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.

2,423 questions
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Injection from cardinal $\lambda$ to cardinal $\kappa$ implies $\lambda\leq\kappa$

I'm trying to prove that if there is an injection $f:\lambda\to\kappa$ (for $\lambda$,$\kappa$ cardinal numbers) then $\lambda\leq\kappa$. This is not true if they are just ordinal numbers, for ...
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Let $\mathfrak F$ be the set of injective mappings $f:\Bbb N\to\Bbb N$. Then $|\mathfrak F|=2^{\aleph_0}$

Let $\mathfrak F$ be the set of injective mappings $f:\Bbb N\to\Bbb N$. Then $|\mathfrak F|=2^{\aleph_0}$. This is an alternative proof to one in my textbook. Does it look fine or contain flaws? ...
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Suppose that $\aleph_0\le |A|<|B|$. Compare $|\mathcal {P}(A)|$ and $|\mathcal {P}(B)|$, $|A^B|$ and $|B^A|$, $|B^A|$ and $|B|$

Suppose that $\aleph_0\le |A|<|B|$. Compare below cardinalities. $|\mathcal {P}(A)|$ and $|\mathcal {P}(B)|$ $|A^B|$ and $|B^A|$ $|B^A|$ and $|B|$ My attempt: From $|A|<|B|$, I ...
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In Hungerford's Algebra, he firstly defines that equipollent is a relation between two sets $A$, $B$ if there exists bijective $f:A\rightarrow B$, then proves Equipollence is an equivalence ...
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Let $X,Y,Z$ be nonempty sets. Then $|(X^Y)^Z|=|X^{Y\times Z}|$

Let $X,Y,Z$ be nonempty sets. Then $\left |\left(X^Y\right)^Z\right|=\left|X^{Y\times Z}\right|$. Please help me verify this proof! Thank you so much! My attempt: We define a mapping $F$ that ...
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The set of sequences of $\Bbb N$ and $\mathcal{P} (\Bbb N)$ are equinumerous.

The set of sequences of $\Bbb N$ and $\mathcal{P} (\Bbb N)$ are equinumerous. My attempt: Let $S,S_1,S_2$ be the sets of sequences, finite sequences, and infinite sequences of $\Bbb N$ respectively....
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Set Representation with tabular form, the number of elements

In set theory, the order in which elements are listed is immaterial but elements must not be repeated. So {1, 1} is not permitted. Then the number of elements in set {1,a} (a is real number) is 1 or ...
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What is the cardinality of the set of equivalence classes of countable order types under bi-embeddability?

Two ordered sets have the same order type if there exists an order isomorphism between them. Now the set $X$ of all order types of countable totally ordered sets has cardinality $2^{\aleph_0}$; see ...
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Cardinality of sets ($\aleph_0$, distinct elements)

Which answers are correct? $| \{1000, 1001 \}| = 2$: True. $| \{1, 2, 2, 3 \}| = 4$: False. There are only $3$ distinct elements. The cardinality of $\mathbb{N} \times \mathbb{Z}$ is $\aleph_0$: I'd ...
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If $\kappa$ is infinite and $\kappa$ is not a sum of $<\kappa$ cardinals each less than $\kappa$, then $\kappa$ is regular.

An infinite cardinal $\kappa$ is regular if $\mathrm{cf}(\kappa) = \kappa$. It is known that if $\kappa$ is regular, then for any family $(\kappa_i)_{i \in I}$ of cardinals $\kappa_i < \kappa$ ...
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What order of infinity is the set of vectors whose components sum to $0$?

I was wondering this question today, but I don't really have the machinery to answer it, and I'm not sure how to search for such an answer. Here's the set up of the problem: What is the order of ...
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ZF with “double powersets”, is there a set with the same size as ℝ?

If you replace the axiom of the powerset, with one guaranteeing the existence of double powersets only ... roughly speaking, what changes? Here's the axiom of power set in ZF**, written out in first ...
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Cardinality of reduced products

Suppose $\mathcal{A}_i (i \in I)$ is a family of $L$-structures and consider the reduced product $\mathcal{A}$ of this family by a filter $F \subseteq \mathcal{P}(I)$. Is there a way to determine the ...