All Questions
Tagged with forcing axiom-of-choice
40 questions
19
votes
2
answers
856
views
Do choice principles in all generic extensions imply AC in $V$?
It's well-known that not all choice principles are preserved under forcing, e.g. in this answer https://mathoverflow.net/a/77002/109573 Asaf shows the ordering principle can hold in $V$ and fail in a ...
- 4,316
18
votes
5
answers
2k
views
Forcing over models without the axiom of choice
In the vast majority of papers forcing is always developed over ZFC.
Not surprisingly too, since infintary combinatorial principles are often used to prove results based on properties such as chain ...
- 39.8k
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
16
votes
1
answer
1k
views
Can there be a global linear ordering of the universe without a global well-ordering of the universe?
This question arose in the answers to Asaf Karagila's
question Does ZFC prove that the universe is linearly orderable?. The answer there was that one can have a ZFC model with no global linear ...
- 236k
13
votes
3
answers
796
views
How to make countably closed forcing "nice" without choice
When working over a model $V$ of $ZFC$, countably closed forcings are extremely nice:
If $\mathbb{P}$ is countably closed, then $V[G]$ has no new $\omega$-sequences of elements of $V$. In ...
- 20.9k
12
votes
1
answer
442
views
Is there a more modern account of the main results of "Adding Dependent Choice" by D. Pincus?
I would like to read Pincus' article Adding dependent choice, where he proves, among other things, the consistency of the theory $\mathsf{ZF+DC+O+\neg AC}$, where $\mathsf{DC}$ stands for Dependent ...
- 2,286
12
votes
1
answer
695
views
A new cardinality living in every forcing extension?
I'm broadly interested in notions of "generic presentability" - when a given object exists in every forcing extension of the universe by some fixed forcing, at least up to the appropriate ...
- 20.9k
10
votes
4
answers
554
views
What are some kinds of models where DC holds?
There are a lot of ways to build a model where DC fails. However, all of them that I'm aware of involve adding at least a messy set of reals (or rather, taking a forcing extension and then passing to ...
- 20.9k
10
votes
1
answer
683
views
Small Violations of Choice: Can we force AC without collapsing the cardinalities of ordinals?
We say that a model $M$ of $\mathsf{ZF}$ satisfies Small Violations of Choice ($\mathsf{SVC}$) if all (any) of the following apply:
There is a model $V\subseteq M$ such that $V\vDash\mathsf{ZFC}$, ...
- 2,206
10
votes
1
answer
535
views
Models of ZF intermediate between a model of ZFC and a generic extension
Let $M$ be a countable model of $ZFC$ and $M[G]$ be a (set) generic extension of $M$. Suppose $N$ is a countable model of $ZF$ with
$$M\subseteq N \subseteq M[G]$$
and that $N=M(x)$ for some $x\in ...
- 1,142
10
votes
1
answer
761
views
Forcing, cuts, and Dedekind-finite cardinalities
Tl;dr version: there are two natural classes of cuts in the nonstandard model of arithmetic consisting of the Dedekind-finite sets (if, in fact, they constitute such a model); both these classes are ...
- 20.9k
9
votes
2
answers
1k
views
Relationship between fragments of the axiom of choice and the dependent choice principles
The dependent choice principle ${\rm DC}_\kappa$ states that if $S$ is a nonempty set and $R$ is a binary relation such that for every $s\in S^{\lt\kappa}$, there is $x\in S$ with $sRx$, then there ...
- 2,240
9
votes
1
answer
313
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{...
- 149
7
votes
2
answers
349
views
Forcing $\neg AC$
Sorry if this sounds like a silly reference request, but I wasn't able to track down any. I'm looking for proof, via forcing, that axiom of choice can fail in a model of $ZF$. All of papers I found ...
- 28.2k
7
votes
2
answers
722
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 ...
- 2,286
7
votes
2
answers
683
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 ...
- 20.9k
7
votes
2
answers
460
views
Possible Choices for Cofinality of $\aleph_n$ without Choice
$\text{ZFC}$ proves that each $\aleph_{n}$ for $n\in \omega$ is a regular cardinal. But it seems without the Axiom of Choice there are many consistent possible choices for cofinality of such cardinals....
user42090
7
votes
1
answer
401
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&...
- 20.9k
7
votes
1
answer
908
views
Symmetric extensions and class forcing
Suppose $V\models ZFC$ and $P\in V$ is a poset of forcing conditions.
It is a basic theorem in forcing that $V[G]\models ZFC$ for any generic extension by a $V$-generic filter $G$.
It is also known ...
- 39.8k
6
votes
2
answers
886
views
Why are some axioms preserved in generic extensions?
It is a known theorem that for a model of $ZF$, $M$, if $M\models AC$ and $G$ is a $P$-generic filter over $M$, for some $P\in M$, then $M[G]\models AC$.
On the other hand, it is long known that ...
- 39.8k
6
votes
1
answer
695
views
A question about the first Cohen model
Consider the first Cohen model, i.e. let $M$ be a countable transitive model of ZFC + $V=L$, let $\mathbb P$ be the poset consisting of finite partial functions from $\omega\times\omega$ to $2$, let $...
- 1,688
6
votes
1
answer
171
views
Function Approximation in c.c.c Forcings without AC in Ground Model
Consider the following basic theorem.
Theorem. If $M$ is a c.t.m of ZFC and $\mathbb{P}$ a c.c.c forcing notion in $M$ and $G$ a $\mathbb{P}$ - generic filter on $M$ then for all $A,B$ in $M$ and for ...
user47679
6
votes
1
answer
337
views
Intermediate submodels which do not satisfy AC
The following is known:
Theorem. Suppose $V[G]$ is a generic extension of $V$ by a set forcing, and let $N$ be a model of $ZFC$ with $V\subseteq N\subseteq V[G].$ Then $N$ is a generic extension of $...
- 32.1k
6
votes
1
answer
546
views
On successive regular cardinals with no ladders
Definition: Let $\kappa$ be an $\aleph$ cardinal, we say that $\langle f_\alpha\colon\alpha\to\kappa\mid\alpha<\kappa^+\rangle$ is a ladder if every $f_\alpha$ is injective.
Equivalently this is ...
- 39.8k
6
votes
0
answers
300
views
What are these sets in Freyd's model?
Recall Freyd's model of $ZF +\neg AC$ (as recounted in MacLane and Moerdijk's book Sheaves in Geometry and Logic): it arises as the Fourman interpretation of the topos of sheaves on a particular ...
- 35.5k
5
votes
1
answer
149
views
Maximality principle in symmetric extensions
Let $M$ be a ctm and $P\in M$ a forcing order.
In regular forcing extensions, we have the following well-known Principle:
$$p\Vdash_{M,P}\exists x\phi[x]\;\Longrightarrow\;\exists\sigma\in M^P\;p\...
- 1,799
5
votes
2
answers
610
views
Forcing the nonexistence of a certain set
I have a certain set-theoretic axiom (WISC) which follows from Choice (this is a nuking a fly BTW), but which I suspect is independent of ZF. To show this I need to show that a certain set does not ...
- 35.5k
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 ...
- 7,545
4
votes
2
answers
753
views
Minimal Generalized Continuum Hypothesis & Axiom of Choice
It is well known that working in the frame of $\text{ZF}$, the Generalized Continuum Hypothesis ($\text{GCH}$) implies the Axiom of Choice ($\text{AC}$), i.e. $\text{ZF}+\text{GCH}\vdash \text{AC}$.
...
user42090
4
votes
2
answers
222
views
Does $\mathsf{ZF}$ prove $\operatorname{Col}(\lambda,\kappa)$ preserves cardinals below $\lambda$?
Let $\lambda<\kappa$ be cardinals and consider the forcing $\operatorname{Col}(\lambda,\kappa)$ adding a generic surjection $\lambda\to\kappa$. More formally, $\operatorname{Col}(\lambda,\kappa)$ ...
- 3,042
4
votes
1
answer
219
views
Generic Absoluteness restricted to formulas with low complexity or to the class of forcings
Ikegami and Schlicht proposed a principle, namely generic absoluteness, which is stated below using Hamkins' and Lowe's terminology:
Working in $ZF$:
(Generic Absoluteness): For all formulas in ...
- 2,381
4
votes
1
answer
441
views
Is the ordering principle preserved in generic extensions?
The ordering principle says that every set can be linearly ordered.
In a previous question Why are some axioms preserved in generic extensions? Asaf Karagila asked which axioms are preserved in ...
- 16.2k
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
4
votes
0
answers
142
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 $...
- 2,286
4
votes
0
answers
379
views
Forcing without choice: when countable sets yield reals
One natural way to show that a forcing adds no new reals is to show that it is countable closed (EDIT: this is somewhat misleading, see Joel's comment below). However, it turns out that this is ...
- 20.9k
4
votes
0
answers
161
views
What is the meaning of restricting a Boolean value to a subalgebra?
$\require{AMScd}$
I am studying the proof that the ordering principle does not imply the axiom of choice in Jech's book "The Axiom of Choice" (Section 5.5). Let $P$ be the set of finite partial ...
- 1,688
3
votes
1
answer
600
views
Does existence of $\omega_1$ subset of reals imply $\omega_1$ choice for subsets of reals?
Suppose there exists a subset of $\Bbb R$ which has cardinality $\omega_1$. Is it then necessarilly true that for every collection of $\omega_1$ subsets of $\Bbb R$ there exists a choice function?
I ...
- 28.2k
2
votes
1
answer
784
views
Subsets of Real Numbers (Edited & Revised Version)
Question 1: Is it consistent with $\text{ZF}$ that only countable subsets of $\mathbb{R}$ are well-orderable?
Question 2: Is it consistent that for some $\lambda$, $\aleph_0 < \lambda < 2^{\...
user42090
2
votes
1
answer
438
views
The patterns of possibility for nontrivial automorphisms and nontrivial elementary embeddings of the universe
In their paper "The Role of the Foundation Axiom in the Kunen Inconsistency" (arXiv:1311.0814 [Math.LO]), Daghighi, Golshani, Hamkins, and Jerabek show that the patterns of possibility for the ...
- 6,132
1
vote
1
answer
156
views
Invariant names and submodels of forcing extensions
EDIT: There are serious problems with the definition below; see the comment thread below for those problems and some thoughts on addressing them. I'm leaving the question up for now since I think the ...
- 20.9k