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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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 ...
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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&...
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$2^{|V|}$ 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 ...
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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 ...
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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 ...
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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 $G$ ...
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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?
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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_{\...
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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 $\...
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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 $...
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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 ...
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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 ...
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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<\...
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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 ...
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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 ...
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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^\...
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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 $...
7
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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$...
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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 ...
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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 ...
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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{...
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$\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 ...
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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$".
...
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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 ...
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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}...
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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 ...
7
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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 ...
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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 ...
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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 ...
11
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1
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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&...
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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$ ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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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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$,...
3
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1
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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 ...
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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 ...
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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 $Q$ ...
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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 ...
4
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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]...
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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 $...
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2
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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 $T$ ...
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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 ...
7
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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 ...
5
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1
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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 ...
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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 ...
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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{...