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.)

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

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. Ari Brodsky says:

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. Micheal Pawliuk says:

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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