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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Distributivity of certain infinite products
Suppose we have a sequence of posets $\{\mathbb P_n : n\in\omega\}$ such that for each $n$, $\mathbb P_{n+1}$ is $|\mathbb P_n|^+$-distributive. Is $\prod_{n>0} \mathbb P_n$ necessarily $|\mathbb ...
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Strong Chains in $^{\omega_1}\omega_1$ of length $\omega_3$
In a previous question, I asked about the impact of strong chains in $^{\omega_1}\omega_1$ (e.g., sequences of functions $\langle f_\alpha:\alpha<\kappa\rangle$ in $^{\omega_1}\omega_1$ that are ...
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On the role of $\diamondsuit$
The well-known axiom $\diamondsuit$ states that there is a sequence $\langle A_\alpha:\alpha<\omega_1\rangle$ (a $\diamondsuit$-sequence) of countable sets with the property that for any $A\...
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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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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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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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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: ...
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On an unpublished result of Magidor
In 1970th, Magidor proved the following important results:
(1) Assuming the existence of a supercompact cardinal, it is consistent that $\aleph_\omega$
is strong limit and $2^{\aleph_\omega}=\aleph_{\...
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321
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preserving saturated ideals
A reliable source made the following claim:
Suppose CH there is an $\omega_2$-saturated ideal on $\omega_1$. Then this is preserved by $\mathrm{Add}(\omega_1,\omega_2)$.
Question 1: How do you ...
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Proving regularity properties from forcing axioms
It's well known that PFA implies projective determinacy. It's also well known that PD implies that all projective sets are Lebesgue measurable, have the Baire property, etc.
Is there a direct proof ...
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Which forcing types preserve the axiom of determinacy?
Do we have some rudimentary understanding of some properties that a forcing can have in order to guarantee that it doesn't violate the axiom of determinacy?
To be more specific, in Which forcings ...
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Consistency strength of saturated ideals on $[\lambda]^{<\kappa}$
In the Handbook of Set Theory, Foreman notes that in models constructed by Magidor from a huge cardinal, there exists a normal ideal on $[\omega_2]^{<\omega_1}$ that is $\omega_3$-saturated. Other ...
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4
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Ultrafilters and the exposition of forcing
I've just been reading Timothy Chow's "A beginner's guide to forcing."
Question: Does an exposition of forcing really need to delve into ultrafilters?
What I have in mind: If ZFC proved CH, the ...
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Tractability of forcing-invariant statements under large cardinals
It is usual to mention theorems of the kind:
Th. Assume there is a proper class of Woodin cardinals, $\mathbb{P} $ is a partial order and $G \subseteq \mathbb{P}$ is V-generic, then $V \models \phi \...
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Probabilities independent of ZFC?
Hi guys,
is it possible to change the probability of an event via forcing? More precisely, is there an innocent looking question on the probability of "something" whose answer is independent of ZFC?
...
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Non-set-theoretic consequences of forcing axioms
This article by Quanta Magazine states:
... forcing axioms ... are workhorses that regular mathematicians “can actually go out and use in the field, so to speak,” ...
What are some examples of uses ...
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History of forcing over admissible sets
In his paper "Forcing in admissible sets", Ershov writes
In unpublished lectures given at Novosibirsk State University in 1976-1977 on the theory of admissible sets, the author
showed that it is ...
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Class forcing: Pelletier vs Friedman
[Apologies in advance for a fluffy question]
I'm reading this old paper by Pelletier, where he gives a Boolean-valued model version of class forcing, assuming that the Boolean algebra in question can ...
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919
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Size of stationary sets
What can we say about the size of stationary subsets of $P_{\kappa}(\lambda)$ for infinite cardinals $\kappa, \lambda,$ especially when $\kappa=\aleph_1.$
Please give me some references, if there are ...
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Statements forced by one condition of a poset, but not the whole thing
In order to get the relative consistency of some statement, it suffices to find a notion of forcing, and a condition $p$ in that forcing, such that $p$ forces the desired statement. It seems to be ...
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centeredness in forcing iterations
Suppose $A$ is a complete subalgebra of a complete boolean algebra $B$, and $B$ is $\kappa$-centered. Let $G \subset A$ be a generic ultrafilter. Is $B/G$ $\kappa$-centered in $V[G]$?
Naively, we ...
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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 ...
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Paul Cohen on genesis of method of forcing and mathematical similarities
We have on record Paul Cohen's comments on being inspired by issues of formalizing algorithms in number theory (this needs to be verified as per comment) as well as related remarks on computability. ...
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Ultrafilters preserved by $\mathbb{P}$ but not by products?
Let $U\in V$ be an ultrafilter on $\omega$. We say $U$ is preserved under forcing with $\mathbb{P}$ if $\Vdash \forall x\subset \omega \ \exists Z\in U \ Z\subset x \vee Z\subset x^c$. In other words,...
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Consistency strength of being strong cardinal and indestructible under collapses
What is the consistency strength of the following statement:
$\kappa$ is a strong cardinals and it is indestructible under $Col(\kappa, <\theta),$ where $\theta> \kappa$ is some fixed ...
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388
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Largeness and arithmetic progression properties of generic reals
Consider the following properties for a subset $A$ of $\mathbb{N}$:
(1) $A$ is large: $\sum_{n \in A}$$ 1\over n$$=\infty,$
(2) $A^\infty=\limsup \frac{|A \cap \{ 1, \dots, n\}|}{n} >0$,
(3) $A_\...
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539
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On $V$-decisive and weakly homogeneous forcings
Suppose that $\Bbb P$ is a forcing in $V$, we say that $\Bbb P$ is $V$-decisive if whenever $\varphi(x_1,\ldots,x_n)$ is a statement in the language of forcing, and $u_1,\ldots,u_n\in V$ then $1_{\Bbb ...
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Inaccessible becomes successor of singular
Is it possible, starting from any large cardinal assumption, to find a countably closed forcing $\mathbb{P}$ such that for some inaccessible $\kappa$, $\Vdash_\mathbb{P} "\kappa = \lambda^+$ and $\...
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460
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Does forcing with recursively pointed perfect trees add a Turing degree that is minimal over $V$?
A tree $T$ on $\omega$ is recursively pointed if it is recursive in each of its branches. We can consider a variant of Sacks forcing where the conditions are recursively pointed perfect trees ordered ...
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Iterated forcing and CH
I need some help with this theorem: if $P_\beta=\langle P_\alpha,\dot{Q}_\alpha:\alpha\leq\beta\rangle$, $\beta<\omega_2$, is a CSI of proper forcings, $P_\alpha\Vdash \lvert \dot{Q}_\alpha\rvert\...
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756
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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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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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321
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ccc after strongly proper forcing
Let $P, Q \in V$ be such that $P$ is strongly proper and $Q$ is ccc. Does $Q$ continue to be ccc after forcing with $P$?
Since strongly proper forcings do not add new branches to $\omega_1$-trees, ...
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415
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Preservation of ultrafilters by Sacks forcing
It is well-known that, if $p$ is a Ramsey (selective) ultrafilter on $\omega$, then after adding a Sacks real $p$ remains an ultrafilter (well, it's really the upwards closure of $p$ the one that's an ...
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600
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Does "antichain" mean something different in set-forcing than in lattice theory?
On page 3 of Introduction to Lattices and Order, Davey and Priestley define an antichain in a poset $\langle P,\leq\rangle$ as a set of pairwise incomparable elements:
The ordered set P is an ...
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Precipitous ideal and inner model
Assume $\kappa$ is measurable, $U$ is its unique normal measure, $V=L[U]$. We levy collapse $\kappa$ to make it become $\omega_1$.
If we don't have the inner model condition, then we only know that $\...
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Example of a distributive forcing which is entirely improper
One of the standard examples of a forcing which is distributive but not proper (equiv. not closed) is a club shooting into a stationary co-stationary set.
But that forcing is $S$-proper for the ...
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Why relative consistency results by forcing arguments are provable in finitistic metatheory
It is claimed in many textbooks that relative consistency results, such as $\text{Con}(\text{ZFC})\rightarrow\text{Con}(\text{ZFC}+2^{\aleph_0}\geq\aleph_2)$, are provable in the finitistic metatheory....
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Different approaches to forcing
There are many different approaches to the forcing method, and I am looking for all known such approaches. So my question is:
Question 1. Which different approaches to set theoretic forcing are ...
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If there is a non-constructible real, is there an $L$-generic real?
If we assume that $\Bbb R^V\neq\Bbb R^L$, can we deduce that there is some $x\in\Bbb R^V$ which is $L$-generic?
Of course if $V$ is a generic extension of $L$ this is true, but if $V=L[0^\#]$ this is ...
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Is $\kappa$-distributive the same as $\kappa$-strategically closed?
References for the definitions are Jech's Set Theory Definition 15.5, and Cummings paper in the Handbook of set theory Definition 5.15.
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Forcing a unique $\Delta_3^1$ generic real
I know Jensen developed a forcing notions in $L$ that adds a unique, minimal and $\Delta_3^1$ $L$-generic real. In his paper Definable sets of minimal degree he says that Solovay had already shown the ...
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Quotable equivalents of Martin's axiom
I am seeking "quotable equivalents" for MA (Martin's axiom). For the continuum hypothesis, examples of such statements are as follows.
(a) (Sierpinski) The (xy) plane can be covered by countably many ...
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Forcing in Constructive Set Theories
I searched on the internet, but I could not find anything useful about applications of forcing in constructive set theories.
Are there any developments of forcing in CZF or IZF?
Thanks in advance.
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Consistency Strength of "HC is elementary in V[G]"
Let $P$ be the Levy-collapse of the ordinals, so $P$ is a class forcing notion that makes every ordinal countable.
Note that since $P$ is weakly homogeneous, for any formula $\phi(\overline{a})$ ...
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Introducing meets while preserving directed closure
A poset $\mathbb{P}$ is called well-met iff every pair of compatible conditions in $\mathbb{P}$ has a greatest lower bound.
Question: Suppose $\mathbb{P}$ is a separative partial order which is $\...
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Is there a version of Miller forcing "guided by" an ultrafilter?
It’s well known that Mathias forcing factors as a two-step iteration $P(\omega)/\mathrm{fin}\ast \mathbb{M}_{\dot U}$, where $\mathbb{M}_{\dot U}$ is Mathias forcing guided by the generic ultrafilter ...
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Making all cardinals countable and its HOD
Suppose that $G$ is $Col(\omega, <Ord)$-generic over $V$ and let $W=HOD^{V[G]}$. Is $CH$ true in $W$? In general, what can we say about the behaviour of the power function in $W$?
Update. Are the ...
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$\mathfrak{c}$-universal linear order
I've been told once or twice that the following holds:
There is a model of $ZFC+MA+\neg CH$ in which there is a $\mathfrak{c}$-universal linear order embedded in $(\omega^\omega, \le^\ast)$
...
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closure of separative quotients
Does there exist a partial order, nontrivial for forcing, that is countably closed, but whose separative quotient is not countably closed? Supposing the answer is yes, then is there a partial order, ...