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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). ...
Dmytro Taranovsky's user avatar
9 votes
2 answers
384 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 ...
Monroe Eskew's user avatar
  • 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 ...
Dmytro Taranovsky's user avatar
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 ...
Peter Gerdes's user avatar
  • 3,029
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 ...
Dmytro Taranovsky's user avatar
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^{+} ...
George Marangelis's user avatar
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 ...
Dmytro Taranovsky's user avatar
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 ...
Dmytro Taranovsky's user avatar
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 $$\{...
Zoorado's user avatar
  • 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 ...
Ember Edison's user avatar
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 ...
Noah Schweber's user avatar
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 ...
Imperishable Night's user avatar
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 ...
Zuhair Al-Johar's user avatar
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 $P_κ(λ)^N$...
Dmytro Taranovsky's user avatar
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 ...
Dmytro Taranovsky's user avatar
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 ...
Dmytro Taranovsky's user avatar
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 $\mathbb{P}$...
Miha Habič's user avatar
  • 2,389
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 ...
user2925716's user avatar
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 ...
Noah Schweber's user avatar
7 votes
2 answers
440 views

On the existence of a real which is not set-generic over $L$

Recall that a real $r$ is set-generic over $L$ if there is a constructible forcing notion $\mathbb{P}$ and some $L$-generic filter $G\subset\mathbb{P}$ such that $r \in L[G]$. I know that Jensen's ...
Lorenzo's user avatar
  • 2,286
8 votes
1 answer
413 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
5 votes
0 answers
136 views

When does an iteration not add functions $\eta\to V$ at the final stage?

I am interested in better understanding the following property: Let us say that an iteration of forcings $\langle\mathbb{P}_\alpha,\dot{\mathbb{Q}}_\beta\mid\alpha\leq\gamma,\beta<\gamma\rangle$ is ...
Calliope Ryan-Smith's user avatar
6 votes
1 answer
431 views

A strange product forcing

Given two forcing notions $\mathbb{P}_0,\mathbb{P}_1$, we know that the product $\mathbb{P_0}\times\mathbb{P_1}$ will produce the following diagram of models ordered by inclusion: where $M$ is the ...
Lorenzo's user avatar
  • 2,286
9 votes
1 answer
434 views

Natural set-theoretic principles implying the Ground Axiom

The Ground Axiom states that the set-theoretic universe is not a set-forcing extension of an inner model. By Reitz, it is first-order expressible and easy to force over any given ZFC model with class-...
Monroe Eskew's user avatar
  • 18.6k
5 votes
1 answer
156 views

Countable closure of quotient forcing

Let us say that a partial order is "countably closed with infima" if every descending $\omega$-sequence has an infimum. Suppose $P$ and $Q$ are posets that are countably closed with infima, ...
Monroe Eskew's user avatar
  • 18.6k
11 votes
3 answers
792 views

When are two forcing posets "the same"?

Let $B$ and $C$ be complete Boolean algebras. To avoid triviality I may also want them to be atomless. For $b\in B$ nonzero, denote $B\upharpoonright b=\{p\in B:p\leq b\}$, which can be viewed as a ...
new account's user avatar
9 votes
1 answer
383 views

Extending Namba forcing to arbitrary lengths

Namba forcing is stationary-preserving and forces $cf(\omega_2^{\mathbf{V}}) = \omega$. Ronald Jensen used $\mathcal{L}$-forcing to iterate Namba posets in order to solve the extended Namba problem: ...
Zoorado's user avatar
  • 1,328
2 votes
1 answer
114 views

Closure properties of elementary embeddings resulting from generic iterations

In the context of generic embeddings, we fix a set $Z$, an ideal $I$ on $Z$, and $G$ a generic ultrafilter on $\mathcal{P}(Z) / I$. In $V[G]$, we can define the generic ultrapower $Ult(V, G)$ and an ...
Zoorado's user avatar
  • 1,328
13 votes
2 answers
598 views

Amoeba collapse

Here is a naive idea for a forcing $\mathbb A(\kappa)$, for an inaccessible cardinal $\kappa$. Conditions are pairs $(P,p)$, where $P \in V_\kappa$ is a partial order and $p \in P$. We define the ...
Monroe Eskew's user avatar
  • 18.6k
17 votes
3 answers
1k views

Minimum transitive models and V=L

Is there a c.e. theory $T⊢\text{ZFC}$ in the language of set theory such that the minimum transitive model of $T$ exists but does not satisfy $V=L$? You may assume that ZFC has transitive models. ...
Dmytro Taranovsky's user avatar
17 votes
6 answers
1k views

Strategic vs. tactical closure

The Banach-Mazur game on a poset $\mathbb P$ is the $\omega$-length game where the players alternate choosing a descending sequence $a_0 \geq b_0 \geq a_1 \geq b_1 \geq \dots$. Player II wins when ...
Monroe Eskew's user avatar
  • 18.6k
6 votes
0 answers
273 views

Forcing Martin's Axiom without cardinal arithmetic

We know that if $\kappa>2^{\aleph_0}$ and $\kappa^{<\kappa}=\kappa$, then there is a c.c.c. forcing which forces $\sf MA+2^{\aleph_0}=\kappa$. Traditionally, we even start with $\sf GCH$, which ...
Asaf Karagila's user avatar
  • 39.7k
5 votes
1 answer
248 views

Highly improper forcings

The following question comes from a typo in an old notebook of mine (I changed what I was calling my forcing notion partway through writing the definition of properness): Say that a forcing $\mathbb{P}...
Noah Schweber's user avatar
7 votes
0 answers
234 views

Is this equivalent to (some version of) Hechler forcing?

Let $\omega^{<\omega}$ be the set of finite strings of naturals, and let $\omega^{<\omega}_{\not=\emptyset}$ be the set of nonempty finite strings of naturals. Consider the following forcing ...
Noah Schweber's user avatar
7 votes
1 answer
400 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
146 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
104 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
345 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
2 votes
0 answers
61 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
161 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
  • 18.6k
7 votes
1 answer
219 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
  • 18.6k
6 votes
1 answer
227 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 E Hanson's user avatar
7 votes
0 answers
313 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
10 votes
3 answers
1k 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
  • 237
5 votes
1 answer
311 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
  • 1,372
12 votes
1 answer
648 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
273 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
317 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
  • 18.6k
6 votes
1 answer
208 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
  • 2,381
8 votes
1 answer
271 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
  • 39.7k

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