All Questions
5
questions
0
votes
1
answer
114
views
Shouldn't ℵ₀ be the cardinality of the reals?
If in ZFC any set can be well ordered, and that $\aleph_0$ is the cardinality of every infinite set that can be well ordered, shouldn't $\aleph_0$ be the cardinality of the real numbers?
I know this ...
0
votes
1
answer
605
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
0
votes
1
answer
77
views
Which stage in the Neumann hierarchy do powers of the reals fit in?
To be more specific than the short title, I try to gauge the size of some "normal" function spaces as e.g. found in functional analysis against set universe sizes at certain stages.
For the sake of ...
- 12.3k
2
votes
1
answer
159
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
6
votes
2
answers
726
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