Questions tagged [forcing]

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/...
Benjamin Dickman's user avatar
19 votes
9 answers
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Forcing as a tool to prove theorems

It is often mentioned the main use of forcing is to prove independence facts, but it also seems a way to prove theorems. For instance how would one try to prove Erdös-Rado, $\beth_n^{+} \to (\aleph_1)...
Rachid Atmai's user avatar
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19 votes
2 answers
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Woodin's unpublished proof of the global failure of GCH

An unpublished result of Woodin says the following: Theorem.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 ...
Mohammad Golshani's user avatar
12 votes
1 answer
579 views

Can $\mathcal{L}_{\omega_1,\omega}$ detect $\mathcal{L}_{\omega_1,\omega}$-equivalence?

Roughly speaking, say that a logic $\mathcal{L}$ is self-equivalence-defining (SED) iff for each finite signature $\Sigma$ there is a larger signature $\Sigma'\supseteq\Sigma\sqcup\{A,B\}$ with $A,B$ ...
Noah Schweber's user avatar
10 votes
2 answers
980 views

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 ...
Victoria Gitman's user avatar
10 votes
2 answers
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Resembling the Levy Collapse

Suppose $\kappa$ is a weakly compact cardinal. Is there a $\kappa$-c.c. forcing $\mathbb{P}$ such that $\mathbb{P} \subseteq V_\kappa$ and $\Vdash_{\mathbb{P}} \kappa = \aleph_1$, where $\mathbb{P}$ ...
Monroe Eskew's user avatar
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16 votes
3 answers
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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 $\...
Monroe Eskew's user avatar
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12 votes
2 answers
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Which forcings preserve (some) determinacy?

The question is exactly as in the title. I'm interested in general in all questions of the form "which forcings preserve property P?" for any P, but determinacy assumptions occupy a special place in ...
Noah Schweber's user avatar
6 votes
0 answers
620 views

Forcing with c.c.c forcing notions, Cohen reals and Random reals

I think the following question is due to Prikry: Question. Is it consistent that any non-trivial c.c.c forcing notion adds a Cohen real or a Random real? Is the question still open? What partial ...
Mohammad Golshani's user avatar
18 votes
10 answers
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Examples of ZFC theorems proved via forcing

This is an old suggestion of Joel David Hamkins at the end of his answer to this question: Forcing as a tool to prove theorems I just noticed it while trying to understand his answer. But indeed it ...
14 votes
3 answers
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Questions about $\aleph_1-$closed forcing notions

"Foreman`s maximality principle" states that every non-trivial forcing notion either adds a real or collapses some cardinals. This principle has many consequences including: 1) $GCH$ fails everywhere,...
Mohammad Golshani's user avatar
12 votes
1 answer
1k views

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 ...
mikeyus's user avatar
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11 votes
1 answer
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A new cardinal characteristic (related to partitions)?

In this post I will discuss some cardinal characteristic of the continuum, related to partitions of $\omega$ and would like to know if it is equal to some known cardinal characteristic. By a partition ...
Taras Banakh's user avatar
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10 votes
1 answer
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Reals added after Cohen forcing

Let $V_1$ be a generic extension of $V\models GCH$ obtained by adding $\aleph_{\omega}-$many Cohen reals. Then we have the following: 1- In $V_1$ there are $\aleph_{\omega+1}-$many reals, 2- In $V_1$...
Mohammad Golshani's user avatar
8 votes
3 answers
1k views

Tractability of forcing-invariant statements under large cardinals

It is usual to mention theorems of the kind: Th. Assume there is a proper class of Woodin cardinals, $\mathbb{P} $ is a partial order and $G \subseteq \mathbb{P}$ is V-generic, then $V \models \phi \...
Marc Alcobé García's user avatar
7 votes
1 answer
1k views

Can the Cohen forcing collapse cardinals?

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 ...
Yair Hayut's user avatar
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7 votes
2 answers
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Mutually generics

Given posets $P,Q\in M$, I would like to know under what circumstances there are mutually generic filters $G\subseteq P$ and $H\subseteq Q$ (generic over $M$). Also, what are the characterizations of ...
kvagk's user avatar
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48 votes
1 answer
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Producing finite objects by forcing!

It is a trivial fact that forcing can not produce finite sets of ground model objects. However there are situations, where we can use forcing to prove the existence of finite objects with some ...
Mohammad Golshani's user avatar
18 votes
2 answers
1k views

Can measures be added by forcing?

The Lévy-Solovay theorem says that small forcings do not create measures. J.D. Hamkins has generalized this to a larger class of forcings called gap forcings. I would assume this cannot be ...
Trevor Wilson's user avatar
18 votes
1 answer
2k views

What is the modal logic of outer multiverse?

The mathematical multiverse could be viewed as a gigantic Kripke model with models of $ZFC$ as possible worlds connected to each other via a certain accessibility relation. The modal logic associated ...
Morteza Azad's user avatar
18 votes
1 answer
2k views

Forcing over set theory versus forcing over arithmetic

I've been trying to understand better some of the research on forcing over bounded arithmetic and its connections with lower bounds in complexity theory. For example, Takeuti and Yasumoto have some ...
Timothy Chow's user avatar
15 votes
4 answers
1k views

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 ...
Yair Hayut's user avatar
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14 votes
4 answers
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Is it possible for countably closed forcing to collapse $\aleph_2$ to $\aleph_1$ without collapsing the continuum?

Suppose the continuum is larger than $\aleph_2$. Does there exist a countably closed notion of forcing that collapses $\aleph_2$ to $\aleph_1$, but does not collapse the continuum to $\aleph_1$? ...
Norman Lewis Perlmutter's user avatar
14 votes
3 answers
2k views

Forcing over an arbitrary model of ZFC

I’m learning set theory and forcing in the (french) book from Jean-Louis Krivine “Théorie des ensembles”. Given a countable transitive model $\mathscr{M}$ of ZFC together with a poset $P$, he ...
Guillaume Brunerie's user avatar
14 votes
1 answer
1k views

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 ...
Noah Schweber's user avatar
14 votes
2 answers
1k views

Failure of diamond at large cardinals

What is known about the failure of $\Diamond_{\kappa}$ (diamond at $\kappa$) for $\kappa$ (the least) inaccessible, (the least) Mahlo and (the least) weakly compact. Remark. The problem of forcing ...
Mohammad Golshani's user avatar
13 votes
1 answer
1k views

Forcing Diamond

It is well known that adding a subset of a regular cardinal $\kappa$ with partial functions of size $< \kappa$ forces $\Diamond_\kappa$. One can also see that if $S \in V$ is a stationary subset ...
Monroe Eskew's user avatar
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13 votes
2 answers
1k views

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, ...
Noah Schweber's user avatar
12 votes
3 answers
1k views

Why do we need a transitive model in forcing arguments?

One major approach to the theory of forcing is to assume that ZFC has a countable transitive model $M \in V$ (where $V$ is the "real" universe). In this approach, one takes a poset $\mathbb{P} \in M$, ...
dorebell's user avatar
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12 votes
2 answers
1k views

Questions about Prikry forcing and Cohen forcing

I have some questions. The first one is about the product of Prikry's forcing. Let $\kappa$ be a measurable cardinal, $U_1, U_2$ be normal measures on $\kappa$ and let $\mathbb{P}_{U_1}, \mathbb{P}...
Mohammad Golshani's user avatar
12 votes
0 answers
719 views

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 well-...
Mohammad Golshani's user avatar
12 votes
1 answer
735 views

Can we change the Lebesgue measure by forcing?

Suppose $M$ is a model of ZFC, and $\mu^M$ is the Lebesgue measure on $\mathbb R^M$ such that $\mu^M(\mathbb R^M)=1$. It is known that if $r$ is a Cohen real over $M$ and $N=M[r]$ then $\mu^N(\mathbb ...
Asaf Karagila's user avatar
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11 votes
1 answer
696 views

Is $\mathfrak j_{2:1}=\mathfrak{j}_{2:2}$ in ZFC?

A function $f:\omega\to\omega$ is called $\bullet$ 2-to-1 if $|f^{-1}(y)|\le 2$ for any $y\in\omega$; $\bullet$ almost injective if the set $\{y\in \omega:|f^{-1}(y)|>1\}$ is finite. Let us ...
Taras Banakh's user avatar
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8 votes
1 answer
504 views

Intersection of two generic extensions

It is well known that the intersection of two models of ZFC does not have to be a model of ZFC (or even ZF). Now what if we restrict ourselves to models $M[G]$, $M[H]$ which are generic over $M$ for ...
Jonathan Schilhan's user avatar
7 votes
1 answer
400 views

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?
Mohammad Golshani's user avatar
7 votes
0 answers
489 views

A proposed axiom of Laver (updated)

A few months back, I posted a question asking about a proposed axiom of Laver's and I, unfortunately, left out a critical piece. Here is the full axiom: (L) Some elementary embedding $j:V_{\lambda+1}...
Everett Piper's user avatar
7 votes
1 answer
282 views

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 ...
user avatar
6 votes
1 answer
258 views

Restricted notions of set-theoretic geology

We say that $W$ is a ground of the universe $V$ if $W$ is a model of ZFC and there is a poset $P\in W$ such that $W[G]=V$ for some $G$ which is $P$-generic over $W$. The Ground Axiom ($\text{GA}$) ...
Seba Thei's user avatar
  • 359
6 votes
2 answers
868 views

Why are some axioms preserved in generic extensions?

It is a known theorem that for a model of $ZF$, $M$, if $M\models AC$ and $G$ is a $P$-generic filter over $M$, for some $P\in M$, then $M[G]\models AC$. On the other hand, it is long known that ...
Asaf Karagila's user avatar
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6 votes
1 answer
259 views

A question on simple $P_{\aleph_2}$-points

This question is motivated by discussion surrounding this MO question. An ultrafilter $U$ on $\omega$ is a simple $P_{\aleph_2}$-point if it is generated by a sequence $\langle X_\alpha:\alpha<\...
Todd Eisworth's user avatar
6 votes
6 answers
2k views

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 ...
Tobias Neukom's user avatar
5 votes
3 answers
313 views

Approximation of infinite set in generic extension

Suppose $M$ is a c.t.m and suppose $P$ is $Fn(I,2)$ where $I$ is infinite. Now suppose $G$ is $P$-generic, and $A \in M[G]$ is infinite set. Is it guaranteed that the exist $B \in M$ such that $B \...
toni's user avatar
  • 51
4 votes
3 answers
925 views

Which notions of forcing add a cofinal branch to an $\omega_1$-tree?

I'd like to know more about forcing to add a cofinal branch to an $\omega_1$-tree. Question 1: What kinds of forcings add cofinal branches to $\omega_1$-trees? What kinds of forcings cannot? ...
user avatar
4 votes
1 answer
425 views

Is the ordering principle preserved in generic extensions?

The ordering principle says that every set can be linearly ordered. In a previous question Why are some axioms preserved in generic extensions? Asaf Karagila asked which axioms are preserved in ...
Stefan Geschke's user avatar
3 votes
1 answer
309 views

Getting measures (especially on $\omega_2$) from potential clubs

This is a spinoff of this earlier question of mine. Short version: What measures in $L(\mathbb{R})$ can be gotten from "potentially club" filters, under appropriate hypotheses? Long version: ...
Noah Schweber's user avatar
72 votes
6 answers
7k views

A better way to explain forcing?

Let me begin by formulating a concrete (if not 100% precise) question, and then I'll explain what my real agenda is. Two key facts about forcing are (1) the definability of forcing; i.e., the ...
Timothy Chow's user avatar
64 votes
3 answers
6k views

Forcing as a new chapter of Galois Theory?

There is a (very) long essay by Grothendieck with the ominous title La Longue Marche à travers la théorie de Galois (The Long March through Galois Theory). As usual, Grothendieck knew what he was ...
Mirco A. Mannucci's user avatar
53 votes
2 answers
3k views

How to add essentially new knots to the universe?

A knot is an embedding of a circle $S^{1}$ in $3$-dimensional Euclidean space, $\mathbb{R}^3$. Knots are considered equivalent under ambient isotopy. There are two different types of knots, tame and ...
Morteza Azad's user avatar
34 votes
5 answers
2k views

Forcing as a replacement of induction and diagonal arguments

Let me give some examples motivating the question. The use of forcing instead of induction: For this consider Cantor's theorem: Theorem 1. Any two countable dense linear orders $I, J$ without end ...
Mohammad Golshani's user avatar
29 votes
2 answers
5k views

What is the dimension of the mathematical universe?

Forcing construction in set theory leads to a new understanding of the mathematical (multi)universe by providing a machinery through which one can construct new models of the universe from the ...
Morteza Azad's user avatar