# Tagged Questions

Forcing is a method first used to prove the continuum hypothesis is independent of the classical axioms of set theory

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### What are the enforceable models of local artinian rings?

I was reading Hodges' "Model Theory" Chapter 8 a propos existentially closed models of $\forall_2$ theories in a countable first order language $L$. He extends the proof of the omitting type theorem ...
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### Similarities between Post's Problem and Cohen's Forcing

Remark: I have since learned that G.H. Moore addresses this question in the third reference listed at the end of this post, beginning on p. 157 in which he cites a letter from Kreisel to Gödel dated ...
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### Random reals and strongly meager sets

Adding a single Cohen real makes the set of reals from the ground model strong measure zero (see this question). The notion of strong measure zero sets has its dual concept in the category branch -- ...
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### Set forcing and ultrapowers

The following is a result of Woodin (the proof is found after Theorem 5 of "Generalizations of the Kunen Inconsistency" by J.D.Hamkins, G.Kirmayer and N.L.Perlmutter): (Woodin) Let $V[G]$ be a ...
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### subalgebra of a simple forcing

Let $\alpha > 0$ be any ordinal. Consider the forcing $Add(\omega,\alpha) * Add(\omega_1,1)$. Let $B$ be its boolean completion. Let $\dot{X}$ be the canonical $B$-name for the generic subset of ...
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### Resembling the Levy Collapse

Suppose $\kappa$ is a weakly compact cardinal. Is there a $\kappa$-c.c. forcing $\mathbb{P}$ such that $\mathbb{P} \subseteq V_\kappa$ and $\Vdash_{\mathbb{P}} \kappa = \aleph_1$, where $\mathbb{P}$ ...
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### Can a model of set theory be realized as a Cohen-subset forcing extension in two different ways, with different grounds and different cardinals?

The question is whether, when you add a Cohen subset to a cardinal $\kappa$, that cardinal becomes a characteristic of the resulting forcing extension $V[G]$. Or can there be strange instances in ...
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### Homogeneous Namba-like forcing

Let $\kappa \ge \aleph_3$ be a regular cardinal that is countably closed ($\alpha^\omega < \kappa$ for every $\alpha < \kappa$.) I'm mostly interested in the case that $\kappa$ is strongly ...
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### Forcing to “minimally” add new reals.

Suppose that I want to force to add a "single" new subset of $\omega$ and not much else. For example, consider the Cohen forcing consisting of finite partial functions from $\omega$ to 2. The forcing ...
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### The canonical forcing of the GCH and direct limits.

The motivation for this question is that I am working through an exercise to force the GCH (generalized continuum hypothesis) over a model of ZFC and obtain a model of ZFC where GCH holds. The ...
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### Forcing in Homotopy Type Theory

I apologize if this question doesn't make any sense. I'll just go ahead and delete it if that's the case. But the question is just the title. Is there a notion of forcing in homotopy type theory? ...
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### Is every class that does not add sets necessarily added by forcing?

We know there are many situations in which we can force over a model $M$ of GBC to add a class $G$ without adding any sets. That is, the extension $M[G]$ satisfies GBC and has the same sets as $M$. ...
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### Forcing over the poset of nonempty open subsets of a nice topological space

Is there anything sensible to be said concerning a notion of forcing given by the poset of nonempty open subsets of the sort of topological space that comes up in ($e.g.$ algebraic) topology? If so, ...
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### How to find a sub-forcing?

Suppose that $M$ is a model of ZFC, $P\in M$ is a notion of forcing and $G$ is a $P$-generic filter over $M$. It is a well-known theorem that if $M\subseteq N\subseteq M[G]$, and $N$ is a model of ...
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### From the product lemma to to a result about powersets

Recall the product lemma from Easton's famous paper, which tells us something about when we have a forcing notion (which may be a proper class) that splits as a product with one factor $\lambda^+$-cc ...
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### Can we weaken GCH in this class forcing?

I've just stumbled across the following theorem (here): Theorem Let $P$ be a partial class order in $M$, a transitive model of ZF (resp. ZFC). Suppose for arbitrarily large cardinals $\kappa$ we have ...
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### Can we prove the completeness of FOL based on forcing?

I asked this question at http://math.stackexchange.com but I didn't get any answer there. In David Marker's Model Theory, there is a exercise said " we can view a countable Henkin construction as a ...
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### I'll admit it: I don't understand the definition of the Easton product.

I'm teaching myself bits and pieces of forcing at the moment, for the purposes of translating them into sheaf-theoretic versions. I'm trying to write down what I feel is a cleaner description of the ...
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### Suslin algebras

A Suslin algebra is a generalization of a Suslin tree: it is a ccc, $(\omega,\infty)$-distributive boolean algebra. Jech proved that every Suslin algebra has size at most $2^{\omega_1}$. A proof is ...
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### Why does the Solovay-Tennenbaum theorem work?

I have decided to learn iterations at long last, and I am reading through Jech's Set Theory for now. The standard example after explaining what is an iteration with finite support is the following ...
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### Forcing with product vs. box product

If we want to add one real number the simplest way to do it is to use Cohen forcing. The poset is $\lbrace p\colon n\to 2\mid n\in\omega\rbrace$ which is a countable set. We can think of this as ...
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### Which of these relations on partial orders allows us to identify forcing equivalence?

Background This question was inspired by Justin Palumbo's excellent question Cantor Bernstein for notions of forcing. In his question, Justin considers a relation $\lhd$ on partial orders (defined ...
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### Infinite products of forcings

Suppose $M \subseteq N$ are models of ZFC such that $(ORD^\omega)^M = (ORD^\omega)^N$. Let $\langle P_n : n \in \omega \rangle$ be a sequence of countably closed partial orders in $M$, and let ...
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### Properness of quotient forcing

It is well known that if $P$ is a proper notion of forcing and $\Vdash_P \dot{Q} \text{ is proper}$ then the iteration $P \ast \dot{Q}$ is proper. Is the converse also true, i.e Suppose that we have ...
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### On intermediate transitive models for ZFC between M an M[G]

Let $P$ be a forcing notion. Let $B(P)$ be the boolean completion of $P$ and $i : P \rightarrow B(P)$ be the corresponding dense embedding (in $B(P)^{+}$). Let $G$ be $B(P)$-generic over $M$, the ...