Set Theory II (Forcing) 2018

Lecture: HGW2403, T 18:30-21
Section: HGW2403, R 18:30-20

You can choose one of the following topics.

An introduction to one specific forcing notion from this list (for Cohen forcing you can introduce an application not discussed in our course) and explain how it works.
On one or more issues concerning the meta-theory of forcing. For example, why and how we can talk about an “outer model” or a “object” not in our universe, why we can and why we need to assume the existence of a transitive model, how to account forcing arguments as purely constructive methods, etc.

Let \(\pi\) be the canonical interpretation of PA into ZF. Can we prove “for each arithmetic formula \(\varphi\), if ZFC \(\vdash\pi(\varphi)\), then PA \(\vdash\varphi\)”? Prove it if we can, explain it if we cannot.
Assume ZF \(\vdash\varphi^L\) for each formula \(\varphi\in\Sigma\), and \(\Sigma\vdash\psi\). Show that ZF \(\vdash\psi^L\).
Why Con(ZF) does not imply there is a countable transitive model of ZF?

Kunen’s set theory (2013) Exercise I.16.6 – I.16.10, I.16.17.

Kunen’s set theory (2013) Exercise II.4.6, 4.8.

Jech’s set theory (2002) Exercise 7.1, 7.3 – 7.5, 7.13, 7.16, 7.18 – 7.20, 7.22 – 7.33.

Let \(M^\mathbb{B}\) be a Boolean valued model. Prove the following statements are valid in \(M^\mathbb{B}\).
- \(\forall y\big(\forall x\varphi(x)\rightarrow\varphi(y)\big)\).
- \(\forall x(\varphi\rightarrow\psi)\rightarrow\forall x\varphi\rightarrow\forall x\psi\).
- \(\alpha\rightarrow\forall x\alpha\), \(x\) does not occur in \(\alpha\) freely.

Jech’s set theory (2002) Exercise 14.12.

Let \(\sigma\) be a \(\mathbb{B}\)-name. Show that \[ |\!|\exists x\in\sigma~\varphi(x)|\!| = \sum_{\xi\in\textrm{dom}\sigma}\sigma(\xi)\cdot|\!|\varphi(\xi)|\!|.\]
For any partial order \(\mathbb{P}\), there is a separative partial order \(\mathbb{Q}\) and a surjection \(h:\mathbb{P}\to\mathbb{Q}\) such that
- \(x\leq y\) implies \(h(x)\leq h(y)\);
- \(x\) and \(y\) are compatible in \(\mathbb{P}\) if and only if \(h(x)\) and \(h(y)\) are compatible in \(\mathbb{Q}\).
Such \(\mathbb{Q}\) is unique up to isomorphism. We call it the separative quotient of \(\mathbb{P}\).

Jech’s set theory (2002) Exercise 14.1, 14.9, 14.14, 14.16. Lemma 14.13.