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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Clarification on the proof of Theorem 8. 11 (Hungerford)

If $\alpha$ and $\beta$ are cardinal numbers such that $0\neq \beta \leq \alpha$ and $\alpha$ is infinite, then $\alpha\beta=\alpha.$ Sketch: Let $A$ be an infinite set with $|A|=\alpha$ and let ...
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16 views

Let $A$ be any non-empty set and $\alpha$ be an infinite cardinal . When can we say $|A^\alpha|=\alpha$ ?

Let $A$ be any non-empty set and $\alpha$ be an infinite cardinal . When can we say that the cardinality of $A^\alpha$ ($A \times A\times ...$ $\alpha$ times ) is $\alpha$ ? When $\alpha > |A|$ , ...
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1answer
60 views

Does $\lambda^2 \leq \kappa^2 \Rightarrow \lambda \leq \kappa$ imply the axiom of choice?

I'm aware that the statement "for all cardinals $\kappa$, $\kappa^2 = \kappa$" is equivalent to the axiom of choice (I believe this was proved by Tarski). More generally, does anyone know if the ...
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3answers
64 views

Let $V,W$ be two countably infinite dimensional vector space over the same field , then are $V,W$ isomorphic as vector spaces?

Let $V,W$ be two countably infinite dimensional vector space over the same field , then are $V,W$ isomorphic as vector spaces ? And please give example of two non-isomorphic uncountable dimensional ...
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1answer
65 views

Enderton's Elements of Set Theory Scott's Trick Exercise (page 207 problem 31)

31. Define kard $A$ to be the collection of all sets $B$ such that (i) $A$ is equinumerous to $B$, and (ii) nothing of rank less than rank $B$ is equinumerous to $B$. (a) Show that kard $A$ is ...
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2answers
62 views

Halmos' Naive Set Theory Cardinal Arithmetic Exercise

On page 95 of Halmos' Naive Set Theory, we get the exercise If $\{a_i\}$ and $\{b_i\}$ are families of cardinal numbers such that $a_i< b_i$, then $$\sum_i a_i<\prod_ib_i$$ I know that we ...
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1answer
40 views

$\operatorname{Card}(X) \leq\operatorname{Card}(Y)$ iff $\aleph (X) \leq \aleph(Y)$

For any two sets $X$ and $Y$, we write $\operatorname{Card}(X)\leq\operatorname{Card}(Y)$ if an injection $X \rightarrow Y$ exist. I have tried Suppose $\aleph (X) \leq \aleph(Y)$, where $\aleph ...
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3answers
46 views

Set cardinality, function onto, open unit square maps into real number set

I have a question in my homework that I have trouble solving it. I'm not sure if I understand the question actually. I'll attach the question below and hope someone could give me any hints. Consider ...
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0answers
37 views

A question regarding Silver's Theorem

I am reading Jech's book Introduction to Set Theory. As an introduction to Silver's theorem, it is stated that: If $\aleph_\alpha$ is a strong limit cardinal and if $k < \aleph_\alpha$ and ...
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1answer
37 views

How to prove $cf(\aleph_{\omega_1})=\omega_1$?

I am reading Jech's & Hrbacek's book "Introduction to set theory" and this is question 2.2 from chaper 9: Prove $cf(\aleph_{\omega_1})=\omega_1$ Isn't it clear from the fact that ...
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1answer
80 views

How do I show the existence of a weakly inaccessible cardinal is not provable in ZFC?

So I know that if we assume the existence of inaccessible cardinals than we can show ZFC is consistent. How do I show that the existence of such cardinals is not provable in ZFC?
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26 views

determine the cardinality of $\{C \subseteq \mathbb N \space|\space \mathbb N - C \text{ is finite}\}$ [duplicate]

what is the cardinality of this set : $\{C \subseteq \mathbb N \mid \mathbb N - C \text{ is finite }\}$ So it must mean that $C$ is infinite, but even though its infinite we know how ...
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1answer
73 views

Why we use $\mathrm{Fn}(\kappa\times\lambda,2,\lambda)$ to force $2^\lambda\ge\kappa$ instead of $\mathrm{Fn}(\kappa\times \lambda,2)$

I am reading Kunen's Set Theory and I learn that, $\operatorname{Fn}(\kappa\times\omega,2)$ forces $2^{\aleph_0}\ge|\kappa|$, where $$\operatorname{Fn}(I,J,\lambda)=\{p:p\text{ ...
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1answer
132 views

ZFC + not-CH, can a set with cardinality between $\aleph_0$ and $2^{\aleph_0}$ be defined?

Continuum hypothesis states, there is no set with cardinality between the integers and the reals. There is a milestone result, that CH is independent from ZFC. That means, both of ZFC + CH, and ZFC + ...
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1answer
35 views

How to determine cardinality of an infinite set using Aleph numbers?

So I was reading a little bit about cardinal infinities, and I thought it was pretty interesting. However I wanted to know a little bit more about how to use them. For example, how would I determine ...
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1answer
53 views

Powers of $\mathfrak{c}^+$

Denote by $\mathfrak{c}^+$ the cardinal successor of continuum. Can we prove in $\mathsf{ZFC}$ that $(\mathfrak{c}^+)^{\aleph_0} = \mathfrak{c}^+$? I guess not. Of course this question is ...
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1answer
24 views

If $f\colon\kappa\rightarrow\kappa$, then the set of all $\alpha < \kappa$ such that $f(\xi) < \alpha$ for all $\xi < \alpha$ is closed and unbounded.

If $f\colon\kappa\rightarrow \kappa$, then the set of all $\alpha < \kappa$ such that $f(\xi) < \alpha$ for all $\xi < \alpha$ is closed and unbounded. This is from Jech's book (page 103) so ...
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0answers
28 views

Ordinals that are not cardinals [duplicate]

I am reading Jech's set theory and he defines a cardinal number as an ordinal $\alpha$ (a cardinal) if $|\alpha| \neq |\beta|$ for all $\beta < \alpha$, and he says that all infinite cardinals are ...
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1answer
23 views

bijective, one-to-one, and number of elements

How does one reconcile the following (seemingly) contradiction in using "number of elements" argument? In the "range" [0,1] in R there are more points than in N, to be shown as "take the inverse of ...
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2answers
30 views

Cardinality of the set of all complex sequences converging to zero.

I was asked to show that the set of all complex sequences converging to zero has the same cardinality as the set [0,1]. This is the only hole in a proof that I am working on. I need to show there ...
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1answer
94 views

A club-guessing exercise

I came across this club-guessing exercise on Cardinal Arithmetic by Abraham and Magidor in the Handbook of Set Theory. Let $\kappa, \lambda$ be regular cardinals $\kappa^{++}<\lambda$ and let ...
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1answer
44 views

Question on Komjáth's “three clouds may cover the plane”

I am reading a wonderful paper by Komjáth, "Three clouds may cover the plane," and am having difficulty proving that certain sets are countable. Assume CH (the continuum hypothesis) holds. Let ...
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59 views

Vector spaces over Z/2Z,countably many predicates added,invariant

Suppose we have a class of vector spaces over $Z/2Z$ with predicates $P_n,n\in N$ added for independent subspaces of co-dimension 2. (BTW, independent in which sense,$P_n$ is independent with ...
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1answer
40 views

Does any set admit a total order? [duplicate]

Is it true that any set $P$ can be endowed with a total order $"\leq" \subseteq P\times P$?
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2answers
37 views

Prove there is either a chain or an antichain of infinite cardinal.

Let $K$ be an set of infinite cardinal, $X$. Let $(K,\le)$ be a partially ordered set. Prove there is either a chain $C$ such that $|C|=X$ or there is an antichain $A$ such that $|A|=X$. I guess I ...
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1answer
41 views

Prove $A$ is either finite or countable.

Let $A$ be a set and let $f:A \to \Bbb{N}$ be an injection. Prove $A$ is either finite or countable. How can one prove such a thing? Since this is so obvious it gets me in a place where I don't know ...
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1answer
20 views

Find $|f^{-1}(\emptyset)|$ where $f: P(\Bbb{R})\to P(\Bbb{N})$, $f(X)=X\cap \Bbb{N}$.

Let $f: P(\Bbb{R})\to P(\Bbb{N})$, $f(X)=X\cap \Bbb{N}$. Find $|f^{-1}(\emptyset)|$. Prove that $|f^{-1}(\emptyset)|=|f^{-1}(\Bbb{N})|$. I am having a difficulty solving 2., but this is what I ...
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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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32 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
47 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
33 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
39 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
44 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
37 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
169 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
55 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
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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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28 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
23 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
46 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
43 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
29 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 = ...
3
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1answer
59 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
64 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
15 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
107 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 ...
3
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3answers
385 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 ...