This tag is for questions about cardinals and related topics such as cardinal arithmetics, regular cardinals and cofinality. Do not confuse with [large-cardinals] which is a technical concept about strong axioms of infinity.

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62
votes
6answers
3k views

Why is $\omega$ the smallest $\infty$?

I am comfortable with the different sizes of infinities and Cantor's "diagonal argument" to prove that the set of all subsets of an infinite set has cardinality strictly greater than the set itself. ...
1
vote
3answers
653 views

what is the cardinality of set of all smooth functions in $L^1$?

What is the cardinality of set of all smooth functions belonging to $L^1$ or $L^2$ ? What is that of set of all integrable or square integrable functions ?
11
votes
2answers
420 views

Cardinality of $H(\kappa)$

Again I have trouble with some exercises in Kunen's set theory. In the following, let $\kappa > \omega$ a cardinal. Then I want to show that 1) $|H(\kappa)| = 2^{<\kappa}$ 2) ...
3
votes
1answer
260 views

set of infinite cardinals admits an injective regressive function

Let $A$ be a set of infinite cardinals. Assume that for every regular $\lambda$, the subset $A \cap \lambda$ of $\lambda$ is not stationary. Then I want to prove that there is an injective function ...
4
votes
2answers
160 views

Number of continuous $[0; 1] \to [0; 1]$ functions for given arc length

Just out of pure curiosity ... Suppose I want to connect the two points $(0,0)$ and $(1,1)$ with the graph of some continuous and differentiable function $$f : [0; 1] \to [0; 1]$$ and let $s$ be ...
3
votes
1answer
102 views

number of regular cardinals in a weakly inaccessible cardinal

Let $\kappa$ ba weakly inaccessible cardinal. Why are there $\kappa$ regular cardinals $\lambda < \kappa$? I've tried a recursive construction, but I don't know what to do in the limit step. ...
2
votes
2answers
152 views

Example of a c.u.b. set

Let $\kappa$ a cardinal of cofinality $\omega$; let $C \subseteq \kappa$ be a unbounded countable subset. Why is then $C$ closed (and thus a c.u.b.)? This means that if $\delta < \kappa$ is a limit ...
9
votes
1answer
543 views

cardinal exponentiation, $k^{<\lambda}$

I have the following well-known exercise in cardinal arithmetics: If $\kappa, \lambda$ are cardinals such that $\lambda$ is infinite, then $\kappa^{<\lambda}$ equals the supremum of the ...
12
votes
4answers
2k views

Cardinality of all cardinalities

Let $C = \{0, 1, 2, \ldots, \aleph_0, \aleph_1, \aleph_2, \ldots\}$. What is $\left|C\right|$? Or is it even well-defined?
27
votes
6answers
5k views

Cardinality of set of real continuous functions

The set of all $\mathbb{R\to R}$ continuous functions is $\mathfrak c$. How to show that? Is there any bijection between $\mathbb R^n$ and the set of continuous functions?