Lecture: HGW2403, T 18:30-21

Section: HGW2403, R 18:30-20

#### Instruction for the final paper

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.

#### Problem set 01

- 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?

#### Problem set 02

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

#### Problem set 03

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.

#### Problem set 04

- 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.

#### Problem set 05

- 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.