This tag is for questions about cardinals and related topics such as cardinal arithmetics, clubs, stationary sets, cofinality, and principles such as $\lozenge$. Do not confuse with [large-cardinals] which is a technical concept about strong axioms of infinity.

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Why are large cardinal axioms actually axioms?

A cardinal is an isomorphism class in ZFC, or a representative of one. I'm not asking what the significant uses of large cardinals are, or why we would want to find them or construct them; I'm ...
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3answers
204 views

Is there a largest large cardinal?

In ZFC, a cardinal is an isomorphism class of sets. However ZFC doesn't explicitly have classes; NBG, which is a conservative extension of ZFC does. There is no largest cardinal by Cantors Theorem ...
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2answers
39 views

Question about partitions in intervals of the real numbers.

I have to prove the following: Let $ \mathcal{D} $ be a partition of $ \mathbb{R} $ in intervals of any kind, except intervals containing a single element. Prove $ \mathcal{D} $ is countable. ...
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3answers
42 views

For all finite sets $A$ and $B$, $\{ f:B\to A \}$ is finite and its (finite) cardinality is $|A|^{|B|}$.

In Halmos' Naive Set Theory (towards the end of the "Arithmetic" chapter) he mentions the titular claim: For all finite sets $A$ and $B$, $\{ f:B\to A \}$ is finite and its (finite) cardinal is ...
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0answers
10 views

A question about countability. [duplicate]

I would like to know if $\mathbb{N}$ has the same cardinality as the set $\mathcal{P}_{0}(\mathbb{N}) = \{ A \subset \mathbb{N}: A \text{ is a finite set } \}.$ My strategy was to prove that $\left| ...
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1answer
44 views

Cardinality of all compact metric spaces

I`m looking for cardinal number of all compact metric spaces. I know that: Cardinal number of compact set is at most $\mathfrak{c}$ (it is a continous image of Cantor set) Compact metric space is ...
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0answers
54 views

Raising a partial function to the power of an ordinal

Consider a set $X$, and let $f : X \rightarrow X$ denote a partial function. Then for natural $n$, we can define $f^n$ as iterated composition, e.g. $f^2 = f \circ f$. Now suppose that $X$ is also ...
3
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2answers
62 views

On those behaviors of continuum function which imply the axiom of choice

It is a folklore fact that within $\text{ZF}$ the generalized continuum hypothesis ($\text{GCH}$) implies the axiom of choice ($\text{AC}$), namely: $$ZF+\forall \kappa\in ...
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2answers
32 views

The product of cardinals

Let $\gamma_i$ be infinite cardinal numbers for $i=1,2,3$ such that $\gamma_i<\gamma_3$ for $i=1,2$. Is it true that $\gamma_1.\gamma_2<\gamma_3$?
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1answer
23 views

Order of cardinal number

I am puzzled. Wikipedia says: "|X| ≤ |Y| means that there exists an injective function from X to Y." Let's see sets A and B: A = {1,2,3} and B = {1,2}. f: A → B: 1 ↦ 1, 2 ↦ 2. f is injective, but |B| ...
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2answers
58 views

Any infinite set partitioned into a set of countably infinite sets?

Prove that if $s$ is infinite, then it can be partitioned into a set of countably infinite sets $\mathcal{A}$. That is: $\bigcup \mathcal{A}=s$ $\forall a\in \mathcal{A}, a$ is countably ...
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2answers
141 views

Is it possible that $2^n=3^n$ for some Dedekind-finite cardinal $n\gt0$?

Is it possible that $2^n=3^n$ for some Dedekind-finite cardinal $n\gt0$? I think the question speaks for itself, but let me try and satisfy the "quality standards" algorithm by padding it. Yes, I ...
1
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1answer
39 views

factorial of infinite Cardinals

Let $S_A$ be set of all bijections over $A$ such that $Card(A)=\kappa$. Define foctorial as $\kappa!:=Card(S_A)$. Show that if $\kappa$ is infinite, then : $\kappa!=2^\kappa$ First, I've ...
4
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2answers
128 views

CH imples the existence of a function

I was studying an article and the author stated that CH implies that there exists a function from $\omega_1 \setminus \omega$ onto the set of all countable subsets of $\omega_1$ such that for each ...
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2answers
41 views

Cardinal of $V_{\omega+\alpha}$

I do not understand why $card(V_{\omega+\alpha})=\beth_\alpha$. The steps in the recursion for $0$ and succesor ordinals are quite easy, but I do not manage to prove it for limit ordinals. I see that ...
0
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1answer
21 views

Seemingly basic cardinality question

If $A\subset A'$, $B\subset B'$, if $card(A)=card(B)$ and $card(A')=card(B')$, why is it that $card(A'\backslash A)=card(B'\backslash B)$ ?
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2answers
28 views

Proove the next cardinal property: $\kappa>1$ $\Rightarrow$ $\lambda \leq \kappa^{\lambda}$

Let $\kappa>1$ and $\lambda$ be cardinals. Proove that: $\lambda \leq \kappa^{\lambda}$
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0answers
21 views

Is 2 a regular cardinal?

there are different definitions of regular cardinals. (1)a cardinal k is regular if cf(k)=k,since 2 is a successor cardinal,cf(2)=1.so cf(2) is not 2,so 2 is not regular. (2)a cardinal k is regular if ...
2
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1answer
36 views

If $|X|<|Y|$ then $|Y|=|Y-X|$ (with $Y$ infinite)

Like the title says, I would like to prove that if $|X|<|Y|$ then $|Y|=|Y-X|$. (with $Y$ infinite) I know I have to use the axiom of choice, but I've no idea about how to proceed. Any help is ...
0
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1answer
39 views

Are these ordinals cardinals?

Consider $\omega_2 \times \omega$ and $\omega \times \omega_2$ with ordinal arithmetic. Then $| \omega_2 \times \omega | =\omega_2$ and $| \omega \times \omega_2 | =\omega_2$ Does this imply ...
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5answers
1k views

Why study cardinals, ordinals and the like?

Why is the study of infinite cardinals, ordinals and the like so prevalent in set theory and logic? What's so interesting about infinite cardinals beyond $\aleph _0 $ and $\mathfrak{c} $? It seems ...
2
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1answer
38 views

Proving $|X^{\mathbb{R}}| \lt |\mathbb{R}^X|$ for $X=\mathbb{R}^{\mathbb{R}}$

Let $X=\mathbb{R}^{\mathbb{R}}$. Claim: $|X^{\mathbb{R}}| \lt |\mathbb{R}^X|$ How can this be shown? Note: We are assuming AC and $A^B$ represents the set of functions from B into A
2
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1answer
37 views

What is the cardinality of Maclaurin series that are entire that induce a bijection between ${\mathbb Z}$ and itself?

So the cardinality of linear polynomials that induce a bijection from ${\mathbb Z}$ to itself is countable, because it is simply the set of linear polynomials of the form $x + n$ where $n$ is an ...
9
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0answers
109 views

Sets of Cardinals without choice

I have a theory that the axiom of choice is equivalent to the statement that sets of distinct cardinals are well ordered by cardinality. I can prove that the axiom of choice implies this. However I am ...
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2answers
54 views

Prove that if an injection exists such that $f: A\to \mathbb{N}$, the set $A$ is countable

I know that since an injection, $f: A \rightarrow \mathbb{N}$ exists, that $|A| \leq |\mathbb{N}|$. That's as far as I've gotten. The definition for "countable" from my book states that "A set is ...
5
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1answer
91 views

Landing between $\beth_\lambda$ and $\beth_{\lambda+1}.$

Main Question. Is it consistent with ZFC that there exists a limit ordinal $\lambda$ and a cardinal number $\nu$ satisfying $$\beth_\lambda < \beth_\lambda^\nu < \beth_{\lambda+1}?$$ I am also ...
3
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1answer
45 views

Is it true that for all infinite cardinal numbers $\nu,$ the set $\{\kappa \mid \kappa^\nu > \kappa\}$ is unbounded?

Bind all lowercase greek letters to cardinal numbers. Question. Is it true that for all infinite cardinal numbers $\nu,$ we have that $\{\kappa \mid \kappa^\nu > \kappa\}$ is unbounded? For ...
0
votes
1answer
30 views

A question about Choice Functions.

Assume the axioms of ZFC. Suppose that X is an infinite set of infinite (and pairwise disjoint) sets, none of which has a cardinal number greater than that of X. Is the cardinal number of the set of ...
5
votes
2answers
129 views

What is the value of $\beth_{\omega_1}^{\aleph_0}$?

It is well known that $\beth_\omega^{\aleph_0} = \beth_{\omega+1}$. This follows since for strong limit $\kappa$, we have $\kappa^\kappa = \kappa^{\mathrm{cf}(\kappa)}.$ Question. To the extent that ...
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1answer
43 views

Is it true that if $\kappa < \kappa^\nu$, then ${\mathrm{cf}(\kappa)} \leq \nu$?

Let $\kappa$ denote an infinite cardinal number. Then we know the following. $$\kappa<\kappa^{\mathrm{cf}(\kappa)}$$ Question. Is it true that if $$\kappa < \kappa^\nu,$$ then ...
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2answers
45 views

Is a limit ordinal necessarily a cardinal?

Maybe this is a trivial question. I see that every infinite cardinal is necessarily a limit ordinal, but is the converse true ?
3
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1answer
64 views

What are the fixed points of cardinal exponentiation?

Whenever $\kappa$ is an infinite cardinal number, write $\mathrm{cl}_\kappa$ for the unique function $\mathrm{Card} \rightarrow \mathrm{Card}$ given by $\mathrm{cl}_\kappa(\nu) = \nu^\kappa.$ It ...
0
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1answer
33 views

$A<_c B \implies P(A)<_c P(B)$

Just started studying set theory. It's seems to me intuitivly correct that if $A<_c B \implies P(A)<_c P(B)$ where $_c$ is the cardinality of a set and $P(\cdot)$ is the powerset. Am I right? I ...
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2answers
62 views

What should Aleph-Two mean?

Just curious, what should Aleph-Two mean? I know that Aleph-One is distinct from Aleph-Null and Aleph-One is not countable, but does Aleph-Two mean?
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1answer
41 views

Is the cardinality of an ordinal, $\alpha$ less than $\alpha$?

Let $k^+=\{\alpha : \alpha \hspace{2mm} \text{is an ordinal,and} \hspace{2mm} |\alpha|<\kappa\}$ where $k^+$ is an infinite cardinal. I am trying to understand the proof the the following ...
2
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1answer
23 views

Given cardinal numbers $\kappa < \nu$ with $\nu$ an aleph fixed-point, do we necessarily have $\aleph_\kappa < \nu$?

Given cardinal numbers $\kappa < \nu$ with $\nu$ an aleph fixed-point, its clear that $\aleph_\kappa \leq \nu$. Is this inequality in fact strict?
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1answer
40 views

Is the following characterization of weak inaccessibility correct?

Let us accept the von Neumann cardinal assignment for this question. Furthermore, given a cardinal number $\kappa$, let us write $2^\kappa$ for the unique cardinal number isomorphic to the powerset of ...
3
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1answer
84 views

Proving König's lemma (technical problems)

the aim of my exercise is to give a proof of the König's lemma. So, let $\kappa, \lambda$, be cardinals such that $cf(\kappa)\leq \lambda$. My professor's suggested us to prove that there exists a ...
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1answer
85 views

Do we need choice to prove that $|\mathbb{N} \times A| = |A|$ for all infinite sets $A$?

I can't think of any way to prove it without choice.
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1answer
41 views

How can we show that $\omega_1$ is a regular cardinal?

A cardinal $\kappa$ is regular if and only if there is no $\lambda<\kappa$ for which there is a function $f:\lambda\rightarrow\kappa$ with range cofinal in $\kappa$. How can we see in ZFC that ...
0
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2answers
47 views

Determining the cardinality of these sets.

I am having trouble with determining the cardinality(finite, denumerable, uncountable) of these two sets: The set of all circles in $\mathbb{R}^2$ in form $(x-a)^2+(y-b)^2=R^2$ with ...
3
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2answers
43 views

Cardinality of the set of at most countable subsets of the real line?

I'm exploring an unrelated question about power series with complex coefficients. While exploring this question, I wondered: What is the cardinality of the set of all such power series? Or with ...
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3answers
47 views

Cardinality of the set of all involutions from $\mathbb N$ to itself

The following is a section in my homework, I couldnt solve it so I'm asking for some help. I have the following set : $\{f:\mathbb N \to \mathbb N | f(f(a)) = a \text{ for all } a\in \mathbb N\}$. I ...
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1answer
41 views

Preserve Cardinals and Adding No Bounded Subsets

In Chapter 15 at the bottom of page 228 of $\textit{Set Theory}$ by Jech, he writes that if $\kappa$ is a cardinal in $V$ and if $\kappa$ has no new bounded subsets in $V[G]$, then $\kappa$ remains a ...
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1answer
61 views

How many ways can we totally order a set $X$ (but not necessarily the whole set $X$, perhaps just a subset) up to isomorphism?

Here's Attempt #2 at asking this question. Suppose $X$ is a set (not necessarily finite), and let $T$ denote the collection of all totally ordered sets $(A,\leq)$ such that $A \subseteq X.$ Now let ...
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3answers
52 views

Prove that $\{(a,b):a,b\in\mathbb N, a\geq b\}$ is denumerable.

If $S=\{(a,b):a,b\in\mathbb N, a\geq b\}$, how do I prove that $S$ is denumerable? Work: Since $S \subseteq\mathbb{N\times N}$ I know that $S$ is denumerable. But I don't know how to structure the ...
2
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2answers
76 views

Proving that (0,1) and [0,1] are numerically equivalent.

as the title suggests, I need help proving that the cardinality of $(0,1)$ and $[0,1]$ are the same. Here is my work: $f:[0,1] \rightarrow (0,1)$ Let $n\in N$ Let $A=\{\frac{1}{2}, \frac{1}{3}, ...
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2answers
42 views

cardinality with finite sets

$A,B,C$ are finite sets. Suppose $A\subseteq B \subseteq C$ and $\#A=\#C$. Prove that $\#A=\#B$ and $\#B=\#C$. Should I prove this by showing that there exist an element in $A$ that exist in $B$ and ...
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2answers
43 views

Cardinality of $\lim_{k\to\infty}\mathbb N^k$ vs. $\mathbb N^\infty$

My friend and I are having a disagreement over whether the number of terms in the following series is countable or uncountable: $$\sum_{i=1}^\infty a_i + \sum_{i=1}^\infty\sum_{j=1}^\infty a_{ij}+ ...
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0answers
15 views

If $X$ is a finite set of cardinality $n$, where $n$ exists in $P$, show that the following conditions on a function $f: X \to X$ are equivalent: [duplicate]

(a) $f$ is an injection (b) $f$ is a surjection (c) $f$ is a bijection I know that (c) implies (a) and (b) and (a) and (b) imply (c). I also have the following definition that I've been playing ...