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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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 ...
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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 ...
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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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Silver-like forcing preserves p-points (reference request)
A Silver forcing "below $2^n$" is defned 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 $P$,...
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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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Examples of ccc forcing similar to Cohen forcing
I want examples of ccc forcing notions that preserve the uniformity of meager sets and that make the covering number of meager sets equal to the continuum.
I know that the forcing notion that ...
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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 ...
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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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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 ...
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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 ...
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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{...
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"Relative plausibility" of some infinitary theories
We work in $\mathsf{ZFC+V=L}$.
Define a plausible theory to be a theory $T\subseteq\mathcal{L}_{\omega_1,\omega}$ in an $\omega_1$-finite language which is $\omega_1$-c.e. and $\omega_1$-finitely ...
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Construction of a model of $ZFC+\neg Con(ZFC)$
By Gödel's second incompleteness theorem, the following assertion is true in ZFC:
$$
Con(ZFC)\rightarrow Con(ZFC+\neg Con(ZFC))
$$
Considering the completeness theorem, this assertion is equivalent to ...
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Grigorieff forcing and destruction of ultrafilters
I was interested in the Grigorieff forcing (you can read the definition here: How "much" does (Grigorieff) forcing destroy an ultrafilter?)
I couldn't prove that it destroys ultrafilters, ...
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Rigid boolean inclusions?
A boolean algebra $B$ is rigid if it has no nontrivial automorphisms and atomless if it has no minimal nonzero elements. $A \subseteq B$ is a complete boolean inclusion if $B$ is complete and $A$ is a ...
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Sheaf-theoretic approach to forcing
Inspired by the question here, I have been trying to understand the sheaf-theoretic approach to forcing, as in MacLane–Moerdijk's book "Sheaves in geometry and logic", Chapter VI.
A general ...
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Martin's Maximum implies stationary/club Chang's conjecture?
Chang's Conjecture (CC) states: for any $f: [\omega_2]^{<\omega} \to \omega_1$, there exists a set $X\subset \omega_2$ of order type $\omega_1$ such that $|f''[X]^{<\omega}|\leq \aleph_0$.
...
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Definability of the ground model in its class-forcing extension
It is known that Laver's ground model definability theorem doesn’t hold for all class forcing notions. That is, if $M$ satisfies ZFC then $M$ is not necessarily definable in $M[G]$, a class forcing ...
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Condensed / pyknotic sets in terms of forcing over Boolean-valued models of set theory / multiverse concepts?
Here is one way of saying what a pyknotic set is. Fix an inaccessible cardinal $\kappa$, and let $Proj_\kappa$ be the category of $\kappa$-small, extremally disconnected compact Hausdorff spaces. ...
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Second order theories for class forcing
We know that if $M$ is a model of ZFC, then taking $\mathcal{C}$ to be the collection of all classes definable in $M$ with set parameters, and taking $\in$ to be the obvious extension of $M$'s epsilon ...
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Can all uncountable (but small) families of sets with positive measure have an uncountable subfamily with an intersection of positive measure?
My general question was is it consistent that any uncountable family of less than $\mathrm{non}(\mathcal{N})$ sets, each with positive measure, has an uncountable subfamily $\mathcal{F}$ s.t. $\bigcap ...
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Complexity of $Σ_n$ theory of $H(ω_2)$ under MM$^{++}$
Under Woodin's $\mathbb{P}_\text{max}$ axiom (which is implied by MM$^{++}$), what is the complexity of the $Σ_n$ theory of $H(ω_2)$? Same question for $Σ_n(I_\text{NS})$ (i.e. using the ...
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Reference for "$\mathrm{PFA}$ implies $L(\mathbb{R}) \cap \bigcup_{1 \leq k < \omega} \mathcal{P}(\mathbb{R}^k)$ is productive"
The preprint of the recent result of Aspero and Schindler, "Martin's Maximum$^{++}$ implies Woodin's Axiom $(*)$", mentions productive pointclasses, and states that "$\mathrm{PFA}$ ...
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Independence through forcing vs generic collapses
Are there known statements in $V_{ω+ω}$ independent through forcing after $\mathrm{Col}(ω,<κ_1)*\mathrm{Col}(κ_1,<κ_2)*\mathrm{Col}(κ_2,<κ_3)*...$ where $κ_1<κ_2<κ_3<...$ are ...
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Are generic filters that produce the same forcing extension related by a ground-model automorphism?
Suppose $M$ is a countable transitive model of some fragment of $\mathbf{ZFC}$, $\mathbb{P}\in M$ is a forcing notion and $G, H$ are $\mathbb{P}$-generic such that $M[G]=M[H]$. Does it then follow ...
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Can we bound $2^{\aleph_\omega}$ without pcf theory?
One of the famous applications of pcf theory is that if $\aleph_\omega$ is a strong limit cardinal then $2^{\aleph_\omega}<\aleph_{\omega_4}$. I'm curious whether any weaker result with the same ...
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Obtaining elements of a generic extension from a Boolean-valued model of ZFC
Let $\mathcal{M}$ be a countable transitive standard-model of ZFC.
Let $B \in \mathcal{M}$ be a boolean algebra that is complete in $\mathcal{M}$.
Further, let $\mathcal{M}^{(B)}$ be the corresponding ...
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Independence result where probabilistic intuition predicts the wrong answer?
In some of his writings, Paul Cohen gave an informal, motivational discussion about the word generic (as it is used in forcing). While very suggestive, the discussion leaves the meaning of the word ...
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What did Paul Cohen mean by saying that generic sets of natural numbers have "no asymptotic density?"
In Paul Cohen's original 1963 paper on forcing, The independence of the Continuum Hypothesis, published in PNAS, he gives his general proof sketch of how he intends to create a model of ZFC that doesn'...
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