Forcing is a method first used to prove the continuum hypothesis is independent of the classical axioms of set theory
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104 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 ...
1
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1answer
111 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
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1answer
176 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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451 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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146 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/...
5
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195 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
197 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?
11
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2answers
222 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
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2answers
264 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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367 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
352 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
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1answer
451 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
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1answer
292 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
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0answers
180 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}...
8
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3answers
269 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 ...
14
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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
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1answer
344 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 ...
4
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131 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 ...
9
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0answers
163 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
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0answers
146 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
480 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
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2answers
290 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
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1answer
366 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 $U$ ...
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2answers
514 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
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1answer
193 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
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1answer
303 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
158 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
277 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
755 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
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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
208 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
61 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 ...
16
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0answers
429 views
Gitik's work on Shelah's weak hypothesis
It seems that Moti Gitik has recently refuted some variants of Shelah's weak hypothesis. For this see the title and abstract of his talk at the Set Theory, Model Theory and Applications conference.
I ...
4
votes
0answers
146 views
Iterations of proper forcings defined in an inner model
Are there any examples where a countable support iteration of proper forcing defined in an inner model does crazy things in an outer model? This is vague, so to narrow it down, are there examples of $...
7
votes
0answers
152 views
Specializing fat trees
The discussion is about trees of height $\omega_1$ that are not necessarily thin, namely, no cardinality constraints on the size of each level. A classcial theorem of Baumgartner states that it is ...
6
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1answer
198 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 ...
5
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0answers
76 views
Chain condition in iterated forcing
(I apologize for the long question, which has no mathematical content. Just looking for the right reference.)
In their celebrated paper [ST1971] introducing iterated forcing, Solovay and Tennenbaum ...
3
votes
1answer
192 views
Measurably-isomorphic subsets of polish spaces and the continuum hypothesis
In Theorem 2.7 in the following notes, we seem to assume the following statement.
Let $(\Omega,\mathcal F)$ be a Polish space, and $A\in\mathcal F$ an uncountable set. Then there exists a bijection ...
6
votes
1answer
390 views
fake and weak cardinals
Suppose $\lambda$ is a successor of a singular cardinal. We will say $\lambda$ fake if there is a transitive set $M$ such that $\lambda \subseteq M$ satisfying $\mathrm{ZFC}^-$ (ZFC without powerset) ...
3
votes
0answers
132 views
distributivity of club shooting forcing destroyed in the limit
Let $S\subset \omega_1$ be a stationary and co-stationary set. Let $CU(S)$ be the standard forcing that shoots a club into $S$. It is a standard fact that $CU(S)$ is countably distributive, i.e. it ...
4
votes
0answers
104 views
weak square and tower forcing
Suppose $\delta$ is a Woodin cardinal and $\kappa < \delta$. If $G$ is generic for the stationary tower $\mathbb Q^\kappa_{<\delta}$, then there is an elementary embedding $j : V \to M \...
14
votes
2answers
372 views
Proper topological spaces
Recall that a topological space is ccc, or has the countable chain condition, if every family of pairwise disjoint open sets is countable.
But equivalently, we can say that the forcing defined with ...
14
votes
2answers
747 views
Is there a universal way to force the Axiom of Choice to be true?
Given a model of set theory $V$ there are various ways to construct a model in which the Axiom of Choice holds, such as Gödel's constructible universe $L^V$ or by using forcing*. I'm wondering if any ...
10
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1answer
129 views
Complexity of $\Vdash_{\mathbb{P}}$ when $\mathbb{P}$ is a class definable iteration
Let $\mathbb{P}$ some class forcing iteration which is absolute enough. For instance, suppose that $\mathbb{P}$ is a $\Delta_1$-definable class iteration. Normally, this is the case for iteration of ...
8
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0answers
197 views
Direct limits of $\sigma$-centered forcing notions
It is quite well known that
Any FS (finite support) iteration of length $<\mathfrak{c}^+$ of $\sigma$-centered posets is $\sigma$-centered (see e.g. here).
Now consider the following question: ...
5
votes
1answer
256 views
How to kill a $\Sigma_{n+1}$-correct cardinal softly(n>1)?
A cardinal $\kappa$ is $\Sigma_n$-correct iff $V_\kappa \prec_n V$. For n>1, how to force a $\Sigma_{n+1}$-correct cardinal to be $\Sigma_{n}$-correct but not $\Sigma_{n+1}$-correct?
For $n=1$, we ...
14
votes
3answers
592 views
Forcings predicted by core model theory $+$$ZFC$ results proved by the method of core model theory
I have two unrelated question.
First question. To motivate the question, let me explain an example. The natural way to force the failure of singular cardinals hypothesis ($SCH$), is to start with a ...
6
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0answers
130 views
Combinatorial characterizations of potentially countably chromatic graphs
Is there a combinatorial characterization of (uncountably chromatic) graphs that are "potentially" countably chromatic? By this I mean: $G=(V,E)$ is a graph such that there exists a cardinal ...
