All Questions
8 questions
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
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
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
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
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
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
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