All Questions
Tagged with real-numbers set-theory
79
questions
-2
votes
2
answers
34
views
Countability of the set [closed]
Let $f$ be differentiable function from $\mathbb{R}$ to $\mathbb{R}$. Consider the set
$$A_y=\{x \in \mathbb{R} : f(x)=y \}$$
I want to know whether $A_y$ is countable for each $y\in \mathbb{R}$. I ...
CommunityBot
- 1
3
votes
2
answers
604
views
Why does axiom of choice not imply the set of real numbers is countable?
The axiom of choice implies all sets can be well ordered. If that is true, you can well order the set of real numbers and the set of the integers. Now, why can one not just pair the set of real ...
- 133k
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 ...
- 397k
3
votes
1
answer
188
views
A closed countable set with Cantor-Bendixson rank of $\omega +1$
I'm looking to find a closed countable set that has a Cantor-Bendixson Rank of $\omega +1$.
I know that $\{0\}\cup\{\frac{1}{x+1}|x\in\omega\}$ has a Cantor-Bendixson Rank of $2$ because we take out
...
- 79.6k
0
votes
1
answer
150
views
Cuts and continuous sets
I'm reading Kuratowski's "Set theory", and here is a question from the Chapter 6.
Consider a linearly ordered set $A$ and its cuts, i.e., such pairs $\langle X,Y\rangle$ of $A$'s subsets that $X=Y^-$ ...
- 1,773
4
votes
1
answer
300
views
Does calculus need choice axioms?
To do calculus, we (presumably) need the real numbers (or perhaps some abstract complete metric space?).
When the real numbers are constructed using Cauchy sequences or Dedekind cuts, does this ...
- 43.9k
1
vote
1
answer
142
views
Maximal Realism and Continuum Hypotehsis
I have read in multiple places, per example here, that there is a unique model of second order logic that axiomatizes the real numbers. If this is true, then we should be able to decide everything ...
- 249k
2
votes
0
answers
265
views
About Bernstein sets
Remember that a subset $X\subseteq \mathbb{R}$ is a $G_{\delta}$-set if $X$ is a countable intersection of open sets in $\mathbb{R}$. For example closed subsets of $\mathbb{R}$ are $G_{\delta}$-sets. ...
- 645
18
votes
2
answers
6k
views
Is there a bijection between the reals and naturals?
I found this pop math article saying that there was a paper published last year that proved that the cardinalities of the reals and naturals are equal. Is this true or is it a misinterpretation of the ...
CommunityBot
- 1
6
votes
0
answers
151
views
Baire space $\mathbb{N}^\mathbb{N}$ written as $\mathbb{R}$ [duplicate]
I'm writing my bachelor thesis on various topics from set theory and descriptive set theory (mainly topological games), and I just read a paper in which the Baire Space $\mathbb{N}^\mathbb{N}$ is ...
- 2,611
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
3
votes
1
answer
449
views
Partition of positive reals with each part closed under addition without choice
It is an easy exercise using transfinite recursion to prove the following (in ZFC):
There exists sets $S,T$ that partition $\mathbb{R}_{>0}$ such that each of $S$ and $T$ is closed under addition.
...
CommunityBot
- 1
3
votes
2
answers
335
views
Is there literature on non-computable numbers that can't be even be identified?
It seems to me that most real numbers can't be calculated by any finite set of instructions, even if we make use of non-computable functions.
First, assume that we have an oracle that will provide ...
- 249k
1
vote
1
answer
242
views
Are there countably many numbers than can be described?
I can explain to you every natural number (in theory) in the sense, that I could describe it and you would know exactly which number I'm talking about, e.g. by writing it down, this can be done in a ...
- 82.2k
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 $\...
- 803