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
76 votes
3 answers
18k views

Czelakowski's claimed proof of the Twin Prime Conjecture

It seems like the article "The Twin Primes Conjecture is True in the Standard Model of Peano Arithmetic: Applications of Rasiowa–Sikorski Lemma in Arithmetic (I)" by Janusz Czelakowski ...
Glycerius's user avatar
  • 1,023
7 votes
1 answer
336 views

How hard is it to get "absolutely" no amorphous sets?

A beautiful and surprising (to me at least) result around the axiom of choice is that, while full $\mathsf{AC}$ is preserved by forcing, a model of $\mathsf{ZF}$ + "There are no amorphous sets&...
Noah Schweber's user avatar
5 votes
0 answers
129 views

$2^{|V|}$ class cardinalities without global choice

Is it consistent with Morse-Kelley set theory without global choice (but with choice for sets) that there are $2^{|V|}$ proper classes of different cardinalities? Alternative question: Is it ...
Dmytro Taranovsky's user avatar
3 votes
0 answers
101 views

Examples of the use of forcing to build up models of stronger theories?

I'm very new to the subject of forcing. I always got the impression that with forcing we begin with say a model $M$ of a theory $\sf T+I$ and produce another model $M[G]$ that is also a model of $\sf ...
Zuhair Al-Johar's user avatar
6 votes
1 answer
335 views

Is there a proof of independence of AC from Z that is done in Z?

The proof that $AC$ is independent of $\sf ZF$ axioms is done by forcing and constructibility, and these don't beg any consistency strength more than that of $\sf ZF$. Is there a known similar proof ...
Zuhair Al-Johar's user avatar
3 votes
1 answer
107 views

Unbounded subset of $\omega$ in $V[G]$ has an unbounded subset in $V$?

This question is similar to a question I asked last year, but I'm not asking for the same thing. Let $S$ be a set of ordinals, and consider the Levy collapse $\operatorname{Col}(\omega,S)$. Let $G$ ...
Clement Yung's user avatar
2 votes
0 answers
103 views

Any connection between extension of algebraic structure and forcing of set theory?

Any connection between extension of algebraic structure and forcing of set theory? And more, are there any approach from one of the two to other field to solve problem?
XL _At_Here_There's user avatar
2 votes
0 answers
60 views

Can we use forcing to adjoin this set to a model of ZF+j+$\alpha$?

Let $M$ be a countable transitive model of $\sf ZF + j +\alpha$, where $j:V_{\alpha+1} \to V_\alpha$ is an external [not used in separation and replacement] bijection such that for any $S \in V_{\...
Zuhair Al-Johar's user avatar
5 votes
1 answer
149 views

Preservation of stationary sets by Mitchell forcing quotients

It is well-known that Mitchell's forcing $\mathbb M$ for the tree property at $\omega_2$, which turns a weakly compact $\kappa$ into $\omega_2$ while adding many reals, is a projection of adding $\...
Monroe Eskew's user avatar
  • 18.1k
4 votes
0 answers
137 views

Consistency of a strange (choice-wise) set of reals, pt. 2

This is a follow-up on this question. Consider a set $X\subseteq \mathbb{R}$ such that $X$ is not separable wrt its subspace topology Every countable family of non-empty pairwise disjoint subsets of $...
Lorenzo's user avatar
  • 2,134
7 votes
2 answers
704 views

Consistency of a strange (choice-wise) set of reals

Consider a set $X\subseteq \mathbb{R}$ such that $X$ is not separable wrt its subspace topology For all $r\in\mathbb{R}$ there exists a sequence $(x_n)_{n\in\omega} \subset X$ converging to $r$ In a ...
Lorenzo's user avatar
  • 2,134
7 votes
1 answer
207 views

For which class of forcings does the "name dichotomy" hold?

Let $\mathbb P$ be a forcing that does not collapse $\omega_1$, $\theta$ sufficiently large and regular and $X\prec H_\theta$ a countable elementary substructure with $\mathbb P\in X$ as well as $p\in ...
Andreas Lietz's user avatar
7 votes
1 answer
198 views

Understanding descending intersections of generic extensions

Let $B_{0}\supseteq B_{1}\supseteq\dots\supseteq B_{\alpha}\supseteq\dots\,\,\left(\alpha<\kappa\right)$ be a descending sequence of complete Boolean algebras, $B_{\kappa}:=\bigcap_{\alpha<\...
Ur Ya'ar's user avatar
  • 329
10 votes
3 answers
893 views

Why can we assume a ctm of ZFC exists in forcing

Following Kunen's book, it makes clear that countable transitive models (ctm) exist only for a finite list of axioms of ZFC. So, why can we assume a ctm of the whole ZFC axioms exists and use it as ...
Guest's user avatar
  • 101
4 votes
1 answer
181 views

Is every first-countable Lindelof space of cardinality $<\mathfrak c$ a $Q$-space under MA?

Definition. A topological space $X$ is a $Q$-space if every subset of $X$ is of type $G_\delta$. It is clear that every $Q$-space has countable pseudocharacter (= all singletons are $G_\delta$) and is ...
Taras Banakh's user avatar
  • 40.8k
3 votes
0 answers
89 views

The set of ground model reals arbitrarily close to a new real in the forcing extension

Consider a forcing notion $\mathbb{P}$, a condition $p\in\mathbb{P}$, a $\mathbb{P}$-name $\dot{r}$ and a formula (with ground model parameters) $\varphi(x)$ such that $$p \Vdash \dot{r} \in \omega^\...
Lorenzo's user avatar
  • 2,134
2 votes
0 answers
146 views

A property of Levy collapse forcing

Consider the following nice property for a forcing notion $\mathbb{P}$ (in a transitive model $M$ of $\mathtt{ZFC}$): Let $G_1,G_2$ $\mathbb{P}$-generic over $M$ and $M[G_1]$ respectively. Then, if $...
Lorenzo's user avatar
  • 2,134
7 votes
1 answer
213 views

Preservation of projective stationarity

A set $S \subseteq [\kappa]^\omega$ is called projective stationary if for every stationary $A \subseteq \omega_1$, and every algebra $F : \kappa^{<\omega}\to\kappa$, there is $z\in S$ such that $z$...
Monroe Eskew's user avatar
  • 18.1k
5 votes
0 answers
196 views

Does this proof by Shelah use any "hidden assumptions"?

Recall that the approachability ideal for $\kappa$, denoted $I[\kappa]$, consists of all sets $A\subseteq\kappa$ such that there is a sequence $\overline{a}=(a_{\alpha})_{\alpha\in\kappa}$ of bounded ...
Hannes Jakob's user avatar
  • 1,612
4 votes
0 answers
131 views

Necessary and sufficient conditions for a forcing to add a Cohen real

Are there some necessary and sufficient conditions for a forcing notion to add a Cohen real in the generic extension? In other words, given a non-atomic forcing p.o. $\mathbb{P}$, what are the ...
Lorenzo's user avatar
  • 2,134
6 votes
1 answer
197 views

How often are forcing extensions of countable computably saturated models of $\mathsf{ZFC}$ computably saturated?

Recall that given a finite language $\mathcal{L}$, we say that an $\mathcal{L}$-structure is computably saturated (or recursively saturated) if for any computable set $\Sigma(\bar{x},y)$ of $\mathcal{...
James Hanson's user avatar
  • 10.3k
3 votes
1 answer
113 views

$\mathtt{PSP}$ holding only for sets of cardinality $\mathfrak{c}$

Consider the sentence $\mathtt{PSP}_\mathfrak{c}$: "Every subset of $\mathbb{R}$ having the cardinality of the continuum contains a Cantor set". A priori this sentence is weaker than the ...
Lorenzo's user avatar
  • 2,134
3 votes
0 answers
76 views

Forcings that preserve $\mathtt{PSP}$

By $\mathtt{PSP}$ I mean the statement "every subset of the reals has the perfect set property, i.e. either is countable or it contains an homeomorphic copy of the Cantor space $2^\omega$". ...
Lorenzo's user avatar
  • 2,134
2 votes
0 answers
178 views

Generalized models of set theory

The forcing method can be viewed as building a Boolean-valued model of set theory. Some generalizations include Heyting algebra/sheaf/lattice-valued model. However, it seems these generalizations are ...
Kushi's user avatar
  • 227
14 votes
2 answers
644 views

Are there interesting examples of theorems proved using ‘height’ extensions?

It's well known that forcing is more than a tool for proving independence: We can prove theorems and formulate axioms in theories like $\mathsf{ZFC}$ by moving to forcing extensions (e.g. $\mathfrak{p}...
Neil Barton's user avatar
7 votes
0 answers
311 views

An uncountable structure with unusual "relatively-computable shadow"

Below, all structures are infinite and in a finite language. Given a structure $\mathcal{A}$ with domain $\omega$, we conflate $\mathcal{A}$ with some reasonable encoding of its atomic diagram for ...
Noah Schweber's user avatar
7 votes
1 answer
231 views

Forcing out of L[U] when we have a precipitous ideal in V

The following theorem of Jech, Magidor, Mitchell and Prikry is well-known. Theorem. (1) If $\kappa$ is a regular cardinal that carries a precipitous ideal, then $\kappa$ is a measurable cardinal in an ...
Toby Meadows's user avatar
  • 1,111
9 votes
3 answers
924 views

Philosophy of forcing and ctm

I asked a similar question on SE before and received an answer. Not completely convinced, I decided to ask it here with some modifications. Note: I understand how forcing works and how it proves ...
Kushi's user avatar
  • 227
5 votes
1 answer
301 views

Locating generic filters in the Lévy collapse

Let $\operatorname{Col}(\omega,<\kappa)$ denote the Lévy collapse of an inaccessible cardinal $\kappa$. A variant of the Factor Lemma is as follows: Lemma. Suppose that $\kappa$ is an inaccessible ...
Clement Yung's user avatar
11 votes
1 answer
335 views

Is every topos a sheaf topos with values in a well-pointed one?

Here's a mix of heuristic and precise questions as I try to grapple with topos theory. I try to think of topoi as two notions of "$1$" being glued at the hip. One is the "building block&...
user475672's user avatar
12 votes
1 answer
580 views

Can $\mathcal{L}_{\omega_1,\omega}$ detect $\mathcal{L}_{\omega_1,\omega}$-equivalence?

Roughly speaking, say that a logic $\mathcal{L}$ is self-equivalence-defining (SED) iff for each finite signature $\Sigma$ there is a larger signature $\Sigma'\supseteq\Sigma\sqcup\{A,B\}$ with $A,B$ ...
Noah Schweber's user avatar
5 votes
1 answer
254 views

Silver-like forcing preserves p-points (reference request)

A Silver forcing "below $2^n$" is defined e.g. in Definition 7.4.11 of [Tomek Bartoszyński and Haim Judah, Set Theory: on the structure of the real line, A. K. Peters, Wellesley, 1995.]. It ...
Andrzej's user avatar
  • 233
2 votes
1 answer
209 views

Continuum function maximum

Easton's theorem can give a very weak nontrivial constraint on continuum function, but it does not hold for singular cardinals. So: What are the non-trivial constraints on continuum function in ...
Ember Edison's user avatar
5 votes
1 answer
247 views

Can the forcing-absolute fragment of SOL have a strong Lowenheim-Skolem property?

Previously asked and bountied at MSE: Let $\mathsf{SOL_{abs}}$ be the "forcing-absolute" fragment of second-order logic - that is, the set of second-order formulas $\varphi$ such that for ...
Noah Schweber's user avatar
9 votes
0 answers
310 views

Non-closed Neeman forcing

This question is something of a follow-up to this one: Iterating Neeman's forcing It regards the work of Itay Neeman, MR3201836. Neeman formulates his two-type models forcing seemingly in greater ...
Monroe Eskew's user avatar
  • 18.1k
6 votes
1 answer
204 views

Properness for uncountable models

There are certain generalizations of the notion of a proper forcing to uncountable cardinals in the context of forcing iterations. For example, the ones introduced by Eisworth, and by Roslanowski and ...
Rahman. M's user avatar
  • 2,341
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
2 votes
0 answers
80 views

proper : (proper + $\omega^\omega$-bounding) = generic : x

If $P$ is a forcing notion, $A \subseteq P$ (usually an antichain), $q\in P$, then I write $A\cap q$ for the set of all conditions in $A$ which are compatible with $q$. For a proper forcing notion $P$,...
Goldstern's user avatar
  • 13.9k
3 votes
1 answer
273 views

Adding a closed unbounded set containing of only limit ordinals with a special property

The following theorem and proof are in Applications of the proper forcing axiom, the Baumgartner's paper in the book Handbook of Set-theoretic topology. $3.6$ THEOREM. Assume PFA. Suppose that for ...
Rouholah Hoseini Nave's user avatar
7 votes
0 answers
279 views

Generic behavior of "polynomialish" models of $\mathsf{Q}$

(This question was originally asked and bountied at MSE - with different notation, some more explicit arguments, and topology in place of forcing.) Suppose $\mathcal{R}=(R_i)_{i\in\mathbb{N}}$ is a ...
Noah Schweber's user avatar
6 votes
1 answer
226 views

Extension of a sequence of complete embeddings

Suppose for $\langle P_i : i < \omega \rangle$ is a sequence of countably closed partial orders. Suppose for each $n$, there is a complete embedding $$e_n : \prod_{i<n} P_i \to B(Q),$$ where $Q$ ...
Monroe Eskew's user avatar
  • 18.1k
4 votes
1 answer
321 views

Which step is wrong in the following simplification of Silver's forcing?

Theorem: If M is a countable transitive model of ZFC, and $\kappa$ is a supercompact cardinal in M, and $2^\kappa=\kappa^+$. Then there exists a forcing extension M[G] such that $\kappa$ becomes a ...
Reflecting_Ordinal's user avatar
4 votes
1 answer
289 views

Unbounded set in $V[G]$ has an unbounded subset in $V$?

This is a repost of the same question on math.SE, which received several comments but no answers/comments on the first question. Suppose $\kappa$ is a cardinal preserved in the generic extension $V[G]...
Clement Yung's user avatar
3 votes
0 answers
236 views

Borel equivalence relations on Ellentuck cubes

Is there a Borel equivalence relation $E$ on $[\omega]^\omega$ such that $E \not \leq_B E_0$ and for any $a \in [\omega]^\omega$ we have that $E \upharpoonright [a]^\omega$ is Borel bireducible with $...
daljnovod's user avatar
  • 101
12 votes
2 answers
900 views

Is this definability principle consistent?

(Below I'm thinking only about computably axiomatizable set theories extending $\mathsf{ZFC}$ which are arithmetically, or at least $\Sigma^0_1$-, sound.) Say that a theory $T$ is omniscient iff $T$ ...
Noah Schweber's user avatar
6 votes
2 answers
239 views

Relation between Laver generic reals

Suppose we have a ctm $M$ and $x, y$ Laver generic reals over $M$ so that $M[x] = M[y]$ (recall that Laver forcing is minimal, so that if $x \in M[y]$ then we already have $M[x] = M[y]$). Is there any ...
daljnovod's user avatar
  • 101
7 votes
2 answers
662 views

Can second-order logic identify "amorphous satisfiability"?

Recall that a set is amorphous iff it is infinite but has no partition into two infinite subsets. I'm interested in the possible structure (in the sense of model theory) which an amorphous set can ...
Noah Schweber's user avatar
5 votes
1 answer
292 views

Continuity of real functions

The following question concerns that without $ZF+DC$, can every function be "a little bit" continuous? Question Is it consistent with $ZF+DC$ that for any function $f:[0,1]\to [0,1]$ and ...
喻 良's user avatar
  • 4,191
14 votes
1 answer
784 views

What is the "Prikry–Silver collapse" when CH fails?

We all know and love Cohen reals, and we can (and often do) define the Cohen forcing as partial functions $p\colon\omega\to 2$ with finite domain. The Prikry–Silver forcing is defined as partial ...
Asaf Karagila's user avatar
  • 38.1k
9 votes
1 answer
278 views

Does ZF + BPI alone prove the equivalence between "Baire theorem for compact Hausdorff spaces" and "Rasiowa-Sikorski Lemma for Forcing Posets"?

Rasiowa-Sikorski Lemma (for forcing posets)is the statement: For any p.o. $\mathbb{P}$ (i.e. $\mathbb{P}$ is a reflexive transitive relation) and for any countable family of dense subsets of $\mathbb{...
Samuel G. Silva's user avatar

1
2
3 4 5
17