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

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Does ZF have an initial model?

Definition 1: A class $\mathcal{K}$ of countable transitive models of $\text{ZF}$ has an "initial member" $M$ if each member of $\mathcal{K}$ is a forcing extension of $M$ for some partial order ...
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capturing small sets in small factors

Suppose $\kappa$ is a regular cardinal and $P$ is a $\kappa$-c.c. partial order. I want to know when are small sets added by subforcings of size $<\kappa$. The following seems well-known: ...
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Prevalent singular cardinals hypothesis

The following notion is introduced by Assaf Rinot: Definition. A singular cardinal $\kappa$ is a prevalent singular cardinal iff there exists a family $\mathbb{A}\subset P(\kappa)$ with ...
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On the number of models between a ground model and its forcing extension

Let $\kappa$ be a (finite or infinite) cardinal. Assuming consistency of $\text{ZFC}$ (and probabely some additional assumptions) is the following consistent with $\text{ZFC}$? There is a countable ...
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Joint Forcing Extension Property

Definition 1: A class $\mathcal{K}$ of countable transitive models of $\text{ZF}$ has strong "joint forcing extension property" (JFEP) iff for all $M,N\in \mathcal{K}$ there are forcing notions ...
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A special c.c.c forcing notion and adding minimal generic reals

This question is related to my question "Forcing with c.c.c forcing notions, Cohen reals and Random reals". A natural way to answer Prikry's conjecture is to build a c.c.c. forcing notion which adds ...
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Is the tree of large cardinals linear?

Kanamori in the introduction of his book "The Higher Infinite" says: "The investigation of large cardinal hypotheses is indeed a mainstream of modern set theory, and they have been found to play a ...
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213 views

Forcing Language

I asked the following question (slightly paraphrased) about a week ago in Stack exchange but no one knew the answer to the particular question. I was hoping someone here might be able to help me. ...
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A New Continuum Hypothesis (Revised Version)

Define $N_n$ as $n$ th natural number: $N_0=0, N_1=1, N_2=2, ...$. What happens after exponentiation? We have the following equation: $2^{N_n}=N_{2^{n}}$. (Which says: For all finite cardinal $n$ ...
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$\Sigma_1$ Statements and Forcing Extensions

Assuming consistency of $\text{ZFC}$ (with some large cardinal axiom), is the following statement consistent with $\text{ZFC}$? Any $\Sigma_1$ statement with parameters $\omega_1,\omega_2$ which ...
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247 views

Consistency Strength of the Failure of Square on Singular Cardinals

Q1. What is the consistency strength of the failure of square on singular cardinals? Q2. What are known as partial results in this direction?
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Possible Choices for Cofinality of $\aleph_n$ without Choice

$\text{ZFC}$ proves that each $\aleph_{n}$ for $n\in \omega$ is a regular cardinal. But it seems without the Axiom of Choice there are many consistent possible choices for cofinality of such ...
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1answer
183 views

An explicit construction of reals added after some forcing notions

Consider the forcing notion(s) introduced by Friedman (or Mitchell or Neeman) for adding a club subset of $\omega_2$ by finite conditions. In the generic extension CH fails, but I can't see the reals ...
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Impact of Supercompacts on Measurables

It is consistent that the least measurable cardinal can carry exactly one normal measure but in almost all models for this theory there is no supercompact cardinal. It seems existence of a ...
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4answers
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A Special Pair of Models for ZFC (New Version)

Are there two models $M$ and $N$ for $\text{ZFC}$ such that: (1) $M\subseteq N$ (2) $\aleph_{1}^{N}=\aleph_{1}^{M}$ (3) $\aleph_{2}^{N}=\aleph_{\omega +1}^{M}$ Update: According to Peter's useful ...
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367 views

The First Failure of GCH in Large Cardinals Smaller than Measurables

A well known theorem by Scott says: If $\kappa$ is a measurable cardinal and $\mu$ a normal measure on it and $\mu (\lbrace\lambda\in\kappa~|~2^{\lambda}=\lambda^{+}\rbrace)=1$ then ...
7
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1answer
264 views

Countable Product of Class Forcing Notions

Is the following consistent? There are definable class forcing notions $\lbrace \mathbb{P}_{n}\rbrace_{n\in \omega}$ such that: The product of any finitly many of them preserves $\text{ZFC}$ and ...
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random real forcing, independent real

Assume all independent reals that are added by random real forcing. Take enumeration of each independent real. Is the family of all enumerations dominating?
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What is the role of absoluteness in existence of a non-trivial self elementary embedding on an inner model?

ِDefinition (1): Call an inner model M "rigid" if there are no non-trivial elementary embeddings $j: M\longrightarrow M$. It is possible that $L$ is not rigid, while the problem is open for $HOD$. ...
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1answer
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Adding large sets not containing countable ground model sets

The question is motivated by Toni's question "Approximation of infinite set in generic extension" (see Approximation of infinite set in generic extension). Before I state the question, let me add ...
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Question about Woodin's stationary tower

Suppose that $\delta$ is a Woodin cardinal and that $\kappa$ is the critical point of the generic embedding $j:V\rightarrow M$ after forcing with the stationary tower ($\kappa$ can be $\omega_1$ or ...
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1answer
203 views

Are measures of a measurable cardinal measurable? (Edited and Updated Version)

Update: Regarding to Prof. Hamkins's guidance I restricted the questions to the "normal" measures to avoid trivial answers. Definition: Let $\kappa$ be a measurable cardinal. Define: ...
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A Question on Special Forcings

The usual use of forcing begins with a "countable" and "transitive" ground model of $ZFC$ and reaches to a "countable" and "transitive" generic model of $ZFC$ with the "same ordinals". In a discussion ...
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what kind of ordinal is the degree of strongness of a partially strong cardinal (Edited and revised)

For an infinite cardinal $\kappa$ and an ordinal $\lambda>\kappa,$ $\kappa$ is called $\lambda-$strong, if there is a non-trivial elementary embedding $j: V \rightarrow M$ with $crit(j)=\kappa$ ...
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Can we force with Fraisse filters to solve Vaught's conjecture?

Around the classic Fraisse amalgamation theorem in model theory we have the following notions: Definition (1): If $M$ be an $\mathcal{L}$-structure then define: $age(M):=\lbrace N~|~N~\text{is ...
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Is there a forcing closure?

The main theorem of forcing says that for any c.t.m of $ZFC$ like $M$ and for all partial order $\mathbb{P}$ and $\mathbb{P}$-generic $G$ over $M$, there is a c.t.m of $ZFC$, like $N$ such that $N$ is ...
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Can we always permute Cohen reals?

Consider the Cohen forcing, and suppose that $\dot x,\dot y$ are names for reals, which are not in the ground model (i.e. $1$ forces that neither is in the ground model). Can we always find an ...
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1answer
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Why does an $\varepsilon$-incomplete $Q$ remain so in the forcing extension of any $\varepsilon$-complete $P$?

This is Shelah's simpler notion of 'completeness' for not adding reals via forcing. Here $P$ and $Q$ are forcing notions. Let $S_{\aleph_0}(\kappa)$ be the set of all countable subsets of a cardinal ...
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1answer
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How Random is Cohen?

Suppose that $V$ is a universe of $\sf ZFC$, and $c$ is a Cohen generic real over $V$. Is it possible that $c$ is also generic in other senses? I know that it can't be random or Sacks or whatnot ...
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The origins of forcing in mathematical logic and other branches of mathematics

As everyone knows, forcing was created by Cohen to answer questions in set theory. Question 1. What are the first applications of set theoretic forcing in other branches of mathematical logic, like ...
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Separation of almost disjoint families by ground model almost disjoint families

Suppose that $V$ is a model of $\sf ZFC$, and for concreteness I should point that at this point I am interested in $V=L$ as a ground model. Suppose that $V[c]$ is a Cohen extension of $V$ where $c$ ...
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Singularizing forcing of “small” cardinality?

Can there be a large cardinal $\kappa$ and a forcing of size $\kappa$ that makes $\kappa$ a singular cardinal? The motivation is that the standard Prikry forcing does not have a dense set of size ...
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Applications of forcing in model theory

What are the major applications of (set theoretic) forcing in model theory?
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Consistency results using nonstandard models

Are there any consistency results in set theory (or in mathematics) that can be proved using nonstandard models of ZFC but not using transitive models of ZFC?
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Kaplansky's conjecture and Martin's axiom

Recall Kaplansky's conjecture which states that every algebra homomorphism from the Banach algebra C(X) (where X is a compact Hausdorff topological space) into any other Banach algebra, is ...
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1answer
138 views

Generalization of Levy-Solovay theorem to kappa-c.c. forcings

The Levy-Solovay theorem says that if $\kappa$ is measurable, then it remains measurable in the extension by a small forcing ($|\mathbb{P}|<\kappa$). Is still true if we replace ...
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Is there a monster behind the trees?

‎First Fix the following notation:‎ ‎ $‎‎\forall ‎\kappa‎\in Card~~~Tp(‎\kappa‎):="‎\kappa‎~has~tree~property"$‎ ‎‎‎‎ The large cardinals as "monsters of heaven" live everywhere in the land of ...
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Countable support iterated forcing of length $\alpha$ which forces $|\alpha| > \aleph_2$

Let $cf(\alpha) > \omega$, and $P_\alpha := \langle P_{\beta}, \dot{Q}_{\beta} : \beta < \alpha \rangle$ be a countable support iterated forcing construction (so for each $\beta$, $P_{\beta} = ...
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The Ground Axiom for special statements of set theory

The Ground Axiom (GA), introduced by Hamkins and Reitz, asserts that the universe is not a nontrivial set forcing extension of any inner model, and it is known that GA is consistent relative to ZFC. ...
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Making all cardinals countable and its HOD

Suppose that $G$ is $Col(\omega, <Ord)$-generic over $V$ and let $W=HOD^{V[G]}$. Is $CH$ true in $W$? In general, what can we say about the behaviour of the power function in $W$? Update. Are the ...
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Consistency strength of saturated ideals on $[\lambda]^{<\kappa}$

In the Handbook of Set Theory, Foreman notes that in models constructed by Magidor from a huge cardinal, there exists a normal ideal on $[\omega_2]^{<\omega_1}$ that is $\omega_3$-saturated. Other ...
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If there is a non-constructible real, is there an $L$-generic real?

If we assume that $\Bbb R^V\neq\Bbb R^L$, can we deduce that there is some $x\in\Bbb R^V$ which is $L$-generic? Of course if $V$ is a generic extension of $L$ this is true, but if $V=L[0^\#]$ this is ...
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How to make countably closed forcing “nice” without choice

When working over a model $V$ of $ZFC$, countably closed forcings are extremely nice: If $\mathbb{P}$ is countably closed, then $V[G]$ has no new $\omega$-sequences of elements of $V$. In ...
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Does forcing generally go one way?

Question Is there any forcing free proof for hard independence results? talks about the use of forcing for independence results such as: $Con(ZFC)\longrightarrow Con (ZFC+\neg CH) $. For that case ...
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Is there any forcing free proof for hard independence results?

We are forced to use forcing for almost all "hard" independence results such as: $Con(ZFC)\longrightarrow Con (ZFC+\neg CH) $. The question simply is: Primary Question: Is there any "forcing free" ...
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Is Prikry forcing minimal?

Let $M$ be a model of $\sf ZFC$ in which $\kappa$ is a measurable cardinal, and $\cal U$ is a normal measure on $\kappa$. We can define the Prikry forcing (the most simple one) as the poset: $$\Bbb ...
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Woodin's unpublished proof of the global failure of GCH

An unpublished result of Woodin says the following: Theorem. Assuming the existence of large cardinals, it is consistent that $\forall \lambda, 2^{\lambda}=\lambda^{++}.$ In the paper "The ...
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On free limits of iterations

Shelah isolates the notion of "$\aleph_1$-free iteration" in the first two sections of Chapter IX from Proper and Improper Forcing, and he proves there that properness is preserved by this sort of ...
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Large cardinals and mild extensions

It is known that for many large cardinals $\kappa$ (like weakly compact, measurable,...) , $\kappa$ remains large of the same type after forcings of size $<\kappa$. Now the questions are: Question ...
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Some questions about $0^{\sharp}$ and forcing over $L$

1-Let $P\in L$ be a nontrivial set forcing or even a tame class forcing (tameness in the sense of Sy Friedman; see for example his Handbook paper), and let $G$ be $P-$generic over $L$. Then it is ...