All Questions
3
questions
6
votes
2
answers
729
views
Well-orderings of $\mathbb R$ without Choice
The question is about well-ordering $\mathbb R$ in ZF. Without the Axiom of Choice (AC) there exists a set that is not well-ordered. This could occur two ways: a) there are models of ZF in which $\...
- 3,009
2
votes
1
answer
160
views
What is the smallest $i$ such that ZFC proves $\mathfrak c\le \aleph_i$?
I think we have that $\aleph_{\mathfrak c}\ge\mathfrak c$. But are there tighter upper bounds for $\mathfrak c$ in ZFC or no such bound is dependent on ZFC?
- 411
0
votes
1
answer
610
views
How is the Continuum Hypothesis equivalent to the existence of a well-ordering on $\Bbb R$ whose bounded initial segments are countable?
There exists an well-ordering $(<)$ on $\Bbb R$ such that the set $\{x \in \Bbb R\mid x < y \}$ is countable for every $y \in \Bbb R.$
How to prove that the above statement is equivalent to ...
- 2,013