All Questions
Tagged with axiom-of-choice forcing
39
questions
4
votes
0
answers
144
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 ...
- 6,365
4
votes
2
answers
207
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, $\operatorname{Col}(\lambda,\kappa)$ ...
- 2,910
9
votes
1
answer
622
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 $V\vDash\mathsf{ZFC}$, ...
- 1,479
12
votes
1
answer
434
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 ...
- 2,236
7
votes
1
answer
369
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&...
- 20.7k
5
votes
0
answers
137
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 ...
- 6,365
4
votes
0
answers
141
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 $...
- 2,236
7
votes
2
answers
714
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 ...
- 2,236
7
votes
2
answers
673
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 ...
- 20.7k
9
votes
1
answer
288
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
5
votes
1
answer
144
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:
$$p\Vdash_{M,P}\exists x\phi[x]\;\Longrightarrow\;\exists\sigma\in M^P\;p\...
- 1,632
10
votes
1
answer
517
views
Models of ZF intermediate between a model of ZFC and a generic extension
Let $M$ be a countable model of $ZFC$ and $M[G]$ be a (set) generic extension of $M$. Suppose $N$ is a countable model of $ZF$ with
$$M\subseteq N \subseteq M[G]$$
and that $N=M(x)$ for some $x\in ...
- 1,132
18
votes
2
answers
817
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 ...
- 4,054
10
votes
4
answers
551
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 ...
- 20.7k
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 ...
- 20.7k