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)

1
vote
0answers
14 views

Show cardinality problem with natural evens, all naturals and all integers

I know how to show the cardinality between 2 of the 3 could work. For example, you can establish a bijection with integers and all naturals through $\mathbb{N}$: s(n) = $\sum_{k =1}^n1$ and ...
-2
votes
1answer
38 views

Prove that (0,1) ⊆ R and (4,10) ⊆ R have the same cardinality [duplicate]

Hows does (0,1) and (4,10) both existing as real numbers make it have the same cardinality?
0
votes
0answers
28 views

Power Set, Theory of Linear Algebra, and Axiomatic Systems

So for homework, my History of Math professor gave us these three questions: Explain why the power set of a set S (collection of all subsets) has the same cardinality as the set [all functions from ...
1
vote
1answer
27 views

Do the adeles $\mathbb{A}$ have the same cardinality as the real numbers $\mathbb{R}$?

My question today is about the Adeles and whether they have the same cardinality as the real numbers. Certainly $\mathbb{R}$ has more elements than $\mathbb{Z}$. This is by Cantor diagonalization. ...
1
vote
1answer
50 views

definition of a $\kappa$-small simplicial set

Let $\kappa$ be a regular cardinal. A set $X$ is $\kappa$-small, if $|X|<\kappa$. What does it mean for a simplicial set $X\colon \Delta^{op}\to Sets$ to be $\kappa$-small. I can imagine at ...
7
votes
2answers
537 views

Cardinality of the Euclidean topology and the axiom of choice

It is relatively straightforward to prove that the Euclidean topology has the same cardinality as the space itself. I have sketched a proof below. The proof seems to rely quite heavily on the axiom of ...
3
votes
1answer
58 views

Trying to understand the slick proof about the dual space

In this famous MO question, a beautiful proof is given of the fact $V\cong V^\ast\iff V$ is finite dimensional. I'm trying to go through it and I'm having some trouble. First of all, I know the in ...
1
vote
0answers
52 views

Examples of uncountable fields of characteristic $p$?

Let $p$ be a prime. The axioms of a field of characteristic $p$ is definable in first order logic and form a satisfiable theory $T$. Indeed, $T$ has arbitrarily large finite models and it also has an ...
1
vote
2answers
70 views

Are cardinal numbers well defined? Or could we have something like $\aleph_{1/2}$? [duplicate]

From what I understood, cardinal numbers are defined as: $\aleph_0$ = the cardinality of $\mathbb{N}$ $\aleph_{n+1}$ = is the least cardinal number greater than $\aleph_n$ The continuum hypothesis ...
0
votes
1answer
26 views

Cardinality of a metric space

Let $X$ be an infinite set .For any two metrics $d_1,d_2 $ on $X$ the identity map $i:(X,d_1)\to (X,d_2)$ is continuous. Prove that $X$ is always countable. I am not getting how to start this ...
-1
votes
1answer
43 views

If a countable union of sets has card $\mathfrak{c}$, prove at least one of them has card $\mathfrak{c}$ [duplicate]

If $A=\bigcup_{n=1}^{\infty}A_n$ and $A$ has cardinality $\mathfrak{c}$, where $\mathfrak{c}$ is the cardinal of the continuum, prove that at least one of the $A_n$ has cardinality $\mathfrak{c}$.
1
vote
1answer
39 views

Comparing Hartogs number to set of all sets of relations on a set

Let $A$ be a set and let $\mathcal{H}(A)$ be the Hartogs number of $A$. Show that $|\mathcal{H}(A)|\neq|\mathcal{P}(\mathcal{P}(A\times A))$. Proof attempt: By contradiction. Suppose that ...
0
votes
2answers
51 views

Show that $\leq$ is transitive over the cardinality of sets

$A, B, C$ are sets. I want to show $|A|\leq|B| \ \text{and} \ |B|\leq|C| \Longrightarrow |A|\leq|C|$. I am confused as to how I would approach this, because the sets in this problem can be either ...
6
votes
4answers
2k views

Taking away infinitely many elements infinitely many times [duplicate]

This is a somewhat hand wavy question but I'm not sure how to ask it more precisely. If we have a countably infinite sequence (or set), can we take away infinitely many elements from the sequence ...
1
vote
1answer
18 views

cardinal of the union of an increasing sequence

Let $A_n\subset \mathbb{R}$ for all $n$. If $\vert A_n\vert=C$ for all $n$ where $C$ is the cardinality of the real numbers and $A_n\subset A_{n+1}$. then Is it true that $\vert \bigcup_n ...
0
votes
2answers
59 views

Show that the set is not countable

To show that a set is countable, you need to show 1 to 1 correspondence, right? So to test if it is 1 to 1 and also onto. So for this example: ...
1
vote
1answer
25 views

cardinality of the set of junctions intervals

Let P$=\{(a,b) : a,b \in \mathbb{R}\}$ and Q$=\{$ countable unions of elements of $P\}$. I 'm interested in knowing the cardinality of Q. I denote with $C$ if the cardinal real , then $C \leq \vert ...
5
votes
1answer
58 views

Cardinals and generic extensions

Let us consider a model of ZFC $M$, a forcing $\mathbb{P}\in M$. If $G$ is a $\mathbb{P}$-generic filter over $M$ we can construct the generic extension $M[G]$ of $M$. Given any cardinal number ...
4
votes
1answer
83 views

Surjective function into Hartogs number of a set

For any set $A$, there is a surjective function $f:\mathcal{P}(A\times A)\longrightarrow\mathcal{H}(A)$. $\mathcal{H}(A)$ is the Hartogs number of $A$. Proof attempt: Suppose $R\subseteq A\times A$. ...
0
votes
4answers
169 views

Proving a set is Uncountable or Countable Using Cantor's Diagonalization Proof Method

I understand the idea that some infinities are "bigger" than other infinities. The example I understand is that all real numbers between 0 and 1 would not be able to "fit" on an infinite list. I have ...
3
votes
3answers
97 views

Example 2, Sec 7 in Munkres' TOPOLOGY 2nd ed: How to show that $\mathbb{Z}_+ \times \mathbb{Z}_+$ is countable?

We need to show that there is a bijection $h \colon \mathbb{Z}_+ \times \mathbb{Z}_+ \to \mathbb{Z}_+$. For this purpose, Munkres define a subset $A$ of $\mathbb{Z}_+ \times \mathbb{Z}_+$ as follows: ...
3
votes
2answers
125 views

Minimal foundations for Cardinal Arithmetic

I would like to develop a theory of cardinal numbers that relies on as weak a basis as possible. Therefore, I would like to know if there is a way to even define a cardinal number for every set ...
0
votes
1answer
27 views

Cardinality of the set of functions which holds the Equality

Let $f$ be a function from $\{1,2,3, \dots ,10 \}$ to $\mathbb R$ such that $$\bigg( \sum_{i=1}^{10}\frac{|f(i)|}{2^i}\bigg)^2 = \bigg( \sum_{i=1}^{10} |f(i)|^2 \bigg) \bigg(\sum_{i=1}^{10} ...
1
vote
3answers
86 views

Let S be the set of all real numbers in the interval (0; 1) whose decimal expansions contain only 0's, 4's and 8's. Prove that S is uncountable.

I just prove $$S_1=\{0.4,0.8\}$$ $$S_2=\{0.04,0.08,0.44,0.48,0.84,0.88\}$$ ... So I can calculate the number of elements in $S_n = 2 \times 3^{n-1}$ I just prove it is countably infinite.
0
votes
2answers
28 views

prove $|Q|$ = $|Q*Q|$

I have this problem: Prove that $|Q|$ = $|Q*Q|$ I know that $Q$ is countably infinite But then how can I prove that $|Q*Q|$ is countably infinite? Thanks you!
0
votes
2answers
23 views

Cardinality proof verification

Problem: Let $C \subset (0,1)$ be the set of all numbers whose unique decimal representation contains the number seven. Show that the number of elements in $C$ must be the same as the number of ...
0
votes
0answers
19 views

Show that |℘(ℕ)| ≤ |Maps(ℕ,ℕ)| [duplicate]

I want to show that there exists an injective map from ℘(ℕ) to Maps(ℕ,ℕ) but I have not an idea of how to go about this. I am thoroughly confused every time I see something like Maps(ℕ,ℕ). Please ...
1
vote
1answer
66 views

In how many ways is my crocodile shaped like itself?

Here is a topologically accurate photograph of my crocodile: In particular, topologically he is a solid sphere. He can be described as being shaped like himself, but that is vague. There are so ...
0
votes
1answer
34 views

Corollary 6.6 in Munkres' TOPOLOGY 2nd ed: Subsets of finite sets are finite? Cardinality of a proper subset is less than that of the set?

Here's Corollary 6.6 in Topology by James R. Munkres, 2nd edition: If $B$ is a subset of the finite set $A$, then $B$ is finite. If $B$ is a proper subset of $A$, then the cardinality of $B$ is less ...
0
votes
1answer
54 views

Showing that $|\mathcal{P}(\mathbb{N})| = |\operatorname{Maps}(\mathbb{N},\mathbb{N})|$

Show that the cardinality of a power set of natural numbers is exactly equal to the map of a set of natural numbers to another, which is $|\mathcal{P}(\mathbb{N})| = ...
0
votes
0answers
27 views

equinumerous cartesian products proofs verification

Prove that $A = (0,1) \times (0,1)$ and $B = (2,4) \times (3, \infty)$ are equinumerous by explicitly define a bijection $h: A \rightarrow B$ Show that h is a bijection. What I have thus far. Let ...
2
votes
1answer
43 views

Proving an intuitively true statement.

Let $X \subseteq Y$ and $X\neq Y$ Also let $f: Y → X$ define a bijection. Prove that $Y$ is infinite. Here's what I have as a proof, but I'm not really sure if it's enough. Let $X \subseteq Y$ ...
0
votes
0answers
46 views

finding counterexamples to proofs.

let f: A → B and let W, X ⊆ A. Prove that if W ⊆ X, then g(W) ⊆ f(X) I don't see how this can work, as I think I've found a counter example. Yet the instructions ask for a proof. Let A = {0,1,2} ...
0
votes
3answers
152 views

Best known upper and lower bounds for $\omega_1$

What are the best known lower and upper bounds for $\omega_1$? Are there sharper lower bounds than $\varepsilon_0$? And are there any known upper bounds which can be explicitly constructed like ...
1
vote
1answer
81 views

Finding bijection from (0,1) → N

How exactly do I go about finding a bijection between (0,1) → N \ {0} so $(0,1) → (1, \infty)$. I figured I could look at this as finding a function from $(0,1) → (0, \infty)$ and just adding 1. ...
1
vote
1answer
54 views

Cantor's bijection between the sets [0,5] and [12,60]

How can I mathematically prove that the sets $[0,5]$ and $[12,60]$ have the same cardinality using Cantor's bijection? While simply drawing a linear function seems tempting, my teacher wants us to do ...
0
votes
1answer
40 views

Cardinality of the set of functions from N to {0,1,2,3,…,9} is equal to card(R)

I know that the set of infinite sequences on {0,1,2,3,4,5,6,7,8,9} is uncountable, but how to show that it has a bijection to R?
-1
votes
3answers
44 views

I dont understand this sets question [closed]

Let $A = \{\{0\}\}$ and $B = \{0\}$. Which of the following statements are true and which are false? Justify each of your answers. $|A| = |B|$ (10 marks) $A \cap B = \emptyset $ (10 marks) $A \cap ...
-1
votes
1answer
83 views

Show that the cardinality of $\mathbb N$ is the same as the cardinality of $\mathbb N \times \mathbb N$ [duplicate]

Show that $| \mathbb N | = | \mathbb N \times \mathbb N |$, i.e., the cardinality of $\mathbb N$ is the same as the cardinality of $\mathbb N \times \mathbb N$. How do I show it using the ...
1
vote
0answers
17 views

Investigating the cardinality of the set of all subsequences of any arbitrary sequence?

This is a non-precisely formulated question recently come to mind: How to investigate the cardinality of the set of all subsequences of an arbitrarily given sequence? Or can we possibly determine the ...
0
votes
0answers
41 views

double horizontal bars notation for sets

I am unsure of the meaning of a specific notation. We have a two-dimansional matrix $I_B$ (representing an image after applying Gaussian blurring to it, but that's just background information). Let ...
1
vote
1answer
16 views

cardinality of a countable sequence of powersets

Given a set $S_1$ of cardinality $\kappa$, we can construct the sequence $\langle S_1, S_2, S_3 ... \rangle$, where $S_i = \wp(S_{i-1})$, for all $i > 1$. If $S$ is finite, so that $\kappa < ...
4
votes
2answers
105 views

On singular products of cardinal numbers

I want to know whether it is possible to show in $ZFC$ that there exist a limit ordinal $\lambda$, a strictly increasing sequence of cardinal numbers $\langle \mu_\alpha : \alpha \in \lambda\rangle$ ...
0
votes
2answers
40 views

Cardinal arithmetic confusion: What is $|\Bbb R|+|\Bbb N|$?

I do not understand how to calculate addition two cardinals. I know that the formula as follows: if $\alpha$ and $\beta$ are two cardinals, then $\alpha + \beta= |\{(a,0):a\in ...
4
votes
2answers
231 views

Is there more than one instance of the Empty Set?

It seems any additional instance would be equivalent in every respect to the first, hence indistinguishable, and arguably identical. I.e., there is only one Empty Set. Correct?
3
votes
2answers
36 views

If one of two sets has larger cardinality, there is a map onto the other set

Let A and B be sets with the cardinality of A less than or equal to B. Show there exists an onto map from B to A. I am struggling with this proof. I don't know how to show this. Any help would be ...
1
vote
2answers
28 views

The cardinality of the set

Let $\mathbb{G} =\{ a^b + \sqrt{c}: a,b,c\in \mathbb Q \}$ I guess the set $\mathbb{G}$ is countable set, but I can't show it properly. How to start the proof?
1
vote
2answers
64 views

König's theorem (set theory) implication

How does König's theorem imply $\quad\aleph_{\omega} \neq \beth_1$?
3
votes
7answers
413 views

Sets with same Cardinality, but no Explicit Bijection?

Are there any good examples of sets where we know that they have the same cardinality, but have not found any explicit bijection between them?
-1
votes
3answers
305 views

The set of all functions from integers to a finite set is uncountable

Show that the set of functions from positive integers to the set $\{0,1,2,3,4,5,6,7,8,9\}$ is uncountable. I suspect I should use the diagonalisation argument but I'm not sure how to approach it. ...