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Forcing is a method first used to prove the continuum hypothesis is independent of the classical axioms of set theory

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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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1answer
305 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 ...
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1answer
424 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 ...
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1answer
275 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 ...
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174 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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3answers
241 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 ...
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1answer
336 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 ...
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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
120 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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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 ...
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0answers
142 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 ...
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0answers
465 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 ...
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2answers
286 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 \...
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1answer
354 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
503 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
190 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 ...
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1answer
299 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
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1answer
154 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
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2answers
268 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 ...
20
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1answer
724 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 (...
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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 ...
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0answers
206 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 $n$-...
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0answers
58 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 ...
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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 ...
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0answers
420 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 ...
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0answers
144 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
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0answers
149 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
188 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 ...
4
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0answers
69 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
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1answer
186 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
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1answer
384 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
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0answers
127 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
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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 \...
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2answers
367 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 ...
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2answers
734 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 ...
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1answer
128 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 ...
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0answers
192 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: ...
4
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1answer
244 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 ...
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3answers
576 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 ...
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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 ...
7
votes
1answer
236 views

Countable support iteration of proper forcings and the tree property

I'm mainly concerned with countable support iterations of proper forcings that add reals of some large cardinal length. It is known that countable support iteration of Sacks forcing/Cohen forcing of ...
6
votes
2answers
347 views

Characterizations of infinite compact Abelian groups and probability spaces based on the forcing notion they give

Let $G$ be an infinite compact Abelian group with the collection $\mathcal{B}$ of Borel subsets of $G$, and $m$ the (unique) normalized Haar measure on $\mathcal{B}$. This gives a natural forcing ...
5
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1answer
168 views

preservation of forcing rigidity in iterations

Say that a partial order $P$ is forcing-rigid in a model $V$ if whenever $G \subseteq P$ is generic over $V$, then in $V[G]$, $G$ is the only filter which is $P$-generic over $V$. This implies there ...
7
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1answer
209 views

What are the typical forcings to shoot a club through a stationary subset of $[\lambda]^\omega$

Let $\lambda\geq \omega_2$ be a regular cardinal and $S\subset[\lambda]^\omega$ be a stationary set. I'm looking for a property of $S$, say "shootable", such that there exists a forcing extension ...
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1answer
707 views

Is the axiom $\Diamond\Box\varphi\to\Box\Diamond\varphi$ in c.c.c. forcing potentialism equivalent to the productivity of c.c.c. forcing?

This question arose in connection with a lecture series on Potentialism that I have just completed here in Hejnice in the Czech Republic at the Winter School 2018 (see Slides). Several of us discussed ...
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0answers
186 views

Iterated forcing with distributive forcing notions

Assume $(\kappa_n| n<\omega)$ is an increasing sequence of inaccessible cardinals with $\kappa_\omega=\sup_{n<\omega}\kappa_n.$. Let $((\mathbb{P}_n| n \leq \omega), (\dot{\mathbb{Q}}_n | n<\...
7
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1answer
137 views

Restrictions of the Stationary Tower forcing providing various critical points

Following Larson's "The Stationary Tower", let $\mathbb{P}_{<\delta}$ be the full stationary tower on $\delta$, and for a stationary $S\subset \mathcal{P}_\delta(V_\delta)$, $\mathbb{P}_{<\delta}...
10
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1answer
251 views

Ways to add Aronszajn trees which are neither Souslin nor special

By an Aronszajn tree, I mean a tree of height $\omega_1$ with countable levels and no branch. Such a tree is Souslin if it has no uncountable antichains and special if it can be written as the ...
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0answers
144 views

Query about iterated collapse forcing

I have heard it said that it is possible to use Woodin's iterated collapse forcing to prove that the theory $\textsf{ZF}$+"there exists a non-trivial elementary embedding $j:V_{\lambda+2} \rightarrow ...