Linked Questions

19
votes
3answers
2k views

What are disasters with Axiom of Determinacy?

It is well-known that Axiom of Choice has several consequences which might be viewed as counter-intuitive or undesirable. For example, existence of non-measurable sets or Banach-Tarski Paradox. H. ...
15
votes
3answers
3k views

Construct a bijection from $\mathbb{R}$ to $\mathbb{R}\setminus S$, where $S$ is countable

Two questions: Find a bijective function from $(0,1)$ to $[0,1]$. I haven't found the solution to this since I saw it a few days ago. It strikes me as odd--mapping a open set into a closed set. $S$ ...
12
votes
3answers
2k views

Does a countable set generate a countable group?

Let $G$ be a group and let $A\subseteq G$ be infinite. Do $A$ and $\left< A\right> $ have the same cardinality?
9
votes
6answers
3k views

The cardinality of $\mathbb{R}/\mathbb Q$

How to prove the cardinality of $\mathbb{R}/ \mathbb Q$ is equal to the cardinality of $\mathbb{R}$
6
votes
4answers
1k views

Cardinality of Vitali sets: countably or uncountably infinite?

I am a bit confused about the cardinality of the Vitali sets. Just a quick background on what I gather about their construction so far: We divide the real interval $[0,1]$ into an uncountable number ...
5
votes
3answers
423 views

The real line has cardinality at most $\aleph_2$, but transfinite ordinal space has arbitrarily high cardinality: what is wrong?

In the context of supertasks, people and mathematicians are comfortable with the idea of transfinite ordinal time, that is, that time can be divided into an arbitrarily high number of steps. In most ...
5
votes
2answers
1k views

Existence of surjection implies existence of injection? [duplicate]

Let $A$ and $B$ be sets. If there exists a surjection $f : A \to B$ then there exists an injection $g : B \to A$. Proof: given $b \in B$ select an element $a \in f^{-1}(b)$. Denote this element by $...
-1
votes
2answers
265 views

Cardinality of “$x-y\in\Bbb Q$”-equivalence class of $1/\sqrt2$

For $x,y\in I:=[0,1]$ define the relation on $I$ as $x-y\in \Bbb Q$. How big (using cardinal number) is the cardinality of the equivalence class $[1/\sqrt2]$? I have tried to solve it by finding ...
2
votes
1answer
118 views

Do the adeles $\mathbb{A}$ have the same cardinality as the real numbers $\mathbb{R}$?

My question today is about the Adeles and whether they have the same cardinality as the real numbers. Certainly $\mathbb{R}$ has more elements than $\mathbb{Z}$. This is by Cantor diagonalization. ...
0
votes
0answers
70 views

Is there a bijection between $\Bbb{R}$ and $\Bbb{R} / \Bbb{Q}$? [duplicate]

$\Bbb{R} / \Bbb{Q}$ is a quotient set of $\Bbb{R}$ with the following equivalence relation $\sim$ : $$r \sim s \Longleftrightarrow r-s \in \Bbb{Q}$$ Then is there a bijection between $\Bbb{R}$ and ...