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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3answers
126 views

Please attempt to fault my proof of the continuum hypothesis [on hold]

I have put this proof of the continuum hypotheses to both a Dr. of Maths and a Professor of Logic, and neither has demonstrated a flaw - although I doubt the professor (who shall remain nameless) ...
0
votes
0answers
16 views

Maximal Sets and Bijections

I'm struggling with this question (The function $f(x) = x^2 -3$): Let $A = \{x \in R : x \geq 0\}$. Determine a maximum set $B$ such that $f : A \rightarrow B$ is a bijection. Let $g : B \rightarrow ...
5
votes
2answers
61 views

Why can't you count up to aleph null?

Recently I learned about the infinite cardinal $\aleph_0$, and stumbled upon a seeming contradiction. Here are my assumptions based on what I learned: $\aleph_0$ is the cardinality of the natural ...
1
vote
0answers
52 views

Ultraproduct with no long descending sequence

I have a countably infinite well-ordered structure $M$ (over a countable language if it helps), and an uncountable regular cardinal $κ$, and I wanted to construct an elementarily equivalent structure ...
1
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1answer
33 views

Can the cardinality of a strictly ordered set exceed the cardinality of the natural numbers?

I'm putting some thought into the CH at the moment and a proof of the answer to this question would be most helpful if anybody would be so kind as to help me out: Can the cardinality of a strictly ...
7
votes
3answers
2k views

What are Aleph numbers intuitively?

I cannot get my head around the concept of the `types' of Aleph infinity. What is an easy intuitive way to see when you are given the integer numbers $\aleph_0$ the $\aleph_1$ will follow?
2
votes
1answer
47 views

If $p_n$ is the $n^{th}$ prime, is it ever appropriate to speak of $p_{\aleph_0}$?

If $p_n$ is the $n^{th}$ prime, is it ever appropriate to speak of $p_{\aleph_0}$? I'm no math student. Your pardon if this is just some clearly obvious and easy answer, I'm just not seeing it. ...
-5
votes
0answers
23 views

Proof by induction of for the cardinality of finite sets A and B [on hold]

Can someone please help me with this proof? Proof by induction that for finite sets, A and B, an injection $f: A \rightarrow B$ exists if and only if A is finite and $|A| \le |B|$.
3
votes
1answer
55 views

Well-ordering of sets of cardinal numbers

Proposition For every cardinal number $m$ there is a definite next larger cardinal number. This proposition is proved on page 136 of "Proofs from the Book" using the fact that any set of ordinal ...
3
votes
2answers
34 views

“Proof” that $\text{cof}(\omega_\lambda)<\omega_\lambda$ if $\lambda$ is a nonzero limit ordinal

The following lemma is from this introduction to cardinals. Lemma 2.7. Let $\omega_\alpha$ be a limit cardinal. Then $\alpha$ is a limit ordinal and $\text{cof}(\omega_\alpha)=\text{cof}(\alpha)$. ...
1
vote
1answer
42 views

For every cardinal $\kappa$, $\kappa^+$ is regular

Again I'm struggling with a proof from this introduction to cardinals. Lemma 2.6. For every cardinal $\kappa$, $\kappa^+$ is regular. Proof. If not, then there would be a cofinal map ...
1
vote
1answer
28 views

Using the axiom of choice to choose bijections

I couldn't think of a better question title. I am trying to understand the proof of theorem 1.8 in this introduction to cardinals. Theorem 1.8. Let $\kappa\in CARD$. Let ...
2
votes
2answers
74 views

What are some applications of large cardinals?

Most mathematicians don't often encounter cardinalities larger than that of the continuum, it seems? What are some results outside of pure set theory/logic that rely on the properties of larger ...
3
votes
2answers
32 views

Another characterization of the cofinality?

Is it true that $cf(\kappa)=\min \{\lambda:\ \kappa^{\lambda}>\kappa\}$? $cf(\kappa)$ is certainly $\geq$ than that minimum since $\kappa^{cf(\kappa)}>\kappa$, but I don't know how to tackle ...
0
votes
2answers
23 views

Validity of certain arguments about the countability of infinite sets

I am trying to get an understanding, in layman's terms / on an intuitive level, why some arguments about the countability of infinite sets are valid, and some arguments which seem almost identical on ...
7
votes
1answer
175 views

Showing a cardinal is regular

I've been thinking about this question, but to no avail and I've got to ask. How to show that for $\kappa\geq\aleph_0,$ $\mu=\min\{\lambda: \kappa^{\lambda} > \kappa\}$ is regular? If I wanted a ...
2
votes
1answer
138 views

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 ...
0
votes
0answers
29 views

Equivalence Relations and Cardinality

I'm looking at the question below from a past paper: What is an equivalence relation? Say that two sets $X$ and $Y$ are related via the relation $\rho$ if $X$ and $Y$ have the same cardinality. Prove ...
3
votes
1answer
61 views

Without the Axiom of Choice, $\aleph_0<2^{\aleph_0}$ implies $\aleph_1\le 2^{\aleph_0}$?

Question: In ZF (so AC does not necessarily hold) does the following claim hold? $\aleph_0<2^{\aleph_0}$ implies $\aleph_1\le 2^{\aleph_0}$ This question arose to me when reading the top ...
3
votes
1answer
15 views

Comparability of a set and a subset of power set.

It's well known that for any set $A$, $A < P(A)$. But now, I have some question that, WITHOUT AC, can we guarantee that $A \leq X$ or $X \leq A$ whenever $X \subseteq P(A)$? Thank you.
1
vote
1answer
36 views

Equinumerousity of operations on cardinal numbers

I want to prove for all Cardinal numbers $a$, $b$, $c$ that: $(a \cdot b)^c =_c a^c \cdot b^c$ $a^{(b+c)} =_c a^b \cdot a^c$ $(a^b)^c =_c a^{b \cdot c}$ I know that for 1. it's enough to show ...
28
votes
1answer
4k views

Overview of basic results on cardinal arithmetic

Are there some good overviews of basic formulas about addition, multiplication and exponentiation of cardinals (preferably available online)?
6
votes
3answers
453 views

The cardinality of a countable union of sets with less than continuum cardiality

Can continuum be a countable union of sets with cardinality, less than continuum? I can prove it for a finite union by mathematical induction from this: $$\mathbb R = A_1 + A_2 \implies |\mathbb R| = ...
3
votes
3answers
51 views

Question about $\aleph$-fixed point

I am working through a proof on cardinals I found and can't reason some of the steps. The proposition is that there is an $\aleph$-fixed point, i.e. there is an ordinal $\alpha$ (which is ...
37
votes
9answers
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
95 views

Regarding the axiom $2^\kappa = 2^{\kappa^+}$ for regular cardinals $\kappa$, and its relationship to a couple of other axioms.

(Take ZFC as background.) The following two statements both follow from GCH: ICF. Injective continuum function. The continuum function (i.e. $\kappa \mapsto 2^\kappa)$ is injective. ...
0
votes
2answers
49 views

What is $n^{\aleph_0},n\in\mathbb N$

Can I say that $$n^{\aleph_0}=2^{\aleph_0\log_2n}=2^{\aleph_0}=\aleph_1$$
7
votes
1answer
65 views

Does $\operatorname{card}(X) < \operatorname{card}(Y)$ imply $\operatorname{card}(X^2) < \operatorname{card}(Y^2)$ without choice?

I looked to see if this question was already posted, but did not find anything. Please let me know if this is a duplicate. Assume $X, Y$ are infinite sets such that there is an injection $X \to Y$ ...
4
votes
1answer
75 views

Cardinality of equivalence relations in $\mathbb{N}$

I came across this long proof on this site: Cardinality of relations set But I would like to know whether my direction can work. Say we want to find the cardinality of all equivalence relations in ...
-3
votes
0answers
37 views
1
vote
1answer
35 views

Number of subsets with the same cardinality

Suppose we have a set $S$ with cardinal number $n$, such that $n+n = n$. Consider the set, T, of all subsets with cardinality $n$. How can I show that the cardinality of $T$ is $2^n$? (Without the ...
0
votes
1answer
18 views

Constructing $2^\kappa$ vectors with a certain property

Take an infinite rectangular array with $\kappa$ columns and $2^\kappa$ rows where $\kappa$ is some infinite cardinal. Can you fill it up with at most $\kappa$ different elements in such a way that ...
0
votes
2answers
40 views

Proving cardinality of an uncountable sum

I'm trying to prove the following thing: For a family $\mathcal{A}$ of countable sets such that $|\bigcup\mathcal{A}|$ is uncountable and such that $\big|\{A\in\mathcal{A}: x\notin A\}\big| ...
-1
votes
1answer
33 views

Set of increasing functions from N to N is uncountable

I know it is uncountable. But what is wrong with this proof? I use the lemma that a countable union of countable sets is countable. Let $f(0)=0$. Then for each function $f$, construct $[a_0, a_1, ...
1
vote
0answers
33 views

Perfect Sets of real numbers have the same cardinality as the reals

I am currently trying to understand a proof from here that all perfect sets have the same cardinality as $\mathbb{R}$. So given some perfect set $P \subseteq \mathbb{R}$, the identity mapping ...
0
votes
1answer
18 views

The fundamental group of a wedge sum of countably many circles is countably generated, hence countable.

The fundamental group of a wedge sum of countably many circles is countably generated, hence countable. I don't really understand this statement. How can I see that the free product of countable ...
1
vote
1answer
40 views

$y\mapsto 10^y$ Bijection between integers and powers of 10

Say $y$ and $x \in \mathbb N$ such that $10^y = x$ MSE research gives me this: $y\mapsto 10^y$ For some of you to see the bijection between the integers and powers of 10 will be obvious, but I ...
7
votes
2answers
183 views

What would a world where $\mathsf{CH}$ is false look like?

My question is a little more specific than the title may lead to believe. In the article The set-theoretic multiverse (J.D. Hamkins), the author writes the following: [...] the continuum is ...
2
votes
1answer
50 views

Attempt at proving the class of all cardinals is a proper class

Define $C=\{\alpha:\alpha=|x|$ for some set $x$$\}$ as the class of all cardinals. ($|x|$ being the cardinality of the set $x$) It will be enough to prove $C$ is a proper class by showing ...
5
votes
1answer
70 views

Continuum hypothesis : Number of connected components of $A^c$ in $\Bbb R$

Let $A \subset \Bbb R$ with a countably infinite number of connected component (for the usual topology). What can be the cardinal of the set of the connected components of $A^c$? With continuum ...
1
vote
1answer
135 views

Show that the set of powers of ten is countably infinite

My question asks: An $x \in \mathbb{N}$ is called a power of ten if there exists a $y \in \mathbb{N}$ such that $10^y = x$. Show that the set of powers of ten is countably infinite. I understand that ...
2
votes
1answer
37 views

Exponential of cardinal numbers

there is two wrong statement that I want to find counterexample for them. if $\alpha$ and $\beta$ and $\gamma$ be infinite cardinals then show that these two statements are wrong $\alpha < \beta ...
0
votes
1answer
22 views

How to create a bijection from the union of the set natural numbers and square root 2 to the set of natural numbers?

Since $\Bbb N$ and ${\sqrt2}$ are each countable sets, I see that the union is also countable. From this and the fact that $\Bbb N$ union ${\sqrt2}$ is infinite we know there exists a bijection to ...
0
votes
2answers
12 views

Finding the cardinality of a collection of lines?

I'm trying to find the cardinality of the set of lines such that the lines are not vertical, the slopes are a natural number, and the line has a natural number for a $y$-intercept. Because there are ...
0
votes
1answer
12 views

Showing the cardinality of a bounded shape in the xy plane?

I am trying to show that the cardinality of the space between $x^2+y^2<1$ and $x+y>1$ is the same as the cardinality of real numbers I haven't a clue where do begin with something like this. I ...
1
vote
1answer
22 views

Cardinal exponentiation question of infinite cardinals.

I got a little confused with a question about cardinal exponentiation: Let $\beta$ be an ordinal, and let $K_\alpha$ , $\alpha < \beta$, be infinite cardinals with $K= ...
0
votes
0answers
22 views

Impossibility of constructing a continuum-size linearly independent set in $\Bbb R$ [duplicate]

This is a response to the following exchange at Is there a quick proof as to why the vector space of $\mathbb{R}$ over $\mathbb{Q}$ is infinite-dimensional? [Bill constructs a $\aleph_0$ ...
0
votes
1answer
27 views

Construct an injective function $f:[a,b]\times[c,d]\rightarrow\mathbb{R}$ for some $a,b,c,d\in\mathbb{R}$

Let $a,b,c,d\in\mathbb{R}$ such that $a<b$ and $c<d$ be given. Construct an injective function $f:[a,b]\times[c,d]\rightarrow\mathbb{R}$. My intuition is to construct a function ...
0
votes
0answers
32 views

Having trouble understanding aspects of cardinality.

I am having trouble understanding the meaning of $\omega_\alpha$, I thought it simply meant that it was the initial $\alpha$ segment of $\omega$, but then that wouldnt make sense if $\aleph_0$ is ...
0
votes
1answer
34 views

Cardinality of a line segment

If ${(a,b)}$ and ${(c,d)}$ are points in $ℝ^2$ then let $S$ be the set of point on the line segment that joins ${(a,b)}$ and ${(c,d)}$. Show $|S|= |ℝ|$ I can see this is similar to how the tangent ...