All Questions
Tagged with forcing set-theory
825 questions
2
votes
0
answers
42
views
Diamonds on supercompact $\kappa$ after a $\kappa$-c.c. forcing
Let $\kappa$ be supercompact. Then the (supercompact) Laver diamond holds at $\kappa$: There is $f:\kappa\to V_\kappa$ such that for all $\lambda\geq \kappa$ and $x\in H(\lambda^+)$ there is $j:V\to M$...
- 51
2
votes
0
answers
62
views
Consistency of Sigma-V-2 uniformization with AD
Is ZF + AD consistent with: For every real $r$, every true $Σ^V_2(r)$ statement has a $Δ^V_2(r)$ example?
DC is provable in ZF + every true $Σ^V_2$ statement has a $Δ^V_2$ example (i.e. witness). ...
- 7,545
3
votes
1
answer
132
views
Forcing equivalence and equal generic extensions
Two forcing notions $\Bbb P$ and $\Bbb Q$ could be defined to be forcing equivalent if the associated complete Boolean algebras are isomorphic (so, the CBA's formed by considering the regular opens of ...
- 393
6
votes
1
answer
549
views
Destroying scales
Suppose $\lambda$ is a cardinal with uncountable cofinality and $C\subseteq\lambda$ is a club of ordertype cf$(\lambda)$. Let $f=\langle f_\xi\mid\xi<\lambda^+\rangle$ be a scale consisting of ...
- 533
9
votes
2
answers
383
views
Iteration of $\aleph_2$-properness
Let us say a forcing $P$ is proper for a class of models $\mathcal C$, if for large enough regular $\theta$ and $M \prec H_\theta$ in $\mathcal C$ with $P \in M$, every $p \in P \cap M$ can be ...
- 18.6k
4
votes
0
answers
107
views
Partial uniformization under AD
Under ZF + AD, and especially $\text{AD}^+$, even if uniformization fails for reals, in some ways it must almost hold.
For a notion of small, we say that uniformization holds on a co-small set of ...
- 7,545
2
votes
1
answer
161
views
Are Cohen Generics Minimal Covers?
Are Cohen generics (in $2^\omega$) minimal covers?
I'm ultimately interested in this question for some more effective notion of forcing but I realized I wasn't sure how to show this even assuming full ...
- 3,029
4
votes
1
answer
144
views
Stably embedded clone
Let $M$ be a first-order structure, considered as an element of the ambient set-theoretic universe $V$.
Clearly, for any $L_{\infty,\infty}$-formula $\varphi(\bar x)$ (with $\bar x$ finite, say) in ...
- 1,338
8
votes
0
answers
258
views
Forcing axiom for Mathias forcing
Given a poset $\mathbb{P}$, let $\mathsf{FA}(\kappa,\mathbb{P})$ denote the assertion that for every family of dense sets $\mathcal{D}$ with $|\mathcal{D}| = \kappa$, there is a filter $G \subseteq \...
- 1,372
7
votes
1
answer
308
views
Proper Forcing Axiom for $|\mathbb{P}| \leq \mathfrak{c}$
Let $\mathsf{PFA}(\mathfrak{c})$ denote the Proper Forcing Axiom (PFA) restricted to posets $|\mathbb{P}| \leq \mathfrak{c}$. I think $\mathsf{PFA}(\mathfrak{c}) \implies \mathfrak{c} = \aleph_2$, but ...
- 1,372
7
votes
0
answers
142
views
What is the forcing $\bf U$ from Bartoszyński-Judah?
In Set Theory - on the structure of the Real Line by Bartoszyński & Judah, a forcing notion $\bf U$ is mentioned on page 339, allegedly corresponding to $\rm{cof}(\cal N)$ as it has several ...
- 393
7
votes
0
answers
260
views
A version of determinacy for all sets
Under ZF + AD, some games are undetermined because of lack of choice. In fact, the axiom of choice is equivalent to determinacy of games of length 2. However, we can ask whether the lack of choice ...
- 7,545
2
votes
0
answers
232
views
Is the poset in the following construction stationary $\aleph_{\alpha + 2}$-linked?
Definition: A poset P is $\mathbf{stationary}$ $\kappa^{+}$-$\mathbf{linked}$ if for every sequence of conditions $(p_{\gamma} | \gamma < \kappa^{+})$, there is a regerssive function $f: \kappa^{+} ...
6
votes
0
answers
179
views
$Δ^1_3$ reals in transitive models
Every real number is $Δ^1_2$ in some $ω$-model of ZFC\P. I am looking for analogous statements for $Δ^1_3$ definability and transitive models, where the situation is more subtle.
What is the ...
- 7,545
18
votes
1
answer
554
views
When can we add choice to a model of ZF
For countable transitive models of ZF, is existence of a ZFC extension with the same height a first order property?
In other words, is there a statement $τ$ (in the language of set theory) such that ...
- 7,545
14
votes
1
answer
642
views
Example of a forcing notion with finite-predecessor condition that does not add reals
This question seems very basic but I cannot seem to find any literature on it.
Let $\mathbb{P}$ be a forcing notion. If $p$ is a condition of $\mathbb{P}$, define the predecessor set of $p$ to be $$\{...
- 1,328
3
votes
0
answers
211
views
Intuitionistic set-theoretic geology
Work in ZF, if there are proper class many supercompact cardinals, then all grounds are uniformly definable. Hence under reasonable assumption, we can have choiceless set-theoretic geology.
But can we ...
- 695
8
votes
1
answer
228
views
Does this mixed-support product have the countable approximation property?
Recall that a forcing order $\mathbb{P}$ has the countable approximation property if for any $\mathbb{P}$-generic filter $G$ and any $x\in V[G]$, if $x\cap y\in V$ for any countable $y\in V$, $x\in V$....
- 1,799
6
votes
1
answer
199
views
$L(\mathbb{R})$-absoluteness from a proper class of Woodins: source?
For a paper I'm writing I need to use (as a blackbox) the following theorem: if there is a proper class of Woodin cardinals and $G$ is set-generic, then $L(\mathbb{R})$ and $L(\mathbb{R})^{V[G]}$ are ...
- 21.2k
9
votes
0
answers
177
views
Inner model of "CH + large cardinals" that satisfies MM?
I've been told that if $V$ has CH and large cardinals, then there is an inner model in which MM holds. The sketch is as follows:
Any $\Sigma^2_1$ sentence that can be forced is already true in such $V$...
- 335
4
votes
1
answer
140
views
Coherent sequence of ultrafilters in iterated forcing extensions
Remember that if $\kappa$ is strongly compact, then any ${<}\kappa$-complete filter extends to a ${<}\kappa$-complete ultrafilter.
Let $\Bbb P_\delta=\langle\Bbb P_\alpha,\dot{\Bbb Q}_\alpha\mid ...
- 393
5
votes
0
answers
536
views
A set theoretic approach to the Riemann hypothesis
Let $X$ be an extremally disconnected
(i.e. such that the closure of open sets is open) compact Hausdorff space. Then
$*_1$ $C(X)$
is the space of continuous functions $f: X \to \mathbb{C}$,
$*_2$ $C^...
- 32.1k
9
votes
1
answer
252
views
Copy of $P(\omega)/\mathrm{fin}$ on $\omega_1$
$\newcommand{\fin}{\mathrm{fin}}$Under what hypotheses does there exist a uniform ideal $I$ on $\omega_1$ such that $P(\omega_1)/I \cong P(\omega)/\fin$? What is the consistency strength?
It follows ...
- 18.6k
5
votes
1
answer
201
views
When is $M[\mathscr{U}]\cap2^\omega=M\cap2^\omega$?
Suppose that $M$ is a countable, transitive model of $\mathsf{ZFC}$ and $\mathscr{U}\subseteq\mathscr{P}(\omega)^M$ is an $M$-generic $M$-ultrafilter (say, $\mathscr{U}\in M[G]$ some $(M,\mathbb{P})$-...
- 2,196
6
votes
0
answers
144
views
Alternative proofs of the countable chain condition in forcing
Advance warning: This question is more about history and pedagogy than "hard" mathematics.
I am studying Cohen forcing with the forcing poset $(\operatorname{Fin}(E,2),\supseteq,0)$, and I ...
3
votes
0
answers
149
views
Cohen reals at limit steps in a finite support iteration
Without success, I have been trying to find who was the first to prove the folklore result that any finite support iteration of non-trivial posets adds Cohen reals at limits steps. Does anybody know?
- 791
4
votes
1
answer
533
views
How to settle the Generalized Continuum Hypothesis when there are urelements?
Work in $\sf ZFCA$ and permutation models has preceded forcing by several decades. Was it used to settle the question of the Generalized Continuum Hypothesis $\sf GCH$ when urelements are admitted? I ...
- 11.3k
6
votes
1
answer
227
views
Forcing $\neg\square_{\omega_1}$ from a Mahlo cardinal, reference
In Jensen's The fine structure of the constructible hierarchy, it is stated that Solovay proved the consistency of $\neg\square_{\omega_1}$ by collapsing a Mahlo cardinal to $\omega_2$. I was ...
- 2,286
5
votes
0
answers
150
views
Consistency upper bounds for $\neg\square_{\aleph_\omega}$
In the introduction of Cummings and Friedman's $\square$ on the singular cardinals the following is written:
Failure of $\square_\lambda$ for $\lambda$ singular is stronger and rather more ...
- 2,286
4
votes
0
answers
139
views
Commutativity of a diagram between complete embeddings
Suppose $\mathbb{P}_0$, $\mathbb{P}_1$ and $\mathbb{P}_2$ are separative posets such that $\mathbb{P}_2$ projects into $\mathbb{P}_1$ and $\mathbb{P}_1$ projects into $\mathbb{P}_0$, i.e. there are ...
- 533
4
votes
0
answers
165
views
Looking for reference on Vopenka's theorem on generic extensions of HOD
Chapter 15 of the third edition of Jech's textbook on set theory gives Vopenka's theorem as saying that if $V=L[A]$ where $A$ is a set of ordinals then $V$ is a set generic extension of $HOD$, whereas ...
- 2,125
4
votes
1
answer
148
views
Reference request: $\kappa^+$-saturated ideal from iterated Cohen forcing on measurable $\kappa$
In Kunen [1] the author makes the following note: Let $\kappa$ be measurable with normal measure $\mathscr{U}$ in a model of $\mathsf{GCH}$. Let $\mathbb{P}$ be an iteration of $\operatorname{Add}(\...
- 2,196
6
votes
1
answer
113
views
Projections between complete boolean algebras
Let $P$ and $Q$ be complete boolean algebras. Suppose that $\dot H$ is a $P$-name such that $1_P\Vdash\dot H$ is $Q$-generic. For each $p\in P$, let $A_p$ be the set of $q\in Q$ such that $p\Vdash q\...
- 533
3
votes
0
answers
200
views
Weak extender models for supercompactness without choice
Assume ZFC and a supercompact cardinal $κ$. Is it consistent that there is a weak extender model $N⊨\text{ZF}$ for supercompactness of $κ$ such that the axiom of choice and well-ordering of $P_κ(λ)^N$...
- 7,545
4
votes
0
answers
166
views
Consistency of definability beyond P(Ord) in ZF
Is it consistent with ZF that the satisfaction relation of $L(P(Ord))$ is $Δ^V_2$ definable? More generally, is it consistent with ZF that there is a $Δ^V_2$ formula (taking $α$ as a parameter) that ...
- 7,545
10
votes
1
answer
532
views
Does proper forcing preserve properness under PFA?
I'm interested in forcing classes $\Gamma$ which preserve membership in themselves, i.e. for all posets $\mathbb{P}, \mathbb{Q}\in \Gamma$, we have $\Vdash_{\mathbb{P}}\check{\mathbb{Q}}\in\Gamma$. ...
- 435
6
votes
0
answers
125
views
From HODs to corresponding models of AD
If $M$ is HOD of a model $N$ of $\text{AD}^+ + V=L(P(ℝ))$, what kind of forcing construction in $M$ gives back such an $N$?
HODs for $\text{AD}^+ + V=L(P(ℝ))$ are conjectured (and under anti-large ...
- 7,545
6
votes
1
answer
179
views
An iteration of proper forcing without proper iterands
Famously, a countable support iteration with proper iterands is again proper. This is mostly stated as follows: Suppose $(\mathbb{P}_{\alpha},\dot{\mathbb{Q}}_{\alpha})_{\alpha<\lambda}$ is a ...
- 1,799
8
votes
1
answer
241
views
A reference for forcing projections
The idea of a projection $\pi\colon\mathbb{Q}\to\mathbb{P}$ of forcing notions is something like a combinatorial stand-in for the fact that forcing with $\mathbb{Q}$ produces a generic for $\mathbb{P}$...
- 2,389
11
votes
1
answer
247
views
Is ground model $\Sigma_n$ truth $\Sigma_n$-definable in every forcing extension?
It is well-known (and quite useful) that for any formula $\phi$, forcing poset $\mathbb{P}$, $p\in\mathbb{P}$, and $\mathbb{P}$-name $\dot{a}$, $p\Vdash_{\mathbb{P}}\phi(\dot{a})$ is definable as a ...
- 435
9
votes
0
answers
258
views
Naive way to violate $\mathsf{SCH}$ at $\aleph_\omega$
I asked this question on MSE and got a partial answer. Shamefully I still haven't figured out myself a full answer, so I would like to ask it here. The usual way to get the failure of $\mathsf{SCH}$ ...
- 877
2
votes
0
answers
142
views
Namba forcing, one Cardinal up
The classical Namba forcing collapses $\omega_2$ to have cofinality $\omega$ while preserving the cardinal $\omega_1$. Higher analogs were constructed to (under some additional hypotheses) give a ...
- 1,799
5
votes
1
answer
196
views
Can a generic ultrafilter over $\mathrm{NS}^+_{\omega_1}$ witness $\omega_1$ is Ramsey-like?
Suppose that $\kappa$ is an appropriate large cardinal (preferably a Woodin cardinal, but possibly something stronger) and let $G$ be a $\operatorname{Col}(\omega_1,<\kappa)$-generic filter over $V$...
- 3,042
15
votes
1
answer
615
views
Changing the cofinality of a regular cardinal without collapsing any cardinals?
I have a short but hopefully interesting question on cardinal arithmetic and collapsing cardinals:
Is it possible to change the cofinality of a regular cardinal without collapsing any cardinals?
Is ...
- 307
8
votes
1
answer
260
views
Can we force $\aleph_\omega^\omega<2^{\aleph_\omega}$?
Since $\operatorname{cf}(\aleph_\omega)=\omega$, $\aleph_\omega<\aleph_\omega^\omega$. However, can we force $\aleph_\omega^\omega<2^{\aleph_\omega}$? I am especially interested in models for ...
- 2,196
6
votes
1
answer
156
views
Preservation of cardinals implies preservation of cofinalities when $V=L$?
Kunen mentions the result stated in the title. It would be much appreciated if someone can give a reference for this. Thank you!
- 105
12
votes
0
answers
213
views
Is $\kappa \rightarrow [\kappa]^2_3$ the same as $\kappa \rightarrow [\kappa]^2_2$ for inaccessible $\kappa$
The principle $\kappa \rightarrow [\kappa]^2_\alpha$ states that whenever we have a coloring $c:[\kappa]^2\rightarrow \alpha$ there is $H \subset \kappa$ of size $\kappa$ s.t. $|c"[H]^2|<\alpha$.
...
- 490
6
votes
1
answer
301
views
A variation on pinned equivalence relations
Recall (see e.g. Zapletal, Pinned equivalence relations) that a Borel equivalence relation $E$ on $\omega^\omega$ is pinned iff for every forcing $\mathbb{P}$ and every $\mathbb{P}$-name $\nu$ we have ...
- 21.2k
5
votes
0
answers
212
views
Questions about very fat sets
If $\kappa$ is a regular uncountable cardinal, we call a set $S\subseteq\kappa$ fat if for every $\alpha<\kappa$ and every club $C\subseteq\kappa$, there is a closed subset of $S\cap C$ of ...
- 1,799
3
votes
1
answer
243
views
Can a non-free Whitehead group embed as a discrete subgroup of a normed space?
Every countable discrete subgroup of a normed space is isomorphic to the direct sum of the group of integers. I wonder whether it is possible to push this beyond such direct-sum (free abelian) groups ...
- 11.3k