All Questions
Tagged with independence-results reference-request
9 questions
13
votes
0
answers
325
views
$\mathsf{AC}_\mathsf{WO}+\mathsf{AC}^\mathsf{WO}\Rightarrow \mathsf{AC}$?
Let
$\mathsf{AC}_\mathsf{WO}$: Every well-orderable family of non-empty sets has a choice function.
$\mathsf{AC}^\mathsf{WO}$: Every family of non-empty well-orderable sets has a choice function.
My ...
- 2,286
3
votes
1
answer
975
views
Implications of the existence of a pair of surjective functions, without Axiom of Choice
The classical Cantor-Schroder-Bernstein Theorem says that there exists a bijective function $X\leftrightarrow Y$ if and only if there exist injective functions $X\hookrightarrow Y$ and $Y\...
- 41.8k
2
votes
0
answers
120
views
Two small uncountable cardinals related to Q-sets
A subset $A$ of the real line is called a Q-set if any subset of of $A$ is of type $F_\sigma$ in $A$.
Let $\mathfrak q_0$ be the smallest cardinality of a subset $X\subset\mathbb R$ which is not a Q-...
- 41.8k
5
votes
1
answer
401
views
Variants of reflection principle
This question concerns two definitions of the reflection principle. One of them known to be a consequence of the other one. I would like to understand if the reverse is true.
Let us state the first ...
- 2,381
32
votes
1
answer
2k
views
Should axiomatic set theory be translated into graph theory?
Recently I saw the abstract of a paper by Nash-Williams: ``Should axiomatic set theory be translated into graph theory?''. The abstract, taken from Mathscinet says the following:
The author ...
- 32.1k
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
3
votes
3
answers
477
views
Undecidability and holomorphic functions (Reference request)
The goal of this question is to recall a certain mathematical fact -not in my field- that I was once briefly told and that I have fogotten, and also to collect similar results.
The fact, I think, ...
- 23.3k
15
votes
4
answers
2k
views
Is every p-point ultrafilter Ramsey?
A non-principal ultrafilter $\mathcal{U}$ on $\omega$ is a p-point (or weakly selective) iff for every partition $\omega = \bigsqcup _{n < \omega} Z_n$ into null sets, i.e each $Z_n \not \in \...
- 3,982
9
votes
2
answers
1k
views
Given a cardinal k, what's the biggest dense linear order with a dense subset of size k?
It's not hard to show that for any cardinal $\kappa$, there is no dense linear order without endpoints (DLO) of size greater than $2^{\kappa}$ that has a dense subset of size $\kappa$. But one can ...
- 3,982