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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2
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3answers
324 views

Is the powerset of the reals any “more uncountable” (in some sense) than the reals are?

I know that $\mathbb{N}$ is countable and has cardinality $\aleph_0$, and that $\mathbb{R}$ has cardinality $2^{\aleph_0} = \text{C}$ and is uncountable. Are sets with cardinalities greater than ...
4
votes
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 ...
-1
votes
1answer
29 views

Let $A$ be an uncountable set and let $B$ be a nonempty set. Prove that the cardinality of $A\times B$ is uncountable. [on hold]

If $A$ is an uncountable set and $B$ is a nonempty set, how do I prove that $A\times B$ is uncountable? Also, what is the cardinality of $A-B$? Is it also uncountable?
4
votes
1answer
56 views

Inequality in cardinal function: $|X|\le 2^{s(X)\psi(X)}$

How to prove that $|X|\le 2^{s(X)\psi(X)}$ by using the Erdős-Rado theorem when $s(X)=\psi(X)=\omega$? $s(X)=\sup \{ |D|: D \subset X, D \text{ is discrete} \} + \omega $ $\psi(X)= \sup\{\psi(p,X): ...
0
votes
0answers
24 views

Filter,dual ideal,definition of $\liminf_I \lambda$

We have this http://shelah.logic.at/files/506.pdf definition of $\lim \inf_I \bar{\lambda}$ in 1.1(3): for $I$ a filter on $\kappa$ let $I^+=2^\kappa \setminus I$.$$\lim \inf_I ...
-4
votes
0answers
158 views

Proving that 2 intervals have the same cardinality [closed]

How can I Prove that the invervals [0, 1) and (0, 2] have the same cardinality by finding a bijection between them? And how can I Prove that the intervals (0, 1) and [0, 1] have the same cardinality ...
0
votes
0answers
21 views

What is the Cardinality of all symmetric density function pairs on reals?

$X=$(total number of all pairs of probability density functions $(f_0,f_1)$ on the real numbers) and let $Y=$(total number of all symmetric probability density functions $(f_0,f_1)$ on the real ...
6
votes
1answer
96 views

For an infinite cardinal $\kappa$, $\aleph_0 \leq 2^{2^\kappa}$

I'm trying to do a past paper question which states: $$ \text{For all infinite cardinals $\kappa$, we have } \aleph_0 \leq 2^{2^\kappa}. $$ I'm supposed to be able to do this without the axiom of ...
1
vote
1answer
52 views

Proving $k^{m+l} = k^m k^l$ by constructing a bijective function F : $ ^MK \times ^LK \to ^{L\bigcup M}K $

For cardinals k which is cardinal of K and l which is cardinal of L and m which is cardinal of M. W.T.S [ $ k^{m+l} $ = $ k^m k^l $] by constructing a bijective function F : $ ^MK \times ^LK \to ...
9
votes
1answer
307 views

Is there an easy proof for ${\aleph_\omega} ^ {\aleph_1} = {2}^{\aleph_1}\cdot{\aleph_\omega}^{\aleph_0}$?

The question contains 2 stages: Prove that ${\aleph_n} ^ {\aleph_1} = {2}^{\aleph_1}\cdot\aleph_n$ This one is pretty clear by induction and by applying Hausdorff's formula. Prove ...
1
vote
1answer
42 views

Saturated models and $\kappa=\kappa^{<\kappa}$

Do not assume GCH. Can you characterize the cardinals $\kappa$ such that every theory $T$ with an infinite model has a saturated model of cardinality $\kappa$? I guess these are the cardinals such ...
1
vote
2answers
141 views

Is the set of all pairs of real numbers uncountable?

My hypothesis is that $\mathbb{R \times R}$, the set of all pairs $(r_1, r_2)$, of real numbers is uncountable. I understand that the set of all pairs of natural numbers is countable. But could ...
12
votes
3answers
734 views

Are there more groups than rings?

It seems pretty clear to me that both of these are at least uncountable (which I think I could prove with some work). It also seems that you should be able to make some diagonal argument about the ...
1
vote
1answer
85 views

Is this behavior of the continuum function consistent with ZFC.

I have an axiom, based on how finite cardinalities work. As we know, every ordinal can be written as the sum of a unique limit ordinal $L$ (where $0$ is a limit ordinal axiom for the purposes of this ...
4
votes
1answer
61 views

What is the degree of a real closure of an ordered field?

Given a subfield $F$ of $\Bbb R$, let the real-closure of $F$ be the smallest subfield of $\Bbb R$ which is real-closed and extends $F$. Since the theory of real-closed fields is complete, this is in ...
1
vote
2answers
71 views

How to prove that $C\cdot\aleph_0=C$

How can I prove that $C\cdot\aleph_0=C$? I tried this: Given that $k\cdot 1=k$ and $C\cdot C=C$ if $C\cdot C = C \wedge C\cdot 1 = C \wedge C>|\mathbb N|>1$ then $C\cdot |\mathbb N|= C$ c is ...
2
votes
0answers
27 views

Existence of infinite set and axiom schema of replacement imply axiom of infinity

I'm self-teaching an intro to set theory course, and came across this exercise: Show that the existence of an infinite set is equivalent to the existence of an inductive set. For the notion of ...
13
votes
0answers
119 views

There's no cardinal $\kappa$ such that $2^\kappa = \aleph_0$

I am trying to prove that there is no cardinal $\kappa$ such that $2^\kappa = \aleph_0$ . My attempt: We suppose it exists. Since $\kappa<2^\kappa$, in particular, $\kappa<\aleph_0$. But ...
0
votes
3answers
33 views

Is the collection of all cardinalities a set or a proper class? [duplicate]

Is the collection of all cardinalities a set or a proper class? Does anybody ever think about the problem?
0
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0answers
17 views

Cardinality relation between reducible sets

Suppose we are considering natural numbers, set $A$ and $B$ are two subsets of the natural number set, suppose set $A$ is many-one reducible to set $B$, i.e. there is a total computable function $f$ ...
1
vote
1answer
47 views

If $|A|<|B|$ does $B$ surject onto $\aleph(A)$?

After reading Proving existence of a surjection $2^{\aleph_0} \to \aleph_1$ without AC I became curious if there is a generalization to arbitrary cardinals. That is, if $\frak m<n$, does it follow ...
3
votes
1answer
49 views

Given the axiom of choice, are cardinals ordinals?

Given a model of ZFC, is it correct to talk indistinctly about cardinals and initial ordinals, namely, ordinals $\alpha$ such that for every $\beta < \alpha$, there is no bijection between $\alpha$ ...
1
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3answers
24 views

Cardinality of the set of finite sets in the power set of natural numbers [duplicate]

It is known that $|2^\Bbb{N}|=|\Bbb{R}|$ and that $2^\Bbb{N}$ contains all the subsets of $\Bbb{N}$, just an idea of a question I had and that I would like suggestions on how to tackle. My question ...
6
votes
4answers
95 views

Is the logarithm of $\aleph_0$ infinite?

In classical mathematics $2^{\aleph_0}=\aleph_1$, right? So if $2^x=\aleph_0$, what does $x$ equal? In other words, can we define a logarithm for $\aleph_0$, and what should it be. Is it infinite? ...
6
votes
1answer
148 views

Dominating strategically $\omega_1$ reals

For a given $\kappa > \omega$, define the game $d(\kappa)$ that runs for $\omega$ stages as follows: At stage $n$, player I chooses a sequence of elements of $\omega$, $g_n$ of length $\kappa$, and ...
4
votes
1answer
82 views

How to prove that $\sf CH$ implies $2^{\aleph_0}=\aleph_1$

Of course, most of you will, upon reading the title, exclaim "But isn't that the definition of the continuum hypothesis?" So I need to be a little more careful about the exact definitions. Let ...
2
votes
1answer
15 views

Can every infinite cardinal $\mu$ such that $\kappa\leq\mu\leq2^\kappa$ be expressed as $\kappa^\lambda$?

Let $\kappa$ be an infinite cardinal. Can I reach every intermediate cardinal $\mu$ with $\kappa \le \mu \le 2^\kappa$ as some power $\kappa^\lambda$? If not, is there another construction that ...
2
votes
3answers
52 views

Show that if a metric space is complete, separable and not countable then it has cardinal $\aleph_1$

Show that if a metric space is complete, separable and not countable then it has cardinal $\aleph_1$ I have encountered this exercise and I don't know where to start. There is a lot of important ...
1
vote
3answers
54 views

Are cardinal numbers sets in ZFC?

Are cardinal numbers sets in ZFC, or just proper classes? If they are sets, what is their structure?
3
votes
1answer
31 views

Proving the equivalence of a finite set

Let A be a finite set. Prove that if A≈􏰔n and A≈􏰔m, then n=m. The answer in the book uses a max function, so I was just wondering if there was a simpler way. If not, it would be appreciated if ...
1
vote
1answer
31 views

Question about $\text{non}(\mathsf{nonstat}_\gamma)$

A lemma in Kunen's (2011) set theory states that if $\gamma$ is a limit ordinal with $\kappa=\text{cf}(\gamma)>\omega$, then $\text{add}(\mathsf{nonstat}_\gamma)$ $=$ ...
4
votes
0answers
38 views

finding a bijective function from the real plane to the real line

As part of a HW assignment in the course elementary set theory, I was given the following question: Prove explicitly (don't use any theorems or known facts, but find a bijective function) that ...
0
votes
0answers
22 views

finding an injective function to prove cardinality equality

As part of a HW assignment in the course elementary set theory, I was given the following question: Prove that the set of all binary sequences (sequences of $0$ and $1$) except for the binary ...
5
votes
1answer
63 views

Cardinality of order topology?

Just out of interest: The cardinality of the Euclidean topology on the real line is $c$. In general, if $X$ is totally ordered of cardinality $\alpha$, the order topology on $X$ must have cardinality ...
0
votes
2answers
55 views

The number of subsets of cardinality less than $\kappa$ of a cardinal $\kappa$ is $\kappa$

Let $L=\bigcup_{\alpha \in Ord} L_\alpha$ be Godel's constructible universe and thus $L \models GCH$. Let $\kappa$ be an infinite cardinal and $S:=\{A \subseteq \kappa : \#A < \kappa \}$. Is it ...
2
votes
1answer
66 views

equality of Cardinality of $\mathbb{R}$ and $\mathbb{R^2}$

There was a question in our exam which wanted us to prove that $\mathbb{R}$ and $\mathbb{R^2}$ both have same Cardinality. My approach to prove this problem was to try to make a bijection between ...
4
votes
1answer
39 views

Is there a simple formula for the cardinality of $\{A\subseteq\kappa\mid |A|\leq\lambda\}$ when $\lambda\leq\kappa$?

If $\lambda\leq\kappa$ are infinite cardinals, how many subsets of $\kappa$ of size $\lambda$ are there? And of size $\leq\lambda$? Is there some sort of explicite formula for this? The internet isn't ...
0
votes
1answer
39 views

Prove that if $S\subset \mathbb{R}^n$ is not countable, then there exists $x \in S$ such that $x$ is a condensation point.

Let $S \subset \mathbb{R}^n$ with the usual metric. A point $x \in \mathbb{R}^n$ is said to be a condensation point of $S$ if for all $r>0$, $B(x,r)\cap S$ is not countable. Show that if $S$ is ...
1
vote
3answers
90 views

What does it mean $|A|<|B|$, when $A$ and $B$ are infinite sets?

For infinite sets, $A$ and $B$, what does it mean to say $|A| < |B|$? Does it mean that there is an injection $A \to B$?
2
votes
1answer
65 views

Can we prove AC from the statement “There is no $\aleph$ cardinal strictly between $\operatorname{CARD}(X)$ and $\operatorname{CARD}(2^X)$”?

If $X$ is a set, let $\operatorname{CARD}(X)$ denote the Cardinal number of $X$. Let GCH(1) be the statement "If $K$ is an infinite initial ordinal number, then there exists no initial ordinal number ...
11
votes
3answers
602 views

How to prove that from “Every infinite cardinal satisfies $a^2=a$” we can prove that $b+c=bc$ for any two infinite cardinals $b,c$?

Prove that if $a^2=a$ for each infinite cardinal $a$ then $b + c = bc$ for any two infinite cardinals $b,c$. I tried $b+c=(b+c)^2=b^2+2bc+c^2=b+2bc+c$, but then I'm stuck there.
3
votes
1answer
44 views

Is this proof that $\kappa^{<\kappa}=\kappa$, when $2^{<\kappa}=\kappa$, correct?

Let $\kappa$ be a cardinal number. I want to show that if $\kappa$ is regular and if $2^{<\kappa} = \kappa$ then $\kappa^{<\kappa}= \kappa$. Here is what I got so far: $$ ...
2
votes
1answer
185 views

Cardinal arithmetic: $b\ge x>1$ and $b^2=b$ implies $x^b=2^b$

Let $x$ and $b$ represent cardinals. Assume that $b\geq x > 1$ and $b^2=b$. Prove that $x^b=2^b$. Thanks!
34
votes
8answers
2k views

Refuting the Anti-Cantor Cranks

I occasionally have the opportunity to argue with anti-Cantor cranks, people who for some reason or the other attack the validity of Cantor's diagonalization proof of the uncountability of the real ...
2
votes
1answer
200 views

What is the cardinality of $\omega^\omega$?

I have started reading about Baire Space as a prerequisity for the subject of Borel hierarchies. And if I understand correctly, Baire space is $\omega^\omega$ (The space of all functions from $\omega$ ...
1
vote
1answer
28 views

Compare density of rationals to the density of integers

Is is possible to somehow quantitatively compare the density of rational numbers to the density of integer numbers, ascribing to the both a number characterizing the density?
0
votes
1answer
22 views

Cardinality of Sets and injections

Let A,B,C,D sets. if |A| $\le$|B| and |B| < |C|, show that |A| < |C| Proof: Case1: suppose |A| < |B| then there exists injection f: A$\to$B and |B| < |C| then there exists injection ...
1
vote
1answer
44 views

A question about infinite sets and Cantor's Power Set theorem

Let $\operatorname{Card}(X)$ denote the cardinal number of the set $X$. The standard proof of Cantor's Power Set theorem stating that "$\operatorname{Card}(X) < \operatorname{Card}(2^X)$" is ...
0
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2answers
96 views

Cardinality and Concrete Mathematics

First, let me distinguish Concrete Mathematics ( Mathematic branches that studies fixed structures ) from Abstract Mathematics ( branches that study classes of structures, such as algebra , topology, ...
2
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1answer
31 views

Regularity of $\omega_1$ and axiom of choice

Why is the regularity of the ordinal $\omega_1$ a consequence of the axiom of choice?