Forcing is a method first used to prove the continuum hypothesis is independent of the classical axioms of set theory
10
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305 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
votes
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 ...
4
votes
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 ...
3
votes
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 ...
6
votes
0answers
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}...
8
votes
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
votes
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
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 ...
23
votes
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 ...
3
votes
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 ...
9
votes
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
votes
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 ...
14
votes
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 ...
11
votes
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 \...
6
votes
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 $U$ ...
9
votes
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
votes
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 ...
3
votes
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
votes
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
votes
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
votes
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 (...
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
votes
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 $n$-...
2
votes
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 ...
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
votes
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 ...
4
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
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
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 ...
14
votes
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 ...
10
votes
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 ...
7
votes
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
votes
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 ...
14
votes
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 ...
6
votes
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
votes
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
votes
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 ...
22
votes
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 ...
10
votes
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
votes
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
votes
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 ...
6
votes
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 ...
