# 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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### A question about $2^{\aleph_0}$

Is the cardinality of the infinite sequences consisting of $\left\{0,1\right\}$ or $\left\{0,1,2\right\}$ or $\left\{0,1,2,3,\cdots,9 \right\}$ elements equal to the cardinality of the infinite ...
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### Is there a cardinal function such that the intersection of a family of open sets with the same cardinality has an open intersection? [duplicate]

I am working in some cardinal functions in topological spaces, and I need to either define a new cardinal function or use an existing one that behaves in the following way: Lets say $\varphi$ is my ...
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### Cardinality of the continuum and binary expansion of numbers

The accepted answer to this question regarding the continuum hypothesis contains the sentence: "I am assuming you know that |ℝ|=2^(ℵ0), which can be proven by looking at binary expansions of numbers ...
1answer
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### What is the cardinality of $L^{\infty}(\mathbb{R})$?

I know of this answer for $L^p(\mathbb{R})$ where $p < \infty$: What is the cardinality of $L^p(\mathbb R)$, $1 \le p < \infty$? But, what about $L^{\infty}(\mathbb{R})$, can we prove any ...
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### Show that the cardinality of ($X$ ∪ {$x$}) is equal to the cardianlity of ($X$)+$1$

Let $X$ be a ﬁnite set, and let $x$ be an object which is not an element of $X$. Then $X$ ∪ {$x$} is ﬁnite and #($X$ ∪ {$x$}) = #($X$)+$1$. Note that #-here means cardinality. Suppose the cardinality ...
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### Prove that for any set A and B, the cardinality of the set of all functions mapping A to B is $\vert B \vert ^ {\vert A \vert}$

What I do is for finite sets A, B, let $A={a_1, a_2, ...a_n}$ and $B={b_1, b_2, ...b_m}$ A function f assigns each element $a_i$ of $A$ to an element $b_j = f (a_i)$ of $B$; there are $m$ ...
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### Infinite Time Turing Machine and Hypertask

Good Day, I would like to ask this question. Infinite Time Turing Machine ordinals are countable ordinals. Also I have read that ITTM either halts or repeats itselve after countably many steps. Does ...
1answer
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### On the distributive number $\mathfrak h$

The distributive number $\mathfrak h$ is defined as the least cardinal $\kappa$ such that there exists a family of $\kappa$ open dense subsets in the preordered set $([\omega]^\omega,\subset^*)$ whose ...
1answer
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### Does $|A|^{|A|}=2^{|A|}$ hold for any infinite set $A$? [duplicate]

Because I know the cardinality of all the functions$f:[0,1]\to \mathbb{R}$ is $2^{c}$（c is the cardinality of continuum).I wonder whether this holds generally.
1answer
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### If $X$ is an infinite set, then is $\sum_{i=1}^\infty |X|=|X|$?

I used the above to solve a problem, but I am not sure if this is true (and if it is true does it require a proof or is it obvious). I think it is true because $|X| + |X| = |X|$, so it should follow ...
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### A Set is Infinite if, and only if, it is in One-to-one Correspondence with a Proper Subset of Itself

Can someone explain what that means? How can there exist an injective function from an infinite set to a proper subset of itself. A function from a set A to a set B where B has fewer elements than A ...
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