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.

learn more… | top users | synonyms (1)

0
votes
1answer
28 views

Large ordered family of set inclusions complete?

I'm not an expert on set theory, so this might be trivial: Let $\dots \subseteq X_{\alpha} \subseteq X_{\alpha+1} \subseteq \dots$ be a chain of set inclusions, indexed by ordinals(!), s.t. $X_{\alpha}...
0
votes
1answer
59 views

Cardinality: Set of all binary sequence equal c

How do I prove the cardinality of the set of all binary sequences equal c? I know I have to find a bijective function from (0,1) to the set of all binary sequences. But I can't think of one. Cantor'...
0
votes
0answers
39 views

Cardinal of a set of numbers: naturals, integers, rationals and irrationals [duplicate]

My professor gave us these properties very fast in our class and I can't find a proper explanation for them, can someone help me please? (1) - The cardinal of the set of naturals is the same of the ...
3
votes
1answer
81 views

Does $|X|<|Y|$ imply $\mathcal{P}(X)<\mathcal{P}(Y)?$ [duplicate]

This might be a terribly simple question, but I cannot convince myself whether the answer is yes or no. Maybe I am missing something simple. I am not well-versed in the area of elementary set theory ...
3
votes
1answer
53 views

A Successor Cardinal is Regular

Trying to show that every cardinal $k$ , $k^+$ , its successor, is regular. This is what I've come up with. Thoughts? If this does not hold, then a cofinal map $f: \lambda\rightarrow k$ where $\...
1
vote
1answer
54 views

Problem on infinite cardinal number

If $e$ is an infinite cardinal number and $d$ is a cardinal number satisfing $2 ≤ d ≤ 2^e$. I need to prove the following $$d^e= 2^e$$ Any help will be appreciated. Thank you in advance. .
1
vote
1answer
40 views

Cardinality of almost everywhere continuous functions

The cardinality of continuous real functions is $|\mathbb{R}|$ but I was wondering wether allowing functions to be almost everywhere continuous would increase the cardinality or not. On the one hand, ...
1
vote
0answers
38 views

Is it possible to define a group structure on arbitrary set? [duplicate]

Is it possible to define a group structure on arbitrary set? It is obvious for finite sets and also sets with cardinality |Q| and |R| and also we don't know that is there other cardinality betwen them ...
2
votes
1answer
43 views

for a given $\kappa$, find a model of DLO with a greater cardinality that has a dense subset of size $\kappa$

So given a cardinal $\kappa$, i am trying to find a dense linear order with no end points of cardinality greater then $\kappa$ which has a dense subset of size $\kappa$ I think I'm almost there What ...
3
votes
0answers
57 views

The supremum of any set of cardinals (considered as a set of ordinals) is again a cardinal.

An ordinal $\alpha$ is a cardinal iff no $\xi < \alpha$ is equivalent to $\alpha$. Now, let $A$ be any set of cardinals and $\sup(A)=\alpha$, then for $\xi < \alpha$ there is a $\beta \in A$ ...
11
votes
1answer
110 views

How does equality $\mathfrak{m}^\mathfrak{m} = 2^\mathfrak{m}$ in $\sf{ZF}$ relate to the axiom of choice?

Usually, the equality $\mathfrak{m}^\mathfrak{m} = 2^\mathfrak{m}$ for infinite cardinal $\mathfrak{m}$ is proved like that: $$ 2^\mathfrak{m} \leqslant \mathfrak{m}^\mathfrak{m} \leqslant (2^\...
1
vote
4answers
114 views

Since $[0,1]$ and $\mathbb{R}$ are not homeomorphic, does that mean the cardinality of $[0,1]$ and $\mathbb{R}$ are different?

Given $[0,1]$ a closed interval on $\mathbb{R}$, we know that $[0,1]$ is compact and $\mathbb{R}$ is not, so these two spaces are not homeomorphic to each other. But homeomorphic perserves ...
1
vote
0answers
34 views

Cardinal Addition When At Least One is Infinite

Show that if at least one of κ > 0 and λ > 0 is infinite, then κ + λ = κλ = max{κ, λ}. My proof: Assume without loss of generality, κ > λ. If λ = 1, then by definition that at least one is infinite, ...
2
votes
1answer
26 views

The set of finite unions of intervals with rational endpoints is countable.

I don't know how to prove the following: Let $K:=\lbrace G : G$ is a union of finitely many intervals with rational endpoints$\rbrace$. Prove that $K$ is countably infinite. Here is my approach:...
5
votes
2answers
66 views

What is the result of a number greater than 2 raised to the power of {Aleph-0}?

So, I know that $2^{\aleph_0} = \beth_1$, right? What about another number, say $10$, raised to the power of $\aleph_0$? Is $10^{\aleph_0} = \beth_1$ also true, or is $10^{\aleph_0} > \beth_1$ ...
0
votes
1answer
24 views

Union of a chain of cardinalities?

I was trying to understand the union of a chain of cardinalities and I found this equation $$\kappa=\bigcup_{\alpha<\kappa} \alpha$$ for any cardinal $\kappa$ in the answers to this question. Can ...
-1
votes
2answers
70 views

How many sequences of rational numbers in $[0,1]$ exist?

I was talking with a friend of mine and we wonder how many sequences of rational numbers on $[0,1]$ there exists. My first attempt was to consider that every sequence like that must be a subset of $\...
2
votes
2answers
83 views

Proof of $\aleph_0^{\aleph_0} = \mathbb{c}$ without using Cantor's $2^{\aleph_0} = \mathbb{c}$

Prove that $\aleph_0^{\aleph_0} = \mathbb{c}$ without using Cantor's $2^{\aleph_0} = \mathbb{c}$ Card $\mathbb{N}^\mathbb{N} = \aleph_0^{\aleph_0}$ Card $(0, 1) = \mathbb{c}$ Define: $f: (0, ...
0
votes
0answers
16 views

Family of infinite sets with finite intersections [duplicate]

I read somewhere that there exists a family of infinite sets $F \subset P_{inf}(\mathbb N)$, such that any two $X, Y \in F$ have a finite intersection and $\lvert F \rvert = \mathfrak c$. ...
1
vote
1answer
37 views

Where's the mistake in my reasoning?

Task: Find the cardinality of all such functions $f: P(\mathbb N) \rightarrow P(\mathbb N)$ that $f(\bigcup S) = \bigcup \lbrace f(Z) \mid Z \in S \rbrace$ The answer is: $\mathfrak c$ My ...
1
vote
2answers
63 views

Can you go from $\aleph_0$ to $\aleph_1$ with tetration or other higher order operators?

The paradox of Hilbert's Hotel shows us that you can not get past the cardinality of the natural numbers ($\aleph_0$) by adding a finite number (one new guest), adding an infinite quantity (infinitely ...
-1
votes
1answer
59 views

Cardinal equality: $\;\left|\{0,1\}^{\Bbb N}\right|=\left|\{0,1,2,3\}^{\Bbb N}\right|$

I need to prove the above equality without Cantor-Bernstein Theorem or cardinals arithmetic (i.e., a bijection must be found). I know that for example $\;S\to 1_S=\;$ the indicator function, gives a ...
0
votes
1answer
206 views

Cardinality of the set of all bijections

Let $A$ be an infinite set and let $S$ be the set of all bijections $A \rightarrow A$. Then if $\mid A \mid = \kappa$, then $\mid S \mid = 2^\kappa$. I'm able to prove it for $A = \mathbb{N}$ by ...
6
votes
2answers
108 views

Why was $\aleph$ (aleph) chosen for infinities?

Why did Cantor choose a letter from the Hebrew alphabet to represent infinities, rather than using some Greek letter?
6
votes
1answer
258 views

Well-foundedness of cardinals and the axiom of choice

Without axiom of choice it is not generally true that the class of all cardinal (in this question we consider Scott cardinal rather than cardinals as ordinals) is not well-founded under the ordinary ...
0
votes
1answer
44 views

What result is $\left | \bigcup_{i \in I}A_i \right | =\sum_{i \in I} |A_i|$?

I'm reading a text that uses the following equality for disjoint sets $(A_i)_{i \in I}$: $$\left | \bigcup_{i \in I}A_i \right |=\sum_{i \in I} |A_i|$$ This has to do with disjoint unions, but I'd ...
2
votes
1answer
56 views

Cardinality of the set of functions $f: A \to B$ where where $|A|=\aleph_0$ and $|B|=2^{\aleph_0}$

Let $X$ be the set of all functions $f: A \to B$ where $|A|=\aleph_0$ and $|B|=2^{\aleph_0}$. Using some cardinal arithmetic, one can show that $|X|=2^{\aleph_0}$. However, I wanted to construct a ...
1
vote
3answers
40 views

Infinite-dimensional, countable, rational vector spaces.

Does such a thing exist? More explicitly, can I find a vector space $V$ over $\textbf{Q}$, such that $$\dim_\textbf{Q}V=+\infty$$ and $$\operatorname{card}V=\aleph_0$$?
3
votes
1answer
56 views

Countability of Collection of All Finite Subsets of a Countable Set

Let V be a countable set. Ok, first thing to say is that this isn't a question as to whether $S = \{ A \subseteq V \mid A \ \text{finite} \}$ is countable -- there are plenty of other duplicates on SE ...
1
vote
1answer
43 views

Multiplication Principle Proof

I am trying to prove the following; If $X$ and $Y$ are finite, then $|X \times Y| = |X||Y|$. Now, I'll define a bijection $g:\mathbb{N_{n}} \rightarrow X$ and a bijection $f: \mathbb{N_{m}} \...
0
votes
0answers
17 views

Is this Lindeloff theorem using AC? [duplicate]

Theorem: the following are equivalent: 1) The metric space $X$ is separable. 2) $X$ is second-countable (it has a countable basis) proof: $1 \Rightarrow 2: \lbrace B(d,r) : d\in D, r \in \mathbb{Q}...
2
votes
1answer
38 views

Alternate definitions of width (of a partial order) without Choice?

Say an antichain of a poset $P$ is a set of pairwise incomparable elements of $P.$ Typically, the width of a partial order is defined to be the supremum of the cardinalities of antichains of $P.$ When ...
1
vote
1answer
24 views

Cardinality of collection of subfields of $\mathbb C$

The question is just curiosity on my part. The title says it all. I can see that the cardinality is at least $\aleph_1$ (take simple extensions by an uncountable family of transcendental numbers). But ...
1
vote
3answers
46 views

Disprove bijection between reals and naturals

Coming across diagonalization, I was thinking of other methods to disprove the existence of a bijection between reals and naturals. Can any method that shows that a completely new number is created ...
1
vote
1answer
39 views

$A$ countable, $f :A \rightarrow B$ surjective. Prove $B$ is at most countable

My question is: can this be proven without using the almighty Axiom of Choice? Here's the idea of my proof using the axiom: We need an injective function from $B$ in $A$. Let $e$ be the choice ...
0
votes
0answers
52 views

Cardinality of the union and product of two sets without AC

We have the following results Let $A$ and $B$ be infinite sets s.t. $|A|=|B|$, then $|A\cup B|=|A|$. I was wondering if we can prove that without the Axiom of Choice or without using cardinal ...
0
votes
2answers
49 views

Cardinal of an infinite set

In our course about combinatorics, our maths teacher recently introduced to us the notion of cardinality with the following definition: Let $E$ be a set. If there exists an integer $n$ and a ...
1
vote
2answers
48 views

Lowering the cardinality of a set?

Given a set X with a certain cardinality, there are explicit constructions for getting a set with the "next bigger" cardinality, e.g. constructing the power set. Does some analogous construction ...
3
votes
0answers
84 views

Minimal Possible Ordinal for a Cardinal

For a given $\aleph_\alpha$, what is the minimal ordinal possible such that $|\beta| = \aleph_\alpha$? More precisely, assume we have a model $N$ of ZFC. $N$ could be an extension of many different ...
2
votes
2answers
91 views

More numbers between $2$ and $4$ than between $2$ and $3$? (I am not a mathematician.) [duplicate]

Between $2$ and $3$ there are infinite numbers and between $2$ and $4$ there are infinite numbers. So which "infinity" is greater?
0
votes
1answer
46 views

Need a formal proof?

If A and B are two equipotent sets (they have 1-1 correspondence). Prove that if A is denumerable then B is also denumerable. It is easy to understand by intuition. But I can't understand how to ...
3
votes
1answer
65 views

What do you call a cardinal $\kappa$ that is a limit of $\kappa$-many cardinals?

For instance, $\omega$ is the limit of $\omega$-many cardinals. But of course $\omega_1$ is not the limit of $\omega_1$-many cardinals. 1) Are there cardinals other than $\omega$ with this property? ...
1
vote
3answers
176 views

Ordinal with given cardinality (without AC)

Is it possible to show that every cardinality has an ordinal with this cardinality (without the axiom of choice)? If so, how?
4
votes
2answers
201 views

Can every set be expressed as the union of a chain of sets of lesser cardinality?

If a set $S$ has countably many elements $\{x_n\}$, it can be expressed as a union of a chain of finite sets $$ \{x_1\} \subset \{x_1,x_2\} \subset $$ But what about a set of arbitrary cardinality $...
1
vote
1answer
56 views

Is this function one to one? Why? [closed]

Is the function $f:P(\mathbb{N})\to\mathbb{R}$ defined by $$f(A)=\sum_{n\in A}\dfrac{2}{3^{n+1}},\quad\forall\,\,A\in P (\mathbb{N}),$$ an one to one function? Please help understand why. For me, $\...
0
votes
2answers
43 views

Cardinal of a difference of power sets

How can the following be calculated? Given the sets $X = \{1, 2, \dots, 10\}$ and $Y = \{1, 2, \dots, 12\}$, compute $| \mathcal P (Y) \setminus \mathcal P (X) |$, where $\mathcal P (X) = \{ A \mid A ...
2
votes
2answers
289 views

Where is the flaw in my Continuum Hypothesis Proof?

I am not a mathematician, but rather a computer engineer with a curious mind. The continuum hypothesis (CH) has gripped my attention today, and I even asked a question about it earlier today. ...
2
votes
1answer
88 views

Is this interpretation of the continuum hypothesis correct?

I am not a mathematician, but rather a computer engineer with a curious mind. The continuum hypothesis states (I believe) that there does not exist a set $S$ such that $\aleph_0 < |S| < 2^{\...
1
vote
2answers
90 views

Why are Aleph numbers by definition of the form $2^x$?

The first Aleph number is $\aleph_0$, and my question is this: why is the second Aleph number defined to be $\aleph_1 = 2^{\aleph_0}$? If I remember correctly, it had something to do with power sets ...
2
votes
2answers
65 views

Cardinality of the set of all real functions which have a countable set of discontinuities

I'm having a trouble calculating the cardinality of the set of all functions $f:\mathbb{R} \longrightarrow \mathbb{R}$ which have at most $\aleph_0$ discontinuities (let's call the set $M$). A hint ...