Articles of forcing

Viewing forcing as a result about countable transitive models

I think of forcing in the following context: Truth and Definability Lemmas. So forcing is a schema in the meta theory. Now in his Set Theory book (the first edition), Kunen claims that setting up the following schema also deals with the mathematics required for showing $ZFC\vdash{\forall{\text{ctm } M \text{ of } {\ulcorner{ZFC}\urcorner}}} \implies \exists{N} […]

exercise VII.G5 in kunen

Recall $Fn(\alpha,\beta,\gamma)$ is the set of partial functions with domain contained in $\alpha$, range contained in $\beta$, size $< \! \gamma$. How do you prove that $Fn(\aleph_\omega,2,\aleph_\omega)$ collapses $\aleph_{\omega+1}$? I see why it collapses $\aleph_\omega$ but I don’t see why it kills the successor too.

An exercise in Kunen

The following exercise appears in Kunen; In $M$, let $\mathbb{P}=Fn(\kappa,\lambda)$ (i.e. the finite partial functions), where $\aleph_0\leq\kappa<\lambda$. Then $\lambda$ is countable in $M[G]$, and all cardinals of $M$ above $\lambda$ remains cardinals in $M[G]$. He gives the hint: If $f=\bigcup{G}$, then $f\upharpoonright\omega$ maps $\omega$ onto $\lambda$. I’m not really sure how you show that. It […]

A question from Kunen's book: chapter VII (H9), about diamond principle

Assume $(\mathbb{P}$ is c.c.c.$)^M$ and $\Diamond$ holds in $M[G]$. Show that $\Diamond$ holds in $M$. Hint: It is sufficient to verify $\Diamond^-$ in $M$. Should I try to create a $\Diamond$ sequence or $\Diamond^{-}$ sequence? Should I work with nice names?

Forcing names, parameters in definitions, and the Iterative Conception of Set

So, I’ve been trying to learn as much as I can about forcing. I know that a model provides its own (trivial) forcing extension. What I’m curious about is whether there is a way to think of the iterative hierarchy in terms of (trivial?) forcing extensions? The iterative hierarchy for $\mathsf{ZF}$ can be given in […]

Countable transitive model of ZFC?

If there is countable transitive model of ZFC, this model cannot capture all ordinals of ZFC. But we use it for stuffs like forcing. But this seems to violate ZFC’s axioms – for example, power set axiom (take, $\omega = \aleph_0$ and apply power set operator.). So how can we use countable transitive model of […]

Why is ZF favoured over NBG

So why is ZF favoured over NBG? Is it historical, but I’ve read after Gödel’s monograph was published, NBG was more prominent. Or is the reason that NBG gets the cold shoulder to do with forcing and all the extra work needed for NBG to handle forcing. I’ve only studied forcing in ZF and have […]

On the number of countable models of complete theories of models of ZFC

This question already has an answer here: Number of Non-isomorphic models of Set Theory 2 answers

If P is k-c.c. and C is club in k in M then C contains a club in M

I’ve seen this written several places without proof, so I assume it’s not difficult, but I am not getting it. Let $\mathbb P$ be a $\kappa$-c.c. notion of forcing, and let $C\in M[G]$ be club in $\kappa$. I want to show there exists $D\in M$ such that $D\subset C$ is club in $\kappa$. Kunen suggests: […]

Forcing Classes Into Sets

I am still studying the topics in forcing and did not yet study much about forcing with a class of conditions. I know from Jech’s Set Theory that you can force that the class of ordinals in the world will be countable in the generic extension, which means that you can take a proper class […]