# Tagged Questions

Forcing is a set theoretic method used mainly for proving independence results. For questions about forcing function please use [tag:differential-equations].

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### Keeping unwanted generic sets out of limit stage of iterated forcing

My question is about keeping unwanted generic sets from appearing at the first limit stage of an iterated forcing. The usual motivation for this is preserving some property $P$ of ZFC models: $P$ ...
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### Cardinals and generic extensions

Let us consider a model of ZFC $M$, a forcing $\mathbb{P}\in M$. If $G$ is a $\mathbb{P}$-generic filter over $M$ we can construct the generic extension $M[G]$ of $M$. Given any cardinal number ...
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### Consistency of the Subcomplete Forcing Axiom (relative to a supercompact cardinal)

In the introduction to his Singapore lecture Jensen mentions that the Subcomplete Forcing Axiom is consistent relative to a supercompact cardinal. Can anyone refer me to a proof of this claim?
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### Transitive models and CH

Suppose $M, N$ are two countable transitive models of ZFC which have same ordinals, cofinalities and reals (but not necessarily same sets of reals!). Suppose $M$ models the continuum hypothesis. Can ...
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### Elementarily equivalent forcing extension?

Is it possible to take a forcing extension which is elementarily equivalent to the ground model? Here I'm assuming the extension is proper, that is, it adds a new set. It's clear it can't be an ...
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### Reflection Principle vs. Löwenheim-Skolem-Theorem

From my undestanding a standard method of deducing relative consistency results is the following: By a combination of the Levy reflection principle, Skolem-Hulls and Mostowski Collapse we show: If ...
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### Forcing, $p \Vdash q \in \dot{G} \Rightarrow p \leq q$

I was wondering if a poset is separative if $p \Vdash q \in \dot{G} ~~ \Rightarrow p \leq q$ I think it's clear that $p, q \in G$ and hence are compatible but I am not seeing why ( if it's true ...
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### Some questions regarding a theorem of Paul J. Cohen

In his paper "Automorphisms of Set Theory", Paul Cohen proved the following theorem: "There exist models of $ZF$ admitting automorphisms of order two. More exactly, If $M$ is any countable model ...
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### Concequences of the result of Woodin's problem

In his book "The Axiom of Determinacy, Forcing Axioms, and the Non-Stationary Ideal" Woodin formulated tho following problem [No 22]: Are there two $\Pi_2$-sentences $\psi_1$ and $\psi_2$ in the ...
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### Good introduction to “forcing” and “inner models”?

I've occasionally come across the use of forcing, e.g., JDH's exploration of the modal logical of forcing. I know it is a massively important proof technique for, e.g., independence proofs. I've also ...
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### Cohen Forcing in Set Theory - Proof that Forcing is Equivalent to intersection of Dense Sets

Cohen's book "Set Theory and the Continuum Hypothesis" on Page 126/127 (see below) shows that the existence of a completed new set a' is equivalent to its intersection with all dense subsets in M. I ...
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### extending automorphisms in complete boolean algebras

Suppose $A$ is a complete subalgebra of a complete boolean algebra $B$. Suppose $f : A \to A$ is an automorphism. Then $f$ can be extended to an automorphism of $B$. I can see this using the fact ...
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### Relation between topological denseness and denseness over poset

In the theory of forcing, the notion of dense set is important. Formally, a subset $D$ of a poset $P$ is dense if, for any $p\in P$ we can find some $q\in D$ with $q\le p$. Intuitively, denseness of ...
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### On the paper “Forcing and the CH” by Aspero/Larson/Moore

Forcing Axioms and the CH by Aspero/Larson/Moore On page 11 of this paper I struggle with the beginning of the proof of Lemma 3.5. Coding: For $r \in 2^\omega$ and $A \in H(\aleph_1)$ let us say ...
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### Why do we define an inner forcing relation?

Studying forcing I came across different definitions of the forcing relation $\Vdash$: the outer forcing relation $\Vdash^M$ where we define $p \Vdash^M \varphi(\tau_0,\dotsc,\tau_n)$ to hold if for ...
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### Splitting condition of forcing posets

I was looking at Wikipedia for brief reminders of what I learned in my elementary set theory class, and discovered the forcing page (which I did not learn): A forcing poset is an ordered triple, ...
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### Quotient of Cohen forcing

How do we know that the quotient of the Boolean algebra associated with Cohen forcing by a generic filter is either atomic or isomorphic to the Cohen forcing? I know that Cohen forcing is the unique ...
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### A road-map through “Combinatorial Set theory: With gentle intro to independence proofs”

I'm going to study independence proofs form Halbeisen's book. It seems that some material is not needed to study independence proofs, so it seems that the book contains more material than my needs. ...
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### $\operatorname{Fn}(\lambda,2,\lambda)$ collapses $\lambda^+$ to $\operatorname{cf}\lambda$ if $\lambda$ is singular?

It is an exercise problem in Kunen (VII G5). I shall show that $\operatorname{Fn}(\lambda, 2, \lambda)$ adds a map from $\theta = \operatorname{cf}\lambda$ onto $\lambda^+$ for singular $\lambda$. ...
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### Good resources for studying independence proofs

I've finished most of Enderton's set theory. And I intend to spend some time studying independence proofs. I'm more interested in independence of axiom of choice not CH. From I know so far, there ...
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### Cohen forcing factoring

I start from $M$ a transitive countable model of $ZFC + \mathbb V= \mathbb L$ and I add a single Cohen generic $G$. Now if $A \in M[G]$ is also Cohen generic over $\mathbb L$ and $M[A] \ne M[G]$, can ...
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### Can every ccc forcing of size $2^{\aleph_0}$ be $\sigma$-linked?

It is well known that $MA_{\aleph_1}$ implies that every ccc forcing of size $< 2^{\aleph_0}$ is $\sigma$-linked (in fact - a countable union of filters). On the other hand, a separative forcing ...
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### What conditions must be checked for that $c$ is Cohen over $V$.

$\textbf{Hechler forcing}$ Let $\mathbb{D}=\{(s,f): s \in \omega^{<\omega},f\in \omega^{\omega} \text{and} s \subseteq f\}$, ordered by $(t,g)\leq (s,f)$ if $s \subseteq t$, $g$ dominates $f$ ...
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### Exercise on forcing

I got this homework in my forcing class: Let $G\subseteq P$ be generic over M. Show that there is a cardinal of M, $\lambda$ such for every set of ordinals $X\in M\left[G\right]$ there is a set ...
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### What are some simple example of “forcing” in set theory?

Can someone illustrate the idea of "forcing" in set theory through some simple examples? The article on forcing on wikipedia goes straight to axiom of choice and continuum hypothesis, I wonder if ...
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### Not meager in $V^{\mathbb{C}_I}$

Assume $A\subseteq 2^\omega$ is not meager in any non-empty open set, in The ground model $V$. Then is not meager in any non-empty open set, in $V^{\mathbb{C}_I}$ where $\mathbb{C}_I$ is Cohen ...
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### P-names are comprehensible in the ground model

If my understanding of $\mathbb{P}$-names $\tau$ is correct, they must be comprehensible by someone living in the ground model $\mathcal{M}$. However, the value $\tau[G]$ of $\tau$ need not be ...
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### show that there is a $\mathbb P$-name $\sigma$ such that $M[G]\vDash \exists x\phi (x) \iff M[G]\vDash \phi (\sigma [G])$

I want to show that there is a $\mathbb P$-name $\sigma$ such that for every $G$ a generic filter we will have $$M[G]\vDash \exists x\phi (x) \iff M[G]\vDash \phi (\sigma [G])$$ while $\phi (x)$ is a ...
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### The Amoeba poset has property Knaster.?

The Amoeba poset has property Knaster.? Amoeba poset $\mathbb{P}=\{p \subseteq \mathbb{R}: p$ $\text{is open} \wedge \mu(p)< \epsilon\}$ for $\epsilon>0$. Any suggestion. Thanks
### Why this set is finite $E_n=\{l \in \omega: \exists{j<n}[r_j \Vdash \check{l} \leq \dot{h}(n)]\}$?
Lemma IV (4.9) (Kunen book). Let $M$ be a countable transitive model of $ZFC$ and fix $\mathbb{Q}\in M$ such that $(|\mathbb{Q}| \leq \aleph_{0})^{M}$. Let $G$ be $\mathbb{Q}$-generic over $M$. Then ...