Questions tagged [forcing]

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

Filter by
Sorted by
Tagged with
9 votes
0 answers
242 views

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 ...
Monroe Eskew's user avatar
  • 18.1k
9 votes
0 answers
160 views

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 ...
Todd Eisworth's user avatar
9 votes
0 answers
371 views

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\...
Todd Eisworth's user avatar
9 votes
0 answers
225 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, ...
Ziemowit Kostana's user avatar
9 votes
0 answers
189 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 ...
Monroe Eskew's user avatar
  • 18.1k
9 votes
0 answers
254 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 $n$-...
Elliot Glazer's user avatar
9 votes
0 answers
347 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: ...
dragoon's user avatar
  • 763
9 votes
0 answers
293 views

On an unpublished result of Magidor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_{\...
Mohammad Golshani's user avatar
9 votes
0 answers
321 views

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 ...
Monroe Eskew's user avatar
  • 18.1k
9 votes
0 answers
273 views

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 ...
Haim's user avatar
  • 421
9 votes
0 answers
265 views

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 ...
Asaf Karagila's user avatar
  • 38.1k
9 votes
0 answers
247 views

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 ...
Monroe Eskew's user avatar
  • 18.1k
8 votes
4 answers
663 views

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 ...
David Feldman's user avatar
8 votes
3 answers
1k views

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 \...
Marc Alcobé García's user avatar
8 votes
1 answer
956 views

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? ...
sebastian's user avatar
  • 165
8 votes
2 answers
863 views

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 ...
Jordan Mitchell Barrett's user avatar
8 votes
2 answers
491 views

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 ...
Noah Schweber's user avatar
8 votes
3 answers
488 views

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 ...
David Roberts's user avatar
  • 33.8k
8 votes
2 answers
919 views

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 ...
Mohammad Golshani's user avatar
8 votes
2 answers
822 views

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 ...
Amit Kumar Gupta's user avatar
8 votes
2 answers
461 views

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 ...
Monroe Eskew's user avatar
  • 18.1k
8 votes
1 answer
505 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 ...
Jonathan Schilhan's user avatar
8 votes
2 answers
731 views

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. ...
Mikhail Katz's user avatar
  • 15.1k
8 votes
2 answers
448 views

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,...
Jing Zhang's user avatar
  • 3,138
8 votes
2 answers
452 views

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 ...
Mohammad Golshani's user avatar
8 votes
1 answer
388 views

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_\...
Mohammad Golshani's user avatar
8 votes
1 answer
539 views

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 ...
Asaf Karagila's user avatar
  • 38.1k
8 votes
1 answer
322 views

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 $\...
Monroe Eskew's user avatar
  • 18.1k
8 votes
2 answers
460 views

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 ...
Trevor Wilson's user avatar
8 votes
2 answers
867 views

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\...
Ruetta's user avatar
  • 81
8 votes
1 answer
756 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 ...
Asaf Karagila's user avatar
  • 38.1k
8 votes
1 answer
334 views

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 ...
Corey Bacal Switzer's user avatar
8 votes
1 answer
321 views

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, ...
tci's user avatar
  • 662
8 votes
1 answer
415 views

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 ...
David Fernandez-Breton's user avatar
8 votes
1 answer
600 views

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 ...
Adam's user avatar
  • 3,247
8 votes
1 answer
386 views

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 $\...
Reflecting_Ordinal's user avatar
8 votes
1 answer
243 views

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 ...
Asaf Karagila's user avatar
  • 38.1k
8 votes
1 answer
719 views

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....
Ruizhi Yang's user avatar
8 votes
1 answer
526 views

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 ...
8 votes
1 answer
396 views

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 ...
Asaf Karagila's user avatar
  • 38.1k
8 votes
1 answer
664 views

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.
Eran's user avatar
  • 649
8 votes
1 answer
313 views

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 ...
Lorenzo's user avatar
  • 2,134
8 votes
1 answer
346 views

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 ...
8 votes
1 answer
344 views

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.
Erfan Khaniki's user avatar
8 votes
1 answer
330 views

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 $\phi(\overline{a})$ ...
Danielle Ulrich's user avatar
8 votes
1 answer
228 views

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 $\...
Sean Cox's user avatar
  • 2,281
8 votes
2 answers
447 views

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 ...
Zach N's user avatar
  • 408
8 votes
1 answer
502 views

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 ...
Mohammad Golshani's user avatar
8 votes
2 answers
486 views

$\mathfrak{c}$-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 $(\omega^\omega, \le^\ast)$ ...
Not Mike's user avatar
  • 1,655
8 votes
3 answers
544 views

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, ...
Norman Lewis Perlmutter's user avatar

1
4 5
6
7 8
17