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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3answers
439 views

Comparing infinite sets (of real numbers)

If $A$ is the set of all real numbers in $(0,1)$ with no $5$ in their decimal representation, and $B$ is the set with no $34$ and no $76446$. Then the set $B$ is in some sense larger then $A$, how can ...
4
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2answers
232 views

If $|X|=|Y|=|X-Y|=\kappa$, can we find a bijection on $X$ that fixes $Y$ only?

in a previous question, I mistakenly attempted to subtract one cardinal number from another. Anyway, this got me to thinking, suppose I have two sets $X$ and $Y$, with $Y\subseteq X$. Suppose also ...
21
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4answers
779 views

Does $k+\aleph_0=\mathfrak{c}$ imply $k=\mathfrak{c}$ without the Axiom of Choice?

I'm currently reading a little deeper into the Axiom of Choice, and I'm pleasantly surprised to find it makes the arithmetic of infinite cardinals seem easy. With AC follows the Absorption Law of ...
5
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1answer
312 views

Cardinality of sets of functions with well-ordered domain and codomain

I would like to determine the cardinality of the sets specified bellow. Nevertheless, I don't know how to approach or how to start such a proof. Any help will be appreciated. If $X$ and $Y$ are ...
6
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2answers
340 views

mahlo and hyper-inaccessible cardinals

Wikipedia states that a Mahlo cardinal is hyper-inaccessible, hyper-hyper-inaccessible, etc. Is this a characterisation of Mahlo? If not what about "alpha = hyper^alpha-inaccessible" with the obvious ...
2
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1answer
186 views

Cardinal arithmetic: $b\ge x>1$ and $b^2=b$ implies $x^b=2^b$

Let $x$ and $b$ represent cardinals. Assume that $b\geq x > 1$ and $b^2=b$. Prove that $x^b=2^b$. Thanks!
3
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2answers
150 views

one more question with cardinality: $(\{1,2,3\}^{\mathbb{N}} - \{1,2\}^{\mathbb{N}})\cap\mathcal{P}(\mathbb{N}\times\mathbb{N}).$

How can I calculate the cardinality of $$\left(\{1,2,3\}^{\mathbb{N}} - \{1,2\}^{\mathbb{N}}\right)\cap\mathcal{P}(\mathbb{N}\times\mathbb{N}).$$ where $A^B$ is the set of all functions $f\colon B\to ...
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2answers
403 views

What does Martin's Maximum imply for $P(\mathbb{R})$?

Prompted by this question: of course Godel's constructibility axiom implies that $P(S)$ is minimal for any set $S$ and so handily answers the question of the size of the power set of the continuum in ...
9
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1answer
310 views

Is there an easy proof for ${\aleph_\omega} ^ {\aleph_1} = {2}^{\aleph_1}\cdot{\aleph_\omega}^{\aleph_0}$?

The question contains 2 stages: Prove that ${\aleph_n} ^ {\aleph_1} = {2}^{\aleph_1}\cdot\aleph_n$ This one is pretty clear by induction and by applying Hausdorff's formula. Prove ...
12
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1answer
784 views

bound on the cardinality of the continuum? I hope not

Suppose we don't believe the continuum hypothesis. Using Von Neumann cardinal assignment (so I guess we believe well-ordering?), is there any "familiar" ordinal number $\alpha$ such that, for ...
17
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4answers
5k views

Cardinality of the set of all real functions of real variable

How does one compute the cardinality of the set of functions $f:\mathbb{R} \to \mathbb{R}$ (not necessarily continuous)?
14
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1answer
2k views

The cardinality of a countable union of countable sets, without the axiom of choice

One of my homework questions was to prove, from the axioms of ZF only, that a countable union of countable sets does not have cardinality $\aleph_2$. My solution shows that it does not have ...
7
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2answers
867 views

Non-existence of a surjection $\aleph_n \to \aleph_{n+1}$, without the axiom of choice

Firstly, let's establish what exactly I mean by these symbols. Let $\omega_0 = \{ 0, 1, 2, \ldots \}$, where $0, 1, 2, \ldots$ are the usual von Neumann representations of the natural numbers. Let $n$ ...
8
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4answers
1k views

Simple example of uncountable ordinal

Can you make a simple example of an uncountable ordinal? With simple I mean that it is easy to prove that the ordinal is uncountable. I know that the set of all the countable ordinals is an ...
3
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2answers
322 views

“Homomorphism” from set of sequences to cardinals?

First off: I barely have any set theoretic knowledge, but I read a bit about cardinal arithmetic today and the following idea came to me, and since I found it kind of funny, I wanted to know a bit ...
8
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4answers
967 views

What's the cardinality of all sequences with coefficients in an infinite set?

My motivation for asking this question is that a classmate of mine asked me some kind of question that made me think of this one. I can't recall his exact question because he is kind of messy (both ...
1
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1answer
244 views

What is the order of the set of distinct (up to similarity) nxn matrices over R?

What is the order of the set of distinct (up to similarity) nxn matrices over $\mathbb{R}$ with determinant equal to some non-zero scalar... say 6? (eg. countable, uncountable etc.) The set of ...
6
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1answer
942 views

Cardinality of a set that consists of all existing cardinalities

I have taken a look at the following topics: number of infinite sets with different cardinalities Cardinality of all cardinalities Are there uncountably infinite orders of infinity? Types of ...
7
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1answer
767 views

Examples of sets whose cardinalities are $\aleph_{n}$, or any large cardinal. (not assuming GCH)

One of the answers to this question indicates that large cardinals are useful for destructive testing of set theory. That aside, and not assuming GCH, are there any sets known that have a cardinality ...
62
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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. ...
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3answers
673 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 ?
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2answers
432 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
266 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
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2answers
161 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
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1answer
107 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
155 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
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1answer
553 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 ...
13
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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
6k 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?