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, ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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Godel and forcing
Are there any forcing based proofs of Godel's first and second incompleteness theorems?
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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 ...
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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 ...
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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 $\...
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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 ...
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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=...
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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 ...
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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 ...
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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 &...
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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 ...
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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 ...
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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 ...
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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 ...
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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
$$L_0(A)=A;$$
$$L_{\alpha+1}(A) = L_\alpha (A)\cup \...
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large ccc forcing that preserves CH
Can you name a ccc forcing with the following properties?
1) Atomless and separative
2) The least size of a dense set is large, say at least $\aleph_3$, hopefully as big as you like.
3) Existence ...
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Characterization of $\kappa$-closed forcings
For the notion of distributivity of forcings, we have equivalent defintions, one combintorial, the other in terms of what the generic extensions look like. For a partial order $\mathbb{P}$, the ...
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Reals added after Cohen forcing II
This is related to my previous question "Reals added after Cohen forcing" Reals added after Cohen forcing
The fact is that what I had in mind by $\lambda-$many Cohen reals was different from what was ...
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Solovay's paper from AD+ that all sets are Ramsey
For my research I've been trying to locate Solovay's proof that AD+ implies that all sets are Ramsey (or completely Ramsey, depending on the terminology). I know that it uses Mathias forcing but I ...
6
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Forcing in Ackermann's Set Theory
How would one do forcing in Ackermann's set theory? C. Alkor published an article entitled "Forcing in Ackermanns Mengenlehre" (Zeitshcr. f. math. Logik und Grundlagen d. Math., Bd. 25,8.265-286 (...
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Equivalence of forcing automorphisms
Suppose that $P$ is a forcing poset in $V$. If $\pi$ is an automorphism of $P$ then $\pi$ extends to automorphisms of the names by induction: $$\pi\dot x = \lbrace(\pi p,\pi\dot y)\mid (p,\dot y)\in\...
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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 $...
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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 ...
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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 ...
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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. ...
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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 ...
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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 ...
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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 ...
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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 $P$} = {$D: D$ ...
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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. ...
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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 $V_1$...
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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 ...
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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.
...
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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\...
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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 ...
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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 ...
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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 ${\mathbb P}$-...
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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?
...
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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
$T$?...
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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 ...
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"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 ...
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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 ...
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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 ...