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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Forcing over the poset of nonempty open subsets of a nice topological space

Is there anything sensible to be said concerning a notion of forcing given by the poset of nonempty open subsets of the sort of topological space that comes up in ($e.g.$ algebraic) topology? If so, ...
Adam Epstein's user avatar
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How to find a sub-forcing?

Suppose that $M$ is a model of ZFC, $P\in M$ is a notion of forcing and $G$ is a $P$-generic filter over $M$. It is a well-known theorem that if $M\subseteq N\subseteq M[G]$, and $N$ is a model of ZFC ...
Asaf Karagila's user avatar
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From the product lemma to to a result about powersets

Recall the product lemma from Easton's famous paper, which tells us something about when we have a forcing notion (which may be a proper class) that splits as a product with one factor $\lambda^+$-cc ...
David Roberts's user avatar
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Can we weaken GCH in this class forcing?

I've just stumbled across the following theorem (here): Theorem Let $P$ be a partial class order in $M$, a transitive model of ZF (resp. ZFC). Suppose for arbitrarily large cardinals $\kappa$ we have ...
David Roberts's user avatar
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Can we prove the completeness of FOL based on forcing?

I asked this question at http://math.stackexchange.com but I didn't get any answer there. In David Marker's Model Theory, there is a exercise said " we can view a countable Henkin construction as a ...
Chao Chen's user avatar
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I'll admit it: I don't understand the definition of the Easton product.

I'm teaching myself bits and pieces of forcing at the moment, for the purposes of translating them into sheaf-theoretic versions. I'm trying to write down what I feel is a cleaner description of the ...
David Roberts's user avatar
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Suslin algebras

A Suslin algebra is a generalization of a Suslin tree: it is a ccc, $(\omega,\infty)$-distributive boolean algebra. Jech proved that every Suslin algebra has size at most $2^{\omega_1}$. A proof is ...
Monroe Eskew's user avatar
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Why does the Solovay-Tennenbaum theorem work?

I have decided to learn iterations at long last, and I am reading through Jech's Set Theory for now. The standard example after explaining what is an iteration with finite support is the following ...
Asaf Karagila's user avatar
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2 votes
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Godel and forcing

Are there any forcing based proofs of Godel's first and second incompleteness theorems?
provocateur's user avatar
6 votes
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Forcing with product vs. box product

If we want to add one real number the simplest way to do it is to use Cohen forcing. The poset is $\lbrace p\colon n\to 2\mid n\in\omega\rbrace$ which is a countable set. We can think of this as ...
Asaf Karagila's user avatar
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Which of these relations on partial orders allows us to identify forcing equivalence?

Background This question was inspired by Justin Palumbo's excellent question Cantor Bernstein for notions of forcing. In his question, Justin considers a relation $\lhd$ on partial orders (defined ...
jonasreitz's user avatar
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Infinite products of forcings

Suppose $M \subseteq N$ are models of ZFC such that $(ORD^\omega)^M = (ORD^\omega)^N$. Let $\langle P_n : n \in \omega \rangle$ be a sequence of countably closed partial orders in $M$, and let $\...
Monroe Eskew's user avatar
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8 votes
1 answer
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Properness of quotient forcing

It is well known that if $P$ is a proper notion of forcing and $\Vdash_P \dot{Q} \text{ is proper}$ then the iteration $P \ast \dot{Q}$ is proper. Is the converse also true, i.e Suppose that we have ...
Stefan Hoffelner's user avatar
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Does strict order-preservation of powerset curtail the candidates for violation of CH?

Thus, let $\mathrm{OPP}$ be the axiom that $|A|\lt|B| \Rightarrow |2^A|\lt|2^B|$ for any sets $A$ and $B$; and, for any ordinal $\alpha$, let $\mathrm{CH}_\alpha$ be the hypothesis that $\aleph_\alpha=...
John Bentin's user avatar
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On intermediate transitive models for ZFC between M an M[G]

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 ...
Zoorado's user avatar
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What goes wrong in Easton forcing if we don't just use regular cardinals?

Recall that Easton forcing was introduced to show that the continuum function at regular cardinals could be anything subject to 'the obvious constraints' (monotonicity etc). However, it is a handy ...
David Roberts's user avatar
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Can Assumptions about forcing produce Mice? [closed]

This is going to take some build up to completely describe what is a very strange question I seem to have walked into by accident: For every partial order $\mathbb{P}$ and regular cardinal $\lambda &...
15 votes
1 answer
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Can there be a global linear ordering of the universe without a global well-ordering of the universe?

This question arose in the answers to Asaf Karagila's question Does ZFC prove that the universe is linearly orderable?. The answer there was that one can have a ZFC model with no global linear ...
Joel David Hamkins's user avatar
18 votes
2 answers
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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
6 votes
3 answers
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Proper class forcing vs forcing with a set of conditions bigger than one's model

This seems like a natural question to ask, but I've not seen it discussed in my reading around (limited to Easton's paper, the third edition of Jech's Set theory and a small handful of articles). What ...
David Roberts's user avatar
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12 votes
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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
10 votes
1 answer
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Generic Extensions and $L(V_{\lambda+1})$

Suppose $\lambda$ is a strong limit cardinal of cofinality $\omega$ and for $A$ a transitive set, define $L(A)$ in the usual fashion by setting $$L_0(A)=A;$$ $$L_{\alpha+1}(A) = L_\alpha (A)\cup \...
Everett Piper's user avatar
9 votes
2 answers
349 views

large ccc forcing that preserves CH

Can you name a ccc forcing with the following properties? 1) Atomless and separative 2) The least size of a dense set is large, say at least $\aleph_3$, hopefully as big as you like. 3) Existence ...
Monroe Eskew's user avatar
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7 votes
2 answers
517 views

Characterization of $\kappa$-closed forcings

For the notion of distributivity of forcings, we have equivalent defintions, one combintorial, the other in terms of what the generic extensions look like. For a partial order $\mathbb{P}$, the ...
Monroe Eskew's user avatar
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7 votes
1 answer
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Reals added after Cohen forcing II

This is related to my previous question "Reals added after Cohen forcing" Reals added after Cohen forcing The fact is that what I had in mind by $\lambda-$many Cohen reals was different from what was ...
Mohammad Golshani's user avatar
7 votes
1 answer
463 views

Solovay's paper from AD+ that all sets are Ramsey

For my research I've been trying to locate Solovay's proof that AD+ implies that all sets are Ramsey (or completely Ramsey, depending on the terminology). I know that it uses Mathias forcing but I ...
Daniel Walker's user avatar
6 votes
1 answer
604 views

Forcing in Ackermann's Set Theory

How would one do forcing in Ackermann's set theory? C. Alkor published an article entitled "Forcing in Ackermanns Mengenlehre" (Zeitshcr. f. math. Logik und Grundlagen d. Math., Bd. 25,8.265-286 (...
Thomas Benjamin's user avatar
5 votes
2 answers
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Equivalence of forcing automorphisms

Suppose that $P$ is a forcing poset in $V$. If $\pi$ is an automorphism of $P$ then $\pi$ extends to automorphisms of the names by induction: $$\pi\dot x = \lbrace(\pi p,\pi\dot y)\mid (p,\dot y)\in\...
Asaf Karagila's user avatar
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9 votes
1 answer
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Can $\omega_1$ be supercompact?

Is "ZF + $\omega_1$ is supercompact" consistent relative to "ZFC + there is a supercompact cardinal"? In particular, if $\delta$ is supercompact, does it remain so in $V(\mathbb{R} \cap V[G])$ where $...
Trevor Wilson's user avatar
7 votes
0 answers
377 views

Sets of reals amenable to each L[x]

If $A$ is a set of reals such that $A \cap L[x] \in L[x]$ for each real $x$, is there a real $z$ such that $A \cap L[x] \in OD_z^{L[x]}$ for a cone of $x$? This can be proved under the Axiom of ...
Trevor Wilson's user avatar
6 votes
2 answers
376 views

Measures that are not OD

Is anything known about the consistency strength of the statement: "There is a normal measure (on a cardinal) that is not ordinal-definable"? In particular, is it consistent relative to the ...
Trevor Wilson's user avatar
4 votes
3 answers
523 views

Is every set class generic over a given inner model?

In a paper by B. Mitchell, I stumbled into the following sentence: "In the summer of 1986 Woodin discovered the second of the forcing orders associated with a Woodin cardinal, the extender algebra. ...
Lstoa's user avatar
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4 votes
2 answers
611 views

The category of Boolean-valued models associated to a model of ZFC

This is related to my previous question here, and indirectly motivated by Andreas Blass intriguing answer therein: start from a ZF transitive model $M$, and consider the category $CBA(M)$ of ...
Mirco A. Mannucci'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
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64 votes
3 answers
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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
2 votes
1 answer
818 views

Absoluteness of Countability

Let M be a countable transitive model for ZFC, P is a partial order in M. Notions like "partial orders" and "dense" are absolute. Consider the following set $S$={$D\in M: D$ is dense in $P$} = {$D: D$ ...
Jing Zhang's user avatar
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2 votes
1 answer
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$(<\kappa)$-closure for Prikry forcing

A p.o. $\mathcal{P}$ is $(<\kappa)$-closure, if for every decreasing sequence of $<\kappa$ conditions in the forcing $p_0 \geq p_1 \geq \cdots$, there is a condition that is below all of them. ...
um Haitham's user avatar
10 votes
1 answer
2k views

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
5 votes
0 answers
1k views

Mathias forcing with Ramsey ultrafilters, and Cohen reals [closed]

Edit/update: One reason this question never received an answer is because it was founded on a faulty premise! The Blaszczyk-Shelah paper I mentioned below does not prove that Mathias forcing with a ...
Justin Palumbo's user avatar
4 votes
1 answer
371 views

a result about Laver property

recently, I am reading Tomek Bartoszynski's book"Set Theory On The Structure Of The Real Line" There is a lemma I don't understand it's proof.(P244 Lamma 4.6.10),it's original expression as follows. ...
Jialiang He's user avatar
10 votes
2 answers
1k views

Difference between Laver's and Mathias's forcing

Mathias forcing consists of conditions of the form $(s,A)$ where $s$ is a finite subset of $\omega$, $A$ is an infinite subset of $\omega$ such that $\max(s) < \min(A)$ and $(s,A)\leq (t,B)$ iff $s\...
David Fernandez-Breton's user avatar
8 votes
4 answers
663 views

Ultrafilters and the exposition of forcing

I've just been reading Timothy Chow's "A beginner's guide to forcing." Question: Does an exposition of forcing really need to delve into ultrafilters? What I have in mind: If ZFC proved CH, the ...
David Feldman's user avatar
9 votes
1 answer
393 views

Intermediate extension of a Prikry-Silver extension?

Prikry-Silver forcing $\mathbb{V}$ (sometimes just Silver forcing) is the forcing notion consisting of all partial functions $p:\omega\rightarrow 2$ with co-infinite domain. In "Combinatorics on ...
Justin Palumbo's user avatar
4 votes
1 answer
321 views

Cohen forcing: why is the cardinality of $\mathcal P \left({\omega}\right)$ in ${\bf L}(\mathcal P \left({\omega}\right))$ independent of G?

This is a part of exercise E4 from Chapter VII of Kunen's Set Theory. The hint (courtesy of A. Miller) goes like this: let ${\mathbb P} = Fn(I,2)$, $(|I| \geq \omega_{1})^M$. Let G be ${\mathbb P}$-...
Z H Lee's user avatar
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4 votes
3 answers
927 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
13 votes
2 answers
892 views

Can there be an almost-special not-fully-special Aronszajn tree?

Question. Can there be an Aronszajn tree $T$, such that no c.c.c. forcing extension adds a cofinal branch to $T$, but there is an $\omega_1$-preserving forcing extension adding a cofinal branch to $T$?...
Joel David Hamkins's user avatar
17 votes
2 answers
2k views

Two versions of "absolutely ccc"

I have recently been slogging my way through Shelah's "Large continuum, oracles". Essentially from the start there has been a question needling me which I cannot seem to answer. In the paper, Shelah ...
user642796's user avatar
10 votes
2 answers
342 views

"generic" in elementary submodels

Suppose you have a Suslin tree $T$ and you have a countable elementary submodel $M$ containing the usual "enough stuff" (including $T$). A comment in Todorcevic's Partition Problems in Topology ...
Judy Roitman's user avatar
21 votes
3 answers
2k views

In What Way are Set Theorists' 'Experiences' in the CH Worlds Flawed, if Any?

This is in regards to Joel David Hamkins' new paper "IS THE DREAM SOLUTION OF THE CONTINUUM HYPOTHESIS ATTAINABLE?" (look under title in arXiv). I quote from the last paragraph of his paper: "My ...
Thomas Benjamin's user avatar
4 votes
1 answer
342 views

How large can the power set P(N) be made via forcing?

Given the Constructable Universe L, is there a forcing extension L[G] of L in which P(N) is the 'size' (so to speak) of ORD, the proper class of all ordinals? I am interested in forcing extensions of ...
Thomas Benjamin's user avatar