Questions tagged [cardinals]
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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Cardinality of set of real continuous functions
I believe that the set of all $\mathbb{R\to R}$ continuous functions is $\mathfrak c$, the cardinality of the continuum. However, I read in the book "Metric spaces" by Ó Searcóid that set of ...
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Cartesian Product of Two Countable Sets is Countable [closed]
How can I prove that the Cartesian product of two countable sets is also countable?
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Proving the Cantor Pairing Function Bijective
How would you prove that the Cantor Pairing Function is bijective? I only know how to prove a bijection by showing that the following holds:
(1) For all $x,x^\prime$ in the domain of $f$, if $f(x) = f(...
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How do we know an $ \aleph_1 $ exists at all?
I have two questions, actually. The first is as the title says: how do we know there exists an infinite cardinal such that there exists no other cardinals between it and $ \aleph_0 $? (We would have ...
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For every infinite $S$, $|S|=|S\times S|$ implies the Axiom of choice
How to prove the following conclusion:
[For any infinite set $S$, there exists a bijection $f:S\to S \times S$] implies the Axiom of choice.
Can you give a proof without the theory of ordinal numbers.
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The cartesian product $\mathbb{N} \times \mathbb{N}$ is countable
I'm examining a proof I have read that claims to show that the Cartesian product $\mathbb{N} \times \mathbb{N}$ is countable, and as part of this proof, I am looking to show that the given map is ...
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Defining cardinality in the absence of choice
Under ZFC we can define cardinality $|A|$ for any set $A$ as
$$
|A|=\min\{\alpha\in \operatorname{Ord}: \exists\text{ bijection } A \to \alpha\}.
$$
This is because the axiom of choice allows any ...
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An infinite subset of a countable set is countable
In my book, it proves that an infinite subset of a coutnable set is countable. But not all the details are filled in, and I've tried to fill in all the details below. Could someone tell me if what I ...
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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 ...
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Cardinality of a Hamel basis of $\ell_1(\mathbb{R})$
What is the cardinality of a Hamel basis of $\ell_1(\mathbb R)$? Is it deducible in ZFC that it is seemingly continuum? Does it follow from this that each Banach space of density $\leqslant 2^{\...
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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)?
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cardinality of all real sequences
I was wondering what the cardinality of the set of all real sequences is. A random search through this site says that it is equal to the cardinality of the real numbers. This is very surprising to me, ...
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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 ...
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Partition of N into infinite number of infinite disjoint sets? [duplicate]
Is it possible to have a countable infinite number of countable infinite sets such that no two sets share an element and their union is the positive integers?
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How to show $(a^b)^c=a^{bc}$ for arbitrary cardinal numbers?
One of the basic (and frequently used) properties of cardinal exponentiation is that $(a^b)^c=a^{bc}$.
What is the proof of this fact?
As Arturo pointed out in his comment, in computer science this ...