# Tagged Questions

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### A question about $\dot{S^Q}$-semiproperness and revised countable support iterated forcing of length a limit ordinal

For a forcing notion $Q$, let $\dot{S^Q}$ be the $Q$-name for the class of ordinals $\{\kappa : \kappa = \omega_1^{V}$ $or$ $\kappa$ $is$ $a$ $regular$ $uncountable$ $cardinal \}$ in $V^Q$. We say ...
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### Different approaches to forcing

There are many different approaches to the forcing method, and I am looking for all known such approaches. So my question is: Question 1. Which different approaches to set theoretic forcing are ...
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### Iterated Forcing and Subposet of Conditions with a Given Support

Let $\mathbb{S}_{\omega_2}$ denote the $\omega_2$-iteration of Sacks forcing. (Sacks forcing is just used as an example, but any other definable forcing and any iteration of the same forcing ...
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### Silver's unpublished work on reverse Easton iteration

Silver was the first person who used the method of reverse Easton iterations in connection with large cardinals, and used it to force the failure of $GCH$ at some measurable cardinal. At most papers ...
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### $RUCar^{V}$-semiproperness implies properness

This is a claim in Shelah's Proper and Improper Forcing, more specifically Claim 2.3(1) of Chapter X (p. 484). The proof of the claim is "Easy" but I cannot quite figure it out. There must be some ...
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### Three old questions on the Sacks forcing

I came across the two following Qs in 1970. Find reals $a,b$ such that $a$ is Sacks over $L[b]$ and vice versa $b$ is Sacks over $L[a]$. Note that a Sacks $\times$ Sacks generic pair definitely does ...
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### Which forcing types preserve the axiom of determinacy?

Do we have some rudimentary understanding of some properties that a forcing can have in order to guarantee that it doesn't violate the axiom of determinacy? To be more specific, in Which forcings ...
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### Iteration of Proper Forcing and Support of Master Conditions

Suppose $\mathbb{P}$ is a definable proper forcing (for instance Sacks forcing). Let $\alpha$ be some ordinal. Let $\mathbb{P}_\alpha$ be the countable support iteration of $\mathbb{P}$ of length ...
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### Relationship between fragments of the axiom of choice and the dependent choice principles

The dependent choice principle ${\rm DC}_\kappa$ states that if $S$ is a nonempty set and $R$ is a binary relation such that for every $s\in S^{\lt\kappa}$, there is $x\in S$ with $sRx$, then there ...
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### Inaccessible becomes successor of singular

Is it possible, starting from any large cardinal assumption, to find a countably closed forcing $\mathbb{P}$ such that for some inaccessible $\kappa$, $\Vdash_\mathbb{P} "\kappa = \lambda^+$ and ...
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### A not defined notion in Friedman's article about Generalized Fubini's Theorem

I intend to study Friedman's article, A Consistent Fubini-Tonelli Theorem for Nonmeasurable Functions (http://projecteuclid.org/download/pdf_1/euclid.ijm/1256047607). I think since I had a modern ...
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### continuum many mutually generic filters

Given a countable model $M$ of set theory and an atomless, separative partial order $\mathbb{P} \in M$, can we construct (in the real universe) $2^\omega$ many pairwise mutually $\mathbb{P}$-generic ...
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### Embeddings of forcing notions - preserve properness?

Let $M$ be a countable, transitive model for $\mathsf{ZFC}^*$, i.e. for a sufficiently large finite fragment of $\mathsf{ZFC}$. Suppose that $\mathbb{P} := (P, {\leq_P}, \mathbb{1}_P) \in M$ ...
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### Question about “Coding the universe”

The following is a result which I know as a weak form of Jensen's coding lemma$^*$ (first published in the book "Coding the universe"; also see http://www.jstor.org/stable/2273986): For any class ...
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### Strongly compact cardinal with bad covering properties

This is a continuation of the question covering properties of strongly compact embedding. Recall that a cardinal $\kappa$ is $\nu$-strongly compact cardinal if there is an elementary embedding ...
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### If all reals are generic, is the set of reals generic?

Let $W\subseteq V$ be two models of $\sf ZFC$ with the same ordinals. Is the following situation consistent: For every $x\in\Bbb R^V$ there is some $P_x\in W$ such that for some $G\subseteq P_x$ ...
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### Does forcing with recursively pointed perfect trees add a Turing degree that is minimal over $V$?

A tree $T$ on $\omega$ is recursively pointed if it is recursive in each of its branches. We can consider a variant of Sacks forcing where the conditions are recursively pointed perfect trees ordered ...
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### Forcing, cuts, and Dedekind-finite cardinalities

Tl;dr version: there are two natural classes of cuts in the nonstandard model of arithmetic consisting of the Dedekind-finite sets (if, in fact, they constitute such a model); both these classes are ...
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### Elements of the method of forcing in some papers of N. N. Luzin

In the paper ElÃ©ments de la mÃ©thode de forcing dans quelques travaux de N. N. Lousin. (French) [Elements of the method of forcing in some papers of N. N. Luzin] Amphora, 469â€“479, BirkhÃ¤user, ...
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### On Consistency of an Existence

Let $\omega \leq \kappa <2^{\omega}$ , $\omega \leq\lambda \leq \kappa$ and $D(\kappa, \lambda)$ be the statement: For all $\mathfrak{B} \subseteq \mathbf{P}(\omega)$ with ...
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### Origin of the term “generic” in set theory

In set theory, in particular the context of forcing, if $M$ is a model of $\sf ZFC$ and $P\in M$ is a partial order, we say that $G\subseteq P$ is a generic filter (or $M$-generic or generic over $M$) ...
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### Ultrafilters of weight $\aleph_2$ in Sacks model

It is well-known that in Sacks model there are P-points and even Ramsey ultrafilters, but what the usual (i.e. findable in the literature) proofs for these facts do is proving that ground model ...
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### Partial interpretation of an iteration

Suppose that $\langle\mathbb{P_\alpha,\dot Q_\beta}\mid \beta<\delta,\alpha\leq\delta\rangle$ is a system of iterated forcing. Let $\dot a$ be a name in $\mathbb P_\delta$, and let $G_\alpha$ be a ...
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### The independence number

I have been reading about cardinal invariants and I have a question about the independence number $\mathfrak{i}$. In Blass's paper (Combinatorial Characteristics of the Continumm) it is mention that ...
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### splitting subsets of cardinals

Suppose $\mathbb{P}$ is a separative partial order of uniform density $\kappa$, i.e. for all $p \in \mathbb{P}$, the least size of dense set below $p$ is $\kappa$. Does forcing with $\mathbb{P}$ add ...
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### Destroying the Mahloness of a cardinal with $\kappa$.c.c. forcing

Question: Is it possible to have a Mahlo cardinal $\kappa$ such that there is a $\kappa$.c.c. forcing that makes it non-Mahlo? If this is possible then this forcing must change the cofinality of all ...
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### Characterization of intermediate submodels of generic extensions

Question 1: Suppose that $V[G]$ is a set generic extension of $V$ by some forcing notion $P\in V,$ and suppose that $W$ is a model of $ZFC, V \subset W \subset V[G].$ Can we find a forcing notion ...
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### What year was Hechler forcing created?

Hechler forcing is described on page 278, Jech. Does anyone know when Hechler forcing was first used in a publication?
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### cardinals below the critical point of a generic embedding

This may be an easy question. Are the cardinals below the critical point of a precipitous ideal embedding always absolute between the generic ultrapower and the generic extension? To focus on the ...
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### Forcing with Nontransitive Models

A common approach to forcing is to use countable transitive model $M \in V$ with $\mathbb{P} \in M$ and take a $G \in M$ (which always exists) to form a countable transitive model $M[G]$. Another ...
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### $\omega$ universally Baire sets, tree representations

I've recently encountered the notion of a universally Baire set, and I've tried to look at the paper by Feng, Magidor and Woodin where this notion is studied. There are several points that confuse me. ...
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### Souslin trees and weakly compact cardinals

In Souslin trees on the first inaccessible cardinal it is asked if it is consistent that there are no $\kappa-$Souslin trees at the least inaccessible cardinal $\kappa$. In this question I would like ...
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### The (non-)absoluteness of second-order elementary equivalence

Elementary equivalence is set-theoretically absolute between any two transitive models of set theory; this is also true for the infinitary logics - e.g., $\mathcal{L}_{\omega_1\omega}$ - at least, ...
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### On the definition of $\alpha$-proper poset

I am reading Uri Abraham's chapter on Proper Forcing in the Handbook of Set Theory and I have a quite trivial question on the definition of $\alpha$-proper forcing. Since there are many equivalent ...
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### Is there a Hotel California of set-theoretic geology?

Is there a universe which can always be forced to, which never can be forced from?
Recall that a topological space $X$ has the fixed point property (FPP) if any continuous function $f: X\to X$ has a fixed point. Is the notion of FPP for topological spaces an absolute notion? More ...