega, \omega_1) we have \vert \mathbb{P} \vert \leq \aleph_1 and \Vdash_\mathbb{P} \vert \omega_1 \text{^} \vert = \aleph_0 .

This tells us an interesting universality property:

Corollary. If \vert \mathbb{Q} \vert \leq \aleph_1 then \mathbb{Q} completely embeds into \text{Col}(\omega, \omega_1) .

proof. Take \mathbb{P} = \mathbb{Q} \times \text{Col}(\omega, \omega_1) . [QED]

Fact (Gaifman). The regular open algebra of this collapsing poset, \text{ro}(\text{Col}(\omega, \omega_1)) is countably completely generated.

“This is one way to collapse. Do you know any others?”

Example 1. Countable support product of Cohen

Exercise. Show (\omega^{<\omega})^\omega \Vdash \vert  \mathbb{R}\text{^}\vert = \aleph_0 .

Hint

\mathbb{R} = 2^\mathbb{N} . “Telling you what ‘the reals’ is is the hint.“

Problem (Foreman). Is there a (nontrivial) forcing notion which does not add reals, nor collapses cardinals?

Note. Under CH or \neg SH there is a positive answer to this.

Theorem (Stevo). Under PFA, Every poset which adds a subset of \omega_1 either adds a real or collapses \aleph_2 .

Example 2. Shoot a club.

I will add details to this soon.

Example 3. Mild variations of Levy Collapse

Example 4. A “drastic” example of type 2.

Take a coherent Suslin Tree T and force with T^2 .

Coherent means that T is a closed subset of \omega^{\omega_1} and \forall s,t \in T we have \{\xi \in \textrm{dom}(s) \cap \textrm{dom}(t) : s(\xi) \neq t(\xi)\} is finite.

“Why can you see [that \omega_1 is collapsed] without proof? A branch in one coordinate is one world, a branch in the other is another world.”

Anti-Suslin

“What is a good example of an Aronszajn tree that is Baire and anti-Suslin?” A tree is anti-suslin if every uncountable set contains an uncountable antichain.

Note. “Special is on the opposite extreme of Suslin.”

Almost Example. Take E \subseteq \omega_1 that is stationary, co-stationary. Take T(E) = \mathcal{K}(E) = \{ compact subsets ordered under end-extension \} .

Note. “Writing down what it means to be anti-Suslin, you will see it is too complex to be a consequence of diamond.”

Note (by Assaf Rinot). Suppose T(E) has a Suslin subtree S . Force with it and this addds a branch and also a club C \subseteq E , but this must contain a club E^\prime in the ground model.

 

 

 

3 thoughts on “Stevo’s Forcing Class Fall 2012 – Class 3”

  1. There’s an = sign missing from the Kripke theorem. Furthermore, on my screen at least, the checks look like hats, and they are displayed to the right of the base character rather than above.

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    1. Fixed the equal sign. The checks really are just carets, as Mathjax doesn’t seem to like accents. I’ll look into it.

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  2. Pingback: Stevo’s Forcing Class Fall 2012 – Class 4

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