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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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 4/...
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Definability of ground model

I have seen mentioned that by a result of Laver, the ground model is definable in any set forcing extension (using parameters). Does the same hold for class forcing? If it does, in order to establish ...
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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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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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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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Bad subforcings of nice forcing notions

Let $\mathbb{P}$ and $\mathbb{Q}$ be two forcing notions. Recall that we say $\mathbb{Q}$ is a subforcing of $\mathbb{P}$ if there exists a regular embedding $\mathbb{Q} \to \text{r.o.}(\mathbb{P}).$ ...
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Formal proof of Con(ZFC) => Con(ZFC + not CH) in ZFC

Is it possible to prove $Con(ZFC) \rightarrow Con(ZFC + \neg CH)$ purely within ZFC? To prove this (using forcing) one seems to need a countable transitive model of ZFC. The texts I am reading avoid ...
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Who proved “sets in every generic are already in the ground model?”

Suppose $\mathbb{P}$ is a notion of forcing in the ground model $V$, and $X$ is a set which is in $V[G]$ for every $\mathbb{P}$-generic filter $G$. Then $X\in V$ already, by a fairly simple (if ...
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Forcing PFA with ccc forcing

Is it consistent (from suitable large cardinals) that there is a ccc poset which forces PFA? This seems quite implausible to me. If we could force PFA via ccc forcing, the ground model would have to ...
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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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Probabilities independent of ZFC?

Hi guys, is it possible to change the probability of an event via forcing? More precisely, is there an innocent looking question on the probability of "something" whose answer is independent of 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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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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A specific Model of ZFC

In his paper "Some Second Order Set Theory", Joel Hamkins asked whether there is a model of set theory $V$ that is elementary equivalent to $V[G]$, Whenever $G$ is $V$-generic for the collapse of a ...
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Universally Baire Tree Representation of Projective Sets

In Feng, Magidor, and Woodin "Universally Baire Sets of Reals", they show that if $A$ is a $\mathbf{\Pi}_2^1$ set and $U$ and $V$ are any pair of trees witnessing the universal baireness of $A$, then ...
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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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Namba forcing and semiproperness

This question is the result of leaving "Proper and Improper Forcing" on my nightstand by accident. Is the statement "Namba forcing is semiproper" known to be equiconsistent with some more standard ...
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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 P=\...
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Products of Cohen forcings

Let $\lambda$ be an infinite cardinal. Does the full support product of $\lambda$ copies of $Add(\omega, 1)$ collapse $2^\lambda$ to $\aleph_0$? For $\lambda = \omega$, it is known to be true (it is ...
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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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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 ...
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 ...
Let $\kappa$ be a regular cardinal, and let $\mathbb{P} = Add(\kappa,1)$ be the standard forcing notion for adding a new subset of $\kappa$ using partial function from $\kappa$ to $2$ with domain of ...