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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Topological applications of $\mathfrak{p}=\mathfrak{t}$
I'm working on the Malliaris-Shelah's recent result of $\mathfrak{p}=\mathfrak{t}$, but I'm more interested in what possible topological applications can derivate from this equality.
Searching in ...
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What is known about topological groups of countable spread in ZFC?
A topological space has countable spread if every discrete subspace is at most countable.
By Theorem 8.10 in Todorcevic's book "Partition Problems in Topology", PFA implies that each regular space $X$...
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measure of generic reals in forcing extensions
It is well-known that if $V[G]$ is a generic extension by adding a Cohen real, then
the set $\{r \in V[G]: r$ is Cohen generic over $V\}$ has measure zero.
On the other hand, if $V[G]$ is a generic ...
9
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Sacks property for higher cardinals
It is well known, that the Sacks forcing has the property that for any $f: \omega \rightarrow \omega$ in generic extension, one can obtain $F:\omega \rightarrow [\omega]^{<\omega}$ in ground model, ...
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Destroying Suslin, nothing special
Recall that a tree on $\omega_1$ is called Suslin if every chain and antichain are countable. If every level is countable and there are no cofinal branches, then it is called Aronszajn (in particular ...
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Is there a minimal extension of $L$ that is not a forcing extension?
It's well known that Sacks forcing constructs a real of minimal constructability degree, i.e. a real $x$ such that for any $y\in L(x) \setminus L$, $L(y) = L(x)$. It's also well known that certain ...
8
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Does $\mathsf{MA}^+(\sigma\text{-closed})$ imply there are no Kurepa Trees?
The question in the title is somewhat self contained but let me make some definitions and remarks to clarify.
Recall that $\mathsf{MA}^+(\sigma\text{-closed})$ is the statement that if $\mathbb P$ is ...
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Reference request: destroying saturation at an inaccessible?
An ideal $I$ on $P(\kappa)$ is said to be $\kappa^+$-saturated if there is no sequence $\langle S_\alpha \mid \alpha<\kappa^+\rangle$ of $I^+$ sets such that $\alpha<\beta<\kappa^+\implies S_\...
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What happens when you iterate Cohen reals?
There are a few classical theorems in set theory:
The finite support iteration of ccc forcing is ccc.
The countable support iteration of proper forcing is proper.
The finite support iteration of ...
10
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stationary reflection in $[\kappa]^\omega$
It is well-known that the following reflection principle is consistent relative to a supercompact:
For all $\kappa \geq \omega_2$ and all stationary $S \subseteq [\kappa]^\omega$, there is $X \...
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A proof of recontruction of Sacks generic filter from it's Sacks real (M[G] = M[f])
Given the Sacks forcing $ (\mathbb{S} = \{T \subset 2^{<\omega} : T \text{ is perfect}\},\subset) $ and $G$ generic over M, we have $f = \bigcup \bigcap G = \bigcup_{T \in G}stem(G) $ a path ...
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The Parovichenko cardinal, is it equal to $\max\{\aleph_2,\mathfrak p\}$?
Let us define the Parovichenko cardinal $\mathfrak{P}$ as the largest cardinal $\kappa$ such that each compact Hausdorff space $K$ of weight $w(K)<\kappa$ is the continuous image of the remainder $...
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O-minimality and forcing
It is well-known that the structure $(\mathbb{R}, +, \cdot, <, 0, 1)$ is an o-minimal structure and hence the set of integers $\mathbb{Z}$ is not definable in it.
In an ongoing project with Will ...
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A ridiculous combinatorial cardinal characteristic of the continuum?
This question assumes familiarity with combinatorial cardinal characteristics of the continuum. It is abstracted out of a question in a joint research with Jialiang He. I hope we've got the ...
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Characterizing "bounded" distributivity in terms of dense open sets
Prikry forcing is a well-known example of a forcing which adds an $\omega$-sequence to some cardinal $\kappa$, but does not add bounded subsets to $\kappa$.
There are other examples of forcings which ...
3
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Modal Principles of Field Extensions
In 2007 (with more work done later), J. Hamkins and B. Löwe found that the ZFC provably valid principles of forcing are the assertions of S4.2. In the introduction, they mention field extension as a ...
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C.c.-ness of a forcing notion based on an atomless complete Boolean algebra
Given $\mathbb{B} = \langle B, \wedge, \vee, \neg, 0, 1 \rangle$ an atomless complete Boolean algebra that has a $< \mkern-4mu \kappa$-closed dense subset and is $\kappa^+$-c.c., we define a ...
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Encoding sets in locally generic sets
Let $\alpha$ be an ordinal, and let $a\subseteq\alpha$ such that $\alpha$ is countable in $L[a]$. Moreover, let $\beta>\alpha$ be an ordinal such that, in $L[a]$, $\alpha$ and $\beta$ have the same ...
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A variant of Radin forcing
Suppose $\kappa$ is a large cardinal (strong cardinal seems to be enough). Is there a forcing notion $\mathbb{R}$ with the following properties:
$(1)$ Forcing with $\mathbb{R}$ adds a club $C$ into $\...
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"Antiforcing" - Is there a method to 'remove' sets from a model of ZF?
Forcing is a method of "adding sets" to a model $M$ of ZF by making a new set $M^{(\mathbb{P})}$ consisting of every set of $M$, but you have the option to add certain sets out of $M^{(\mathbb{P})}$ ...
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Decomposition of forcing iterations
One of the great things about a finite support iteration $\Bbb P_\delta$, is that if $\alpha<\delta$, we can write $\Bbb P_\delta$ as the iteration of $\Bbb{P_\alpha\ast\dot Q_\alpha\ast P_\delta/...
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Bad forcing permutations
Let $P$ be the finite-support product of the Cohen forcing. It adjoins a sequence of Cohen-generic reals say $a_n$, $n<\omega$, which one naturally calls a $P$-generic sequence. Suppose that $\pi$ ...
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Collapse an inaccessible cardinal to a successor of a singular cardinal
Is it possible to turn an inaccessible cardinal in $V$ to a successor of a singular cardinal in some forcing extension?
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Countable support product of Sacks forcings and selective ultrafilters
If $U$ is a selective ultrafilter on $\omega$, then $U$ generates an ultrafilter in $V^{\mathbb S}$, where ${\mathbb S}$ is Sacks forcing. The same is true with ${\mathbb S}$ being replaced by ${\...
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Problem understanding a passage of the proof of $\mathfrak{p}=\mathfrak{t}$ involving forcing
I've a problem with a passage of the proof of Claim 14.7 of the paper "Cofinality spectrum theorems in model theory, set theory, and general topolgy" by Malliaris and Shelah, or equivalently ...
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On the Number of Parallel Automorphism Lines
Given a group $G$, one can define the transfinite line of iterative automorphisms of $G$ to be the following chain of the groups where $G_{\alpha+1}=Aut(G_{\alpha})$ for each ordinal $\alpha$ and the ...
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Do choice principles in all generic extensions imply AC in $V$?
It's well-known that not all choice principles are preserved under forcing, e.g. in this answer https://mathoverflow.net/a/77002/109573 Asaf shows the ordering principle can hold in $V$ and fail in a ...
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What are examples of non-equivalent virtualizations of a large cardinal?
This is a follow up to my previous question concerning virtual large cardinals, that are generally weaker axioms of infinity obtained from ordinary large cardinals through the so-called virtualization ...
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On the Actual Potential of Virtual Large Cardinals
Virtual large cardinals belong to a relatively new breed of strong axioms of infinity. They often appear as statements of the following form:
Definition. Suppose $A$ is a large cardinal property ...
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Does $0^{\#}$ imply that $L$'s set generic multiverse has irreconcilable theories?
Suppose $0^{\#}$ exists. Let $M(L)$ be the generic multiverse over $L$ as viewed in $V$, i.e. $M(L)$ consists of all $L[g]$ where $g \in V$ is $\mathbb{P}$-generic over $L$ for some forcing $\mathbb{P}...
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Locales as spaces of ideal/imaginary points
I posted this question on MSE a few days ago, but got no response (despite a bounty). I hope it will get more answers here, but I'm afraid it might not be appropriate as I'm not sure it's actually ...
18
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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 ...
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Problem on reading Jech's set theory about forcing (of Lemma 15.19)
As the proof in the picture, the author says that we can assume that every condition forces that $\dot{f}$ is a function form $\lambda$ to $A$.
I guess that here he means that we can assume $A=M\cap ...
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Forcing and Family Contentions: Who wins the disputes?
The famous game-theoretic couple, Alice & Bob, live in the set-theoretic universe, $V$, a model of $ZFC$. Just like many other couples they sometimes argue over a statement, $\sigma$, expressible ...
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Non-special $\aleph_2$-Aronszajn trees in the Laver-Shelah model for $\aleph_2$-Souslin hypothesis
Assume $V=L$ and let $\kappa$ be a weakly compact cardinal. Let $G$ be $Col(\aleph_1, < \kappa)$ generic over $V$. Working in $V[G]$ force with a countable support iteration of forcing notions of ...
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Analogue of strong stationary reflection from MM
Foreman-Magidor-Shelah proved that if Martin’s Maximum holds, then for every regular $\kappa>\omega_1$, every stationary $S \subseteq S^\kappa_\omega$ contains a club of ordertype $\omega_1$. This ...
4
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Generic two-cardinal behavior of first-order sentences
This is a hopefully improved version of a question I asked before and then deleted because it was based on some fundamentally incorrect assumptions.
Some first-order theories are able to control the ...
14
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Cohen/Random reals over intermediate models in countable support iterations
Assume that the continuum is $\aleph_2$ in our ground model $V$. Suppose that $(P_n \, : \, n \in \omega)$ is a fully supported iteration of length $\omega$. Suppose further that the factors of the ...
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Does $0^{\#}$ imply the failure of upward directedness in the set generic universe over $L$?
Question 1. Suppose that $0^{\#}$ exists. Are there set generic filters $g,h$ over $L$ ($g,h \in V$) such that for any $f$ ($\in V$) which is generic over $L$ we have that $L[g] \cup L[h] \not \...
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Boolean ultrapower of V[G] by G
In Joel David Hamkins's "Well-founded Boolean Ultrapowers as Large Cardinal Embeddings", it is mentioned that if $U \in \mathbf{V}$ is an ultrafilter of a complete Boolean algebra $\mathbb{B}$ and $U$ ...
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On Applications of Forcing in Domain Theory
An interesting feature of domain theory is to use partial orders in order to provide a mathematical model for the computational approximation in a potentially infinite computational process (e.g. ...
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distributivity of termspace forcing
Laver introduced the concept of termspace forcing. If $\mathbb P * \dot{\mathbb Q}$ is a two-step iteration, then we can order the $\mathbb P$-names for elements of $\dot{\mathbb Q}$ by putting $\dot ...
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Strong Total Failures vs. Weak Instances of the Generalized Continuum Hypothesis
The exponentiation operator inflicts a subtle information loss on the transfinite numerical equations, pretty similar to the case of $a^2=b^2 \nRightarrow a=b$ in real numbers. In fact, for the ...
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Negation of CH implied by lots of special subtrees?
In the following, I focus on trees of height $\omega_1$: if there exists a nonspecial tree any of whose $\aleph_1$-subtrees is special, must CH fail?
Some neither consistent nor coherent thoughts: ...
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Does Easton forcing preserve measurable cardinals?
The question is in the title. For Easton's theorem see Wikipedia. Loosely speaking we can use forcing to manipulate the powerset function on regular cardinals as much as we like given we satisfy the ...
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When will the real numbers be Borel?
In set theory Borel sets are important, but we don't actually care about the sets. We can about the Borel codes. Namely, the algorithm to generate a given Borel set starting with the basic open sets (...
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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 ...
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How many iterations of inner models/generic extensions are sufficient?
Let $M=M_0$ be a ctm of ZF.
If $(M_0, \ldots, M_n)$ is a sequence of ctms of ZF, where for all $i,$ either $M_{i+1}$ is an inner model of $M_i$ or a generic extension of $M_i,$ then call $M_n$ an $n$-...
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Which spaces are still Lindelöf after forcing with a Suslin tree?
Let $T$ be a Suslin tree and $f:T\to Y$ be continuous. ($T$ is endowed with the order topology.) Assume that the image of $T$ is contained in a Lindelöf subset of $Y$. Then, force with $T$. Which ...
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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 ...