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

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A Question on HOD, V and GCH

The theorem 1.1 of the following paper: Mohammad Golshani, V, HOD, and the GCH, Journal of Symbolic Logic. states that: Theorem: Assume $V\models ZFC+GCH+~\text{There exists ...
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Set theory and forcing from the point of view of a formal system $G^+$ of Gentzen type

There are four papers by Vladimir Alfeevich Kuznetsov, which discuss the above titled topic: (1) Some problems in set theory from the standpoint of a formal system G+ of Gentzen type. (Russian) Akad. ...
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Statements that Could be Forced by Ultrapowers

Ultrapower of a structure is a very flexible mathematical creature in comparison with the ground structure and its ordinary products. Depending on the nature of ground structure and the good ...
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Is the Martin's axiom number $\mathfrak m$ regular

The Martin's axiom number $\mathfrak m$ is the least cardinal $\kappa$ for which $\text{MA}_\kappa(\text{ccc})$ is false, i.e. the least cardinal such that there exists a ccc poset $P$ and a family ...
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Reverse-engineer forcing: am I reinventing the wheel?

In the course of a project I’m working on, I’ve started playing around with a sort of “reverse-engineering” forcing. It seems interesting, but I have a sinking feeling I’m reinventing the wheel; does ...
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Can Sacks forcing add a Cohen generic real over $L$?

Motivated by this question Forcing the negation of CH without adding Cohen reals over L and Todd Eisworth's comment, the question is the following: 1) Suppose $V$ has no Cohen generic reals over $L$. ...
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Forcing the negation of CH without adding Cohen reals over L

Suppose CH + "there are no Cohen reals over L". Can we force the negation of CH without adding any Cohen real over L?
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Two questions about the behavior of the continuum function

The first question asks about the global behavior of the power function in the case of finite gaps. Question 1. Fix a natural number $m>1.$ For each limit ordinal $\alpha,$ including $0$, let ...
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Failure of GCH at a strongly compact cardinal

Does Con(ZFC+ there exists a strongly compact cardinal) imply Con(ZFC+ there exists a strongly compact cardinal $\kappa+ 2^\kappa > \kappa^+$)?
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When can Power Sets be Limit Cardinals?

My original question (posted in http://math.stackexchange.com/questions/1584430/can-all-power-sets-be-limit-cardinals) was: Is it possible to create a model of ZFC, so that the cardinality of each ...
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Gleason’s Theorem for non-separable Hilbert spaces: On a theorem of Solovay

Gleason’s theorem plays an important role in the foundations of quantum mechanics. On the positive side it demonstrates how the probabilistic structure of quantum theory follows from its logical ...
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How many closed measure zero sets are needed to cover the real line, really?

This is a refinement of an earlier question. This question assumes familiarity with combinatorial cardinal characteristics of the continuum. For the reader's convenience, I reproduce below the ...
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How many closed measure zero sets are needed to cover the real line?

This question assumes familiarity with combinatorial cardinal characteristics of the continuum. Let $\mathcal{E}$ be the $\sigma$-ideal generated by closed measure zero subsets of the real line. It ...
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Adding minimal subsets to $\aleph_\omega$

Given a cardinal $\kappa,$ recall that $X \subset \kappa$ is called fresh (over $V$), if $X \notin V,$ but $X \cap \alpha \in V$ for all $\alpha < \kappa.$ Question. Is it consistent that there ...
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Boolean-Valued Models: Why is $\| x=y \| \cdot \| \phi(x) \| \leq \| \phi(y) \|$?

Let $B$ be a complete Boolean algebra. Jech defines a Boolean-valued model $\mathfrak{A}$ of the language of set theory to consist of a Boolean universe $A$ and functions of two variables with values ...
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When can you canonically extend an ultrafilter after forcing?

Suppose that $V$ is a model of $\sf ZFC$, and fix some regular $\kappa$, say $\omega_1$ for practical purposes. Let $\cal U$ be an ultrafilter on $\omega_1$ in $V$ which is non-principal and even ...
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Theories introduced by a class of forcing notions

The following notion is introduced by Mohammad Golshani. Let $V$ be a model of set theory and let $\mathcal{P}$ be a class consisting of non-trivial forcing notions in $V$. Let $$Th(V, ...
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Can one force there to be an elementary embedding $j:V_{\lambda}\rightarrow V_{\lambda}$ for some inaccessible $\lambda$?

Is it consistent that there exists an inaccessible cardinal $\lambda$ and a forcing extension $V[G]$ so that $$V[G]\models\text{There is some non-trivial elementary embedding ...
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Transitive models and CH

The following was asked on stackexchange but I think it also belongs here: http://math.stackexchange.com/questions/1513446/transitive-models-and-ch Suppose $M, N$ are two countable transitive models ...
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Sacks minimality without choice

The usual argument for the minimality result for Sacks forcing uses choice. Theorem (Sacks): Let $s \subseteq \mathbb S_\kappa$ be generic for the forcing to add a Sacks subset to $\kappa$, where ...
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Is it consistent with ZFC that no nontrivial forcing notion has automatic mutual genericity?

A nontrivial forcing notion $\newcommand\Q{\mathbb{Q}}\Q$ exhibits automatic mutual genericity, if whenever $G,H\subseteq\Q$ are distinct $V$-generic filters (existing, say, in some forcing extension ...
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Elementary chains in forcing extensions of $M_1$

Let $M_1$ be the canonical inner model with one Woodin cardinal $\delta$. Now suppose that $\mathbb{P}$ is a forcing notion of size $< \delta$, which preserves $\omega_1$ and that $G$ is a generic ...
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$V$ as a $HOD$ of its class generic extension

By an old result of Roguski, The theory of the class $HOD$, any model $V$ of $ZFC$ has a class generic extension $V[G]$ such that $HOD$ of $V[G]$ equals $V$. This result is also stated and generalized ...
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PCF conjecture and fixed points of the $\aleph$-function

Recently Moti Gitik refuted Shelah's PCF conjecture, by producing a countable set $a$ of regular cardinals with $|pcf(a)| \geq \aleph_1.$ See his papers Short extenders forcings I and Short ...
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Where do uncountable models collapse to?

Suppose $T$ is a complete first-order theory (in an finite, or at worst countable, language). Given any model $\mathcal{M}\models T$ of cardinality $\kappa$, we can ask whether $\mathcal{M}$ can be ...
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Can you have many independent reals?

Working in $\sf ZFC$, is it provable, or at least consistent (say, over $L$), that you have $\aleph_1$ forcings, $\Bbb P_\alpha$ such that: $\Bbb P_\alpha$ is c.c.c. $\Bbb P_\alpha$ adds a real ...
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Idea behind the proof of consistency of club filter of $\omega_1$ is ultrafilter + ZF + DC

I've been trying to understand Radin Forcing and some of its applications, one of which is the use of it to prove the consistency of ''Club filter of $\omega_1$ is an ultrafilter + ZF + DC''. However, ...
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Forcings that are not equivalent to Levy collapse

Assume GCH and that $\kappa$ is a regular uncountable cardinal. Let $\mathbb{P}$ be a separative, $<\kappa$-directed closed, nowhere trivial, $\kappa^+$-cc poset of size $\kappa^+$. Must ...
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A question regarding forcing in $NGBC^{-f}$+$BAFA$

Suppose one has a model $M$$\vDash$$NGBC^{-f}$+BAFA. Does there exist a (class) forcing extension $M[G]$$\vDash$$NGBC^{-f}$+$BAFA$ that has a submodel $N$$\vDash$$NGB^{-f}$+$BAFA$+$\lnot$$AC$? Can ...
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Homogeneity of a variant of Prikry forcing

Prikry forcing is easily seen to be cone homogeneous (for any $p, q \in \mathbb{P}$, there are $p' \leq p, q' \leq q$ and an isomorphism $\Phi: \mathbb{P}/p' \simeq \mathbb{P}/q'$); in particular for ...
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Does $ZF$+$LM$+$A_{\lt2^{\aleph_0}}$ imply $\lnot$$WCH$?

In what follows, I will use the following acronyms: $AS$:= Freiling's Axiom of Symmetry $LM$:="Every set of reals is Lebesgue measurable." $WCH$:="every uncountable subset of $\mathbf R$ can be put ...
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Who needs RCS iterations?

According to this paper of Chaz Schlindwein, any countable support iteration of semi-proper forcings is semi-proper. This seems like a breakthrough simplification, and I wonder why it is not more ...
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When does $HOD^{V[G]} \subseteq V$?

Assume that $\mathbb{P}\in HOD$ is non-trivial. It is well-known that if $\mathbb{P}$ satisfies some homogeneity properties, then $HOD^{V[G]} \subseteq V$, where $G$ is $\mathbb{P}$-generic over $V$. ...
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Forcing Extension of Countable Linearly Iterable Structures

Let $V$ satisfy there exists a measurable cardinal. Let $\kappa$ be a measurable cardinal and $U$ be the normal measure on $\kappa$ witnessing this. Let $\mathbb{P}$ be a forcing of size less than ...
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tree properties on $\omega_1$ and $\omega_2$

Are the following mutually consistent (relative to large cardinals)? (1) There are no $\omega_2$-Aronszajn trees. (2) There is an $\omega_1$-Kurepa tree. In the models I know of the tree property ...
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tree property at $\aleph_2$ and $\aleph_4$

It is claimed that if there are two weakly compact cardinals, then there is a generic extension in which $\aleph_2$ and $\aleph_4$ have the tree property. Assuming one knows Mitchell's forcing, what ...
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The least admissible above a dominating real

Let $\mathbb{P}$ be the usual forcing which adds a dominating real: conditions in $\mathbb{P}$ are pairs $(p, f)$ with $p:\omega\rightarrow\omega$ finite partial and $f:\omega\rightarrow\omega$ total, ...
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Does totally proper forcing imply countable distributivity?

For a suitable model $M$ for $Q$ and a condition $q \in Q$ we say that $q$ is $(M,Q)$-generic if whenever $r \leqslant q$, $D \in M$ dense, $D \subset Q$, $r$ is compatible with an element of $D \cap ...
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How “small” can an ordinal be made by forcing?

I know that forcing essentially does not change the ordinals, but by small I mean in comparison with other ordinals whose definition might not be stable under forcing, like the smallest uncountable ...
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On an unpublished result of Magidor

In 1970th, Magidor proved the following important results: (1) Assuming the existence of a supercompact cardinal, it is consistent that $\aleph_\omega$ is strong limit and ...
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On generic forcing conditions

Let $P$ be a forcing poset, and $Q \in V^P$ a forcing poset in $V^P$. Let $M \prec H(\lambda)$ ($\lambda$ sufficiently large) countable with $P,Q \in M$. What I want to know is if then the following ...
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Large Cardinal Principles that Imply $\Sigma_3^1$-Generic Absoluteness

It is known that (light-face) $\Sigma_3^1$ generic absoluteness is consistent with $\mathsf{ZFC}$: Friedman and Bagaria showed that it holds in the $\text{Coll}(\omega, < \kappa)$ extension of $V$ ...
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RCS iteration such that the RCS limit is semi-proper

For a countable ordinal $\eta$ and an ordinal $\gamma$ let $\langle P_\alpha, \dot{Q}_\alpha \colon \alpha < \gamma \rangle$ be an RCS iteration with RCS limit $P_\gamma$, such that ...
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rigidity of $\mathcal P(\omega_1) / NS$ under MA

In Woodin's book, Lemma 5.100 asserts that if $MA_{\omega_1}$ holds and there is an $\omega_2$-saturated ideal $I$ on $\omega_1$, then $\mathcal P(\omega_1)/I$ is a rigid boolean algebra, meaning it ...
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Reverse of a termspace forcing fact

Suppose $\kappa$ is an inaccessible cardinal. Consider the termspace forcing for adding a Cohen subset of $\kappa$ after $\mathbb P= Col(\omega,<\kappa)$. Members are Levy names for countable ...
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Does stationary reflection imply Mahloness?

Suppose $\kappa$ is strongly inaccessible and every stationary subset of $\kappa$ reflects. Must $\kappa$ be Mahlo? Remarks: It is possible for every stationary subset of $\kappa$ to reflect, but ...
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Can we always add sets without collapsing cardinals or adding [very] bounded sets?

Given a model of $\sf ZFC$, and an infinite ordinal $\alpha$. Can we prove that there is always a cardinal $\kappa$, and a forcing $\Bbb P$, such that: $\Bbb P$ does not add sets of rank ...
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Collapsing the cardinals between two singular cardinals

Question 1: Is it consistent that there is a forcing notion collapsing $\aleph_{\omega\cdot 2}$ to $\aleph_\omega$ without collapsing $\aleph_\omega$ or $\aleph_{\omega\cdot 2 + 1}$? If the ...
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Class forcings and elementary embeddings

In the Hamkins-Kirmayer-Perlmutter paper "Generalizations Of The Kunen Inconsistency", they prove the following theorem: "Theorem 7: In any set forcing extension $V[G]$, there is no nontrivial ...
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A question related to Woodin's $HOD$ conjecture

Recall that an $\kappa$ is $\omega$-strongly measurable in $\text{HOD}$ if there exists $\lambda < \kappa$ such that $(2^{\lambda})^{\text{HOD}} < \kappa$ and such that there is no partition of ...