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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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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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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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. ...
user17762
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Overview of basic results on cardinal arithmetic
Are there some good overviews of basic formulas about addition, multiplication and exponentiation of cardinals (preferably available online)?
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Refuting the Anti-Cantor Cranks
I occasionally have the opportunity to argue with anti-Cantor cranks, people who for some reason or the other attack the validity of Cantor's diagonalization proof of the uncountability of the real ...
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Why is the Continuum Hypothesis (not) true?
I'm making my way through Thomas W Hungerfords's seminal text "Abstract Algebra 2nd Edition w/ Sets, Logics and Categories" where he makes the statement that the Continuum Hypothesis (There does not ...
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Doesn't the unprovability of the continuum hypothesis prove the continuum hypothesis? [duplicate]
The Continuum Hypothesis says that there is no set with cardinality between that of the reals and the natural numbers. Apparently, the Continuum Hypothesis can't be proved or disproved using the ...
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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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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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Is symmetric group on natural numbers countable?
I guess it is too difficult a question to ask about the cardinality of $S_{\mathbb{N}}$ so I would like to ask whether it is countable or not.
I tried to prove it is uncountable somewhat mimicking ...
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Is there an infinite topological space with only countably many continuous maps to itself?
Now cross-posted to Mathoverflow.
Is there an infinite topological space $X$ with only countably many continuous functions to itself? Such a space would have only countably many points because the ...
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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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Do the real numbers and the complex numbers have the same cardinality?
So it's easy to show that the rationals and the integers have the same size, using everyone's favorite spiral-around-the-grid.
Can the approach be extended to say that the set of complex numbers has ...
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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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Is cardinality a well defined function?
I was wondering if the cardinality of a set is a well defined function, more specifically, does it have a well defined domain and range?
One would say you could assign a number to every finite set, ...
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