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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0answers
87 views
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 $X$...
6
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0answers
254 views
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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0answers
193 views
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, ...
7
votes
1answer
400 views
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 ...
8
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1answer
130 views
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 ...
7
votes
1answer
144 views
Does $\mathsf{MA}^+(\sigma-{\rm closed})$ 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 $\mathsf{MA}^+(\sigma-{\rm closed})$ is the statement that if $\mathbb P$ is ...
4
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0answers
128 views
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_\...
9
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331 views
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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0answers
149 views
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 \...
1
vote
1answer
145 views
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 ...
11
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1answer
215 views
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 $...
14
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0answers
284 views
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 ...
6
votes
1answer
594 views
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 ...
5
votes
2answers
190 views
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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0answers
96 views
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 ...
2
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0answers
122 views
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 ...
2
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1answer
120 views
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 ...
2
votes
1answer
185 views
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 $\...
3
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0answers
473 views
“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 $M^{(\mathbb{P})}$ ...
5
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0answers
155 views
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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197 views
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 $\pi$ ...
3
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1answer
205 views
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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2answers
236 views
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 ${\...
4
votes
2answers
272 views
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 ...
10
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0answers
377 views
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 ...
13
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1answer
363 views
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 ...
4
votes
1answer
467 views
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 ...
3
votes
1answer
306 views
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 ...
6
votes
0answers
183 views
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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votes
3answers
303 views
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 ...
15
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1answer
1k 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 ...
2
votes
1answer
350 views
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 ...
23
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1answer
1k views
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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0answers
133 views
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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0answers
167 views
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
votes
0answers
149 views
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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0answers
493 views
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 ...
11
votes
2answers
298 views
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 \...
6
votes
1answer
375 views
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 $U$ ...
9
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2answers
519 views
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. ...
6
votes
1answer
199 views
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 ...
3
votes
1answer
308 views
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 ...
5
votes
1answer
161 views
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: ...
9
votes
2answers
287 views
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 ...
21
votes
1answer
789 views
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 (...
28
votes
2answers
4k 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 ...
9
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0answers
211 views
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 $n$-...
3
votes
0answers
63 views
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 ...
48
votes
2answers
2k 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 ...
