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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CH is preserved under a $Fn(\kappa,\lambda)$ forcing? (Kunen IV.7.10)

The following is Exercise IV.7.10 in the 2013 edition of Kunen's "Set Theory": Let $M$ be a countable, transitive model for ZFC. In $M$, let $\aleph_0 \le \kappa < \lambda$ be cardinals and ...
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34 views

Is it possible to create a bijection between all pairs of reals and a real? [duplicate]

The title basically says it all. Is it possible to associate with each pair of reals, another unique real? I guess you could say I'm looking for functions of two real arguments that return a ...
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2answers
56 views

How many infinite subsets of N are there anyway? [duplicate]

I was reading 2 proofs one that the powerset of $ N$ has a higher cardinality than $N$ two a proof that the cardinality of the set of all finite subsets of $N$ has the same cardinality than $N$ ...
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1answer
74 views

Can Aleph Numbers be multiplied?

i.e., does it make sense to say something like $(2 * \aleph_0) > \aleph_0$ ? The original question I was thinking about is: if A = $\mathbb{Z}$ and B = {the set of even integers} is it correct to ...
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1answer
37 views

What is the cardinality of a set of all finite subsets of $\Bbb{N}$? [duplicate]

I'm looking for cardinality of $P_{fin}(\Bbb{N})=\{x|x\subset\Bbb{N}$ and $x$ finite$\}$. I was told in my classes that it's $\aleph_0$, but how to prove it?
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1answer
45 views

Showing that the class of all sets of a particular cardinality is not a set.

How to show that the class of all sets of a particular cardinality ,say $h$ is not a set. My argument: I assume that I've shown the following lemma. Lemma: If $X$ is an infinite set of cardinality ...
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1answer
58 views

Cardinality of the set of all (real) discontinuous functions

This is a question from the book Introduction to Set Theory (Hrbacek and Jech), chapter 5, question 2.6. (Show that) The cardinality of the set of all discontinuous functions is ...
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1answer
37 views

Cardinality of sets: $|A|\le|B|\Rightarrow(|A\cup B|=|B|\land|A\times B|=|B|)$

My book of mathematical logic states the facts that, if we call $|X|$ the cardinality of set $X$, then, for any two sets $A,B$ such that $|A|\le|B|$, $$|A\cup B|=|B|\quad\text{ and }\quad|A\times ...
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1answer
30 views

Proving that if $A\leq_c B$ then $\chi(A)\leq_o \chi(B)$

By Hartog's Theorem we knoe that for every set $A$ There is a definite operation $\chi(A)$ which associates with each set $A$, a well ordered set $\chi(A) = (h(A),{\leq_{\chi(A)}}),$ such that $h(A) ...
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0answers
42 views

Connection beween closure property in forcing and preservation of $H_{\kappa}$

I would like to know about the relation between closure property of forcing notions and preservation of hierarchy of hereditary small sets, $\langle H_\lambda \mid \lambda\in \mathrm{Card}\rangle$, at ...
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2answers
184 views

Understanding proof of Hartogs’ Theorem on Set Theory

I'm trying to understand the proof of the Hartogs’ Theorem on page 100 of this book. My especific question is: If we have for each set $A$ $$\mathrm{WO}(A)=\{ (U,\leq_{U}) \, | \, U\subseteq A \, ...
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2answers
57 views

Injections from all ordinals into a set $X$

We are working in $\mathsf{ZF}$. Let $X$ be a set. Let $A$ be the class of all injections $f: \alpha \to X$ for arbitrary ordinals $\alpha$. I am quite sure that, in fact, $A$ is a set, since if ...
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0answers
25 views

Connectedness ( cardinality and connectedness) [duplicate]

$(X,d)$ metric space and $A\subset X$ and $A$ is connected. $$ \text{Card}(A) > 2 \implies \text{Card}(A) \geq \text{Card}(\mathbb{R}).$$ How do I prove it ?Waiting for your help?
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2answers
31 views

what is the cardinality of powerset of a union set?

Is there exist something like P(X+Y) (P STANDS FOR POWERSET)? I am confuse because power set is the set of all subset of Cartesian product, and X+Y wont give Cartesian product but (x,0) U (y,1), and ...
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1answer
25 views

Find the maximum number of a continuous function

Lets define a function $z:\mathbb{R}^\mathbb{R}\to\mathcal P(\mathbb R)$ that gives you the set of zeros of any $\mathbb R ^\mathbb R$ function. Now, we define a set $S=\{z(f):f\in\mathbb R ^\mathbb ...
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2answers
47 views

$\aleph_0 \aleph_1 =\aleph_1$? But I don’t know any way to prove or disprove it

What is the value of $\aleph_0 \aleph_1$? Clearly $\aleph_0\le \aleph_1$ implies $\aleph_0=\aleph_0\aleph_0\le \aleph_1 \aleph_0$ and again $\aleph_0 \aleph_1\le \aleph_1 \aleph_1=\aleph_1$. But ...
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1answer
49 views

I can't find the mistake in this argument (Cofinality and König's theorem)

I have some trouble explaining this apparent contradiction: we know that given $k>\aleph_0$ an infinite cardinal, $\mu=cof(k)$ is the minimum cardinal such that $k=\sum_{i\in \mu} k_i$ where ...
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1answer
32 views

If $\alpha$, $\beta$ are finite cardinals such that both are greater than $1$ and $\gamma$ is an infinite cardinal then $\alpha^\gamma=\beta^\gamma$.

Can someone suggest a rigorous proof of the following: If $\alpha$, $\beta$ are finite cardinals such that both are greater than $1$ and $\gamma$ is an infinite cardinal then $\alpha^\gamma = ...
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1answer
72 views

Cofinality of $2^{\aleph_\omega}$

Is the following statement correct: $\operatorname{cf}(2^{\aleph_\omega})=\aleph_0$? It appears in the "Jech" book. Wikipedia however states that $\operatorname{cf}(\aleph_\omega)=\aleph_0$. The ...
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1answer
71 views

Is it true that $({\aleph_1})^{\aleph_0} = {\aleph_1}$?

Is it true that $({\aleph_1})^{\aleph_0} = {\aleph_1}$? My scenario is as follows: The cardinal number of $\mathbb R$ is $|\mathbb R|={\aleph_1}$ and the cardinal number of the Cartesian product of ...
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1answer
18 views

closed unbounded set,regular cardinals,cofinality

Given two regular cardinals $\lambda>\mu$, why this set is a closed unbounded set in $\lambda$? {$\alpha$ | cf($\alpha$)=$\mu$ , $\alpha<\lambda$}
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1answer
108 views

Are injections harder to find than surjections?

Given two finite sets $A$ and $B$ with $|A|<|B|$ There are more functions from $B$ to $A$ than from $A$ to $B$ except when $|A|=1$ or $|A|=2,|B|=3,4$. See here for proof. It is also true there are ...
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3answers
399 views

Are there fields and ordered fields of every infinite cardinality?

On the top of my head, I cannot think of any fields of cardinality more than that of the reals. (It is known that the process of algebraic closure does not increase the cardinality of an infinite ...
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0answers
18 views

Mapping vectors to real numbers [duplicate]

Does there exist an invertible mapping that takes n-dimensional vectors (RN) to real numbers (R). Any countable set can be mapped in this way to another countable set. This is the sense in which ...
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1answer
66 views

Notation about cardinals

Does a cardinal $\mathcal{k}$ such that $2^\mathcal{k}=k^+$ have any special name? I never encountered any name for this property, but I think it is possible they have one.
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1answer
23 views

Prove that if A~B then Sym(A)~Sym(B).

I tried to prove it with sets. Really, truly clumsy. I know |A|=|B|. Can I simply conclude that |A|!=|B|! => Sym(A)~Sym(B)?? (Sym(A) for a set A is the set of all bijections from A to A.)
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27 views

Prove $α · β ≤ α · γ$ and $α^ β ≤ α^ γ$ for any three cardinals, where $ β ≤ γ$.

This is what I did: a. let $|A|=α, |B|=β, |C|=γ. |B|≤|C|$ and therefore there is an injection $f: B \to C$, such that $f(b)=c$ for some $b \in B, c \in C.$ Upgrading this function to ...
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3answers
87 views

Prove that $2^\aleph+\aleph$ equals $2^\aleph$.

How do I show this elegantly? I can't seem to find the right sets for it... Maybe there are some substantial laws I could use? I would appreciate your help... What I said is: $2^\mathfrak c$ is for ...
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98 views

Do an infinite set and its double have the same cardinality? [duplicate]

My question was inspired by this answer. Suppose $A$ is an infinite set. Does its double, $A\times\{0,1\}$, always have the same cardinality? In my head I quickly spotted a simple proof that the ...
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1answer
26 views

Transfinite fixed points of a function

Let the function $F\colon On \rightarrow On$ be defined by the following recursion: $F(0) = \aleph_0$ $F(\alpha+1) = 2^{F(\alpha)}$ (cardinal exponentiation) $F(\lambda) = \sup\{F(\alpha): \alpha ...
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4answers
27 views

Why we can conclude immediately that $x \in B$, if $(x, y) \in A \times B = B \times A$

The following statements are part of a proof involving cartesian products, specifically involving this theorem: $A \times B = B \times A \iff$ either $A = \emptyset$, $B= \emptyset$, or $A = B$ ...
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2answers
62 views

Showing the set of functions $\{0, 1\} \to \mathbb{N}$ is countably infinite.

I'm doing a question it asked me to show that $\mathbb{N} \times \mathbb{N}$ was countably infinite but I am stuck on the following part of the question: deduce that the set of all functions $f : ...
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3answers
67 views

Power of sets - $\{0,1\}^\mathbb{N} \simeq \mathbb{N}^\mathbb{N}$ [duplicate]

I've got a problem with prove about cardinality of sets. How can I prove that $\lbrace 0,1 \rbrace^\mathbb{N} \simeq \mathbb{N}^\mathbb{N}$?
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1answer
84 views

Does $\aleph_0\cdot\kappa=\kappa$ for every $\kappa\ge\aleph_0$ hold in ZF?

It is easy to show that for any (Dedekind) infinite cardinal $\kappa$ we have $\aleph_0+\kappa=\kappa$. Definition of an infinite cardinal is a cardinal such that $\aleph_0\le\kappa$. (I believe ...
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2answers
61 views

Cardinality of the countably infinite product of a two-point set $\{0,1\}$?

I'm so confused about cardinalities of some sets. What is the countable infinite product of a two points set $\{0,1\}$? Does it have the same cardinality as the real number $\mathbb R$? Or is the ...
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1answer
78 views

Prove that $\forall\alpha\geq\omega$, $|L_\alpha|=|\alpha|$ without AC

Without using Axiom of Choice, prove that $$\forall\alpha\geq\omega,~~|L_\alpha|=|\alpha|,$$ in which $\alpha$ is an ordinals, $\omega$ is the set of natural numbers, $L_\alpha$ is the $\alpha$-th ...
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3answers
61 views

How to show that if $P\subseteq Q$ are finite sets, and $\#P=\#Q$, then $P=Q$?

Let $P, Q$ two finite sets such that : $$P \subset Q$$ $$\#P = \#Q$$ How do you show that $P = Q$ ? I don't see how I can show that $Q \subset P$
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3answers
86 views

Prove that R and P(R) does not have the same Cardinality.

How can I show such a thing in an elegant, valid way? I know it shouldn't be hard. Well it does have to be plausible and mathematically logic, but I guess I am not expected to rediscover the great, ...
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0answers
67 views

What is an undefinable ordinal? How large are the least undefinable ordinal and supremum of all undefinable ordinals?

What does it mean to say that a particular ordinal/cardinal number is "definable"? Where and how do we define this "definability"? If there are undefinable ordinals/cardinals, how large are the least ...
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1answer
28 views

Are normed spaces isodyne?

In general, do all non-empty open subsets of a normed space necessarily have the same cardinality?
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46 views

Cardinal Arithmetic proof issues.

Let $X$ be a finite set and let $x$ be an object which is not an element of $X$. Then $X \cup \{x\}$ is finite and $|X \cup \{x\}| = |X| + 1$. Proof. Let X be a finite set with cardinality n, ...
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1answer
40 views

What is the cardinality of the class of $0$ in $\mathbb{R}$?

What is the cardinality of the class of $0$ in $\mathbb{R}$? In other words: what is the cardinality of the class of all rational Cauchy sequences that converge to $0$?
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2answers
102 views

How are some infinities larger than other infinities

I heard an expressions, some infinities are larger than others recently, and they stated that it was proved to be so. I haven't been able to find this proof, and ...
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1answer
33 views

Prove that if $\: \forall_{t \in \mathbb{R}} \: \overline{\overline{A}}_t = c$ then $\overline{\overline{\bigcup_{t \in \mathbb{R}} \: A_t}} = c$

Prove that if $\: \forall_{t \in \mathbb{R}} \: \overline{\overline{A}}_t$ is equal to cardinality $c$, then $\:\overline{\overline{\bigcup_{t \in \mathbb{R}} \: A_t}}$ is also equal to cardinality ...
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1answer
23 views

Equalities of cardinal numbers

I need prove that: $2^{\aleph_{0}}=n^{\aleph_{0}}=\aleph_{0}^{\aleph_{0}}=c^{\aleph_{0}}=c$, for $n\geq2$. Where $c$ is the continuum. I know that $2^{\aleph_{0}}\leq ...
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1answer
51 views

Notation for the class of all cardinals

I have seen the notation for the class of all ordinals to be $\rm Ord$ or $\rm On$, is there an analogous notation for the class of all cardinals?
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1answer
40 views

If $2^{\text{cf } \kappa} < \kappa$, then $\kappa^+$ is the least possible value of $\kappa^{\text{cf } \kappa}$

I would like to show that if $2^{\text{cf } \kappa} < \kappa$, then $\kappa^+$ is the least possible value of $\kappa^{\text{cf } \kappa}$ where $\kappa$ is an infinite cardinal. I'm certain it ...
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2answers
51 views

Does ${(x,y)\in\mathbb{R}\times\mathbb{R}:x,y\in\mathbb{Z}}$ have cardinality $\aleph_0$ or $c$?

So here's my intuition. Letting $S=\{(x,y)\in\mathbb{R}\times\mathbb{R}:x,y\in\mathbb{Z}\}$, $\bar S=\aleph_0$ because $(x,y)\in\mathbb{Z}\times\mathbb{Z}$, which can be shown to be equivalent to ...
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1answer
111 views

$H(\kappa)$ a model of all the axioms of ZFC for $\kappa$ not inaccessible

Is this possible? For each cardinal $\kappa$, we can define $H(\kappa) = \{ x \mid |trclx| < \kappa\}$, where trcl is the transitive closure, and $|A|$ is the cardinality of $A$. For every regular ...
3
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1answer
51 views

Cardinality of a set of functions from $\mathbb N$ to a set of real numbers

Let $S$ be the set of functions from $\mathbb N$ to the set $\{\sqrt{2}, \sqrt{3}, \sqrt{5}, \sqrt{7}\}$. Determine $|S|$. I understand how to prove the converse case where $T$ is the set of ...