# Tagged Questions

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### Is the Axiom of Choice necessary to prove $\mathbb R \approx \mathcal P(\omega)$?

I am self-studying Horst Herrlich, Axiom of Choice (Lecture Notes in Mathematics, Vol. 1876), and I'm getting quite confused about cardinal arithmetic without AC. Here (Which sets are well-orderable ...
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### The function space from $n$ to $m$ and the exponent $m^n$ are equinumerous (proof)

Can someone provide a tip with creating the bijection for the titular problem? Any tip is helpful! Update: $n=\{ 0,\ldots,n-1 \}$, $m=\{ 0,\ldots,m-1 \}$, and $m^n=\{0,\ldots,m^n-1\}$. In other ...
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### Prove that the sets $S$ and $D$ have the same cardinality

Prove that the sets $S$ and $D$ have the same cardinality, where $S = \{(x,y)\mid-1\leq x \leq 1\text{ and }-1\leq y\leq 1\}$ and $D = \{(x,y)\mid x^2 + y^2 \leq 1\}$.
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### Prove $|A| = |B|$

Let $A= \{a_1,a_2,a_3,\ldots\}$. Define $B = A − \{a_{n^2} : n \in\mathbb N\}$. Prove that $|A|=|B|$. I would say that $B = \{a_2,a_3,a_5,a_6,\ldots\}$. Thus $B$ is a infinite subset of $A$ and ...
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### What is the cardinality of the set of all higher order functions mapping real functions to real functions?

What is the cardinality of the set of all higher order functions mapping real functions to real functions? To be specific, this set includes all higher order functions with the type signature: ...
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### Show that $\left|X\right|=\left|Y\right|$

Let $A, B$, two sets. $X$ is the set of all relations from $A$ to $B$, and $Y$ is the set of all functions from $A$ to $P(B)$ (power-set of $B$). Prove that $\left|X\right|=\left|Y\right|$. My ...
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### show that for an infinite cardinal $k$, $k + k = k$

Show that for an infinite cardinal $k$, $k + k = k$ So far I have that $k + k = 2k$ Is it possible to somehow show that $2k = k$? I've been trying to understand some cardinal arithmetic, and I ...
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### Behaviour of sum of $2^\kappa$ for all $\kappa<\lambda$ when $\lambda$ is singular [closed]

What can we say about the conditions under which $\sum\limits_{\kappa<\lambda}2^\kappa \leq \lambda$ holds when $\lambda$ is singular?
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### How far is it possible to develop cardinals without ordinals?

I'm wondering which of the usual facts about cardinals in ZFC can be established without using ordinal arithmetic at all. After all the definitions of a cardinal (as a class of equivalence), and also ...
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### Are there any cardinal k with k infinite cardinals below it? [closed]

If there are,please write down the smallest one.
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### Question about partitions in intervals of the real numbers.

I have to prove the following: Let $\mathcal{D}$ be a partition of $\mathbb{R}$ in intervals of any kind, except intervals containing a single element. Prove $\mathcal{D}$ is countable. ...
### For all finite sets $A$ and $B$, $\{ f:B\to A \}$ is finite and its (finite) cardinality is $|A|^{|B|}$.
In Halmos' Naive Set Theory (towards the end of the "Arithmetic" chapter) he mentions the titular claim: For all finite sets $A$ and $B$, $\{ f:B\to A \}$ is finite and its (finite) cardinal is ...