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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3
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1answer
60 views

Cardinals definable using ordinal arithmetics

Let $\kappa$ be an infinite cardinal. $\kappa$ is stable under ordinal addition $+$, ordinal multiplication $.$ and ordinal exponentiation $e: (a,b) \mapsto a^b$, so $\mathcal{K} = ...
1
vote
0answers
36 views

Cardinality of the set of all field automorphisms of $\mathbb C$ [duplicate]

Does $\mathbb C$ have infinitely many field automorphisms? Does it have uncountably many field automorphisms?
0
votes
3answers
299 views

How big is the size of all infinities?

"Not only infinite - it's "so big" that there is no infinite set so large as the collection of all types of infinity..." What does exactly mean? How many infinities are there? I've heard there are ...
4
votes
2answers
107 views

Is the cardinality of $\mathbb{Z^R}$=$\mathbb{R^Z}$?

Previously in this question, we have found that $\mathbb{R^Z}$ is uncountable and its multiset of components, denoted by $$K = \{ (..., 0, 0, w, 0, 0, ... ) : w \in \mathbb{R} \}$$ where for each ...
5
votes
2answers
207 views

Dimension of $\operatorname{Hom}(V, W)$

What is the dimension of $\operatorname{Hom}(V, W)$ if at least one of the two vector spaces $V, W$ is infinite dimensional? In the sense of cardinal numbers. Thanks
0
votes
1answer
25 views

Hilbert space and uncountable cardinal

Given an uncountable cardinal does there exist Hilbert space with orthonormal basis of that cardinality?
1
vote
1answer
18 views

Linear continuum poset with succ and pred

Is there a poset of continuum cardinality that satisfies conditions: it is linear there is a minimum element there is a maximum element each element (apart from maximum) has a successor each element ...
6
votes
2answers
48 views

Let $X \subseteq \mathbb R$ and $X$ has same cardinality as $\mathbb R$ , does there always exist a continuous surjection from $\mathbb R$ onto $X$ ?

Let $X \subseteq \mathbb R$ and $X$ has same cardinality as $\mathbb R$ , does there always exist a continuous surjection from $\mathbb R$ onto $X$ ? ( I know that there need not always be a ...
1
vote
1answer
45 views

Orthonormal Basis and Hamel Basis Cardinality

Will cardinality of orthonormal basis will always be strictly less than cardinality of Hamel Basis. It is true in case of seperable spaces. (Because Hilbert space is always uncountable but ...
6
votes
5answers
2k views

Are there fewer reals on $(0, 1)$ than on $(1,\infty)$?

I know that the cardinality of the sets of real numbers $(0, 1)$ and $(1, \infty)$ are equal. So what is the fallacy in this argument? For every real on $(0, 1)$, we can add any integer $n$ to it ...
0
votes
1answer
49 views

Is this a valid equivalent expression of the twin prime conjecture?

The twin prime conjecture states that it is possible to find two primes $p$, $p+2$ at a distance $2$ that are as big as wanted (Wikipedia). I am learning about the basic properties associated to the ...
3
votes
0answers
49 views

Why is this not a proof of Schroeder-Bernstein?

We can show that if $f: A \rightarrow B$ is injective then $|A| \leq |B|$ and if $g: B \rightarrow A$ is injective then $|B| \leq |A|$ so $|A| = |B|$. By the definition of having equal cardinality, ...
26
votes
8answers
2k views

Is symmetric group on natural numbers countable?

I guess it is too difficult a question to ask about the cardinality of $S_{\mathbb{N}}$ so I would like to ask whether it is countable or not. I tried to prove it is uncountable somewhat mimicking ...
2
votes
6answers
100 views

More numbers between $[0,1]$ or $[1,\infty)$?

There are infinitely many real numbers between any two real numbers, therefore there are infinitely many real numbers in the range $[0,1]$ as there are in $[1, \infty)$. In a mathematical sense, are ...
4
votes
1answer
65 views

First Uncountable Ordinal Cofinality: Needs AC?

Say $\omega_1$ is the first uncountable ordinal. The reason I care about $\omega_1$ is Any countable subset of $\omega_1$ is bounded (or if you prefer, there is no countable cofinal subset). This ...
2
votes
4answers
89 views

Does there exist a path connected metric space , in which at least one open ball is countable ?

Does there exist a path connected metric space with more than one point , in which at least one open ball is countable ?
2
votes
1answer
25 views

Does there exist a connected metric space, with more than one point and without any isolated point, in which at least one open ball is countable?

Does there exist a connected metric space, with more than one point and without any isolated point, in which at least one open ball is countable?
2
votes
0answers
33 views

Why is the weight of a topological space a minimum?

If $(X, \tau_X)$ is a topological space, then the weight is usually defined as follows: $$w(X) = \min \{ \vert B \vert : B \subset \wp(X), B \mathrm{\; is \; a \; basis \; of \;} \tau_X \}$$ I was ...
0
votes
2answers
38 views

What is the cardinality of the power set $P(A \cup B)$

Let $A = \{1, 3, 5\}$ and $B = \{3, 4, 5\}$ be sets. What is the cardinality of the power set $P(A \cup B)$? If i'm not mistaking isn't it all the possible combination of these two: $\{\}, ...
4
votes
1answer
64 views

Singularity of small cardinals under AD

It appears to be quite a folklore result that under AD we know which small well-orderable cardinals (for my purposes I mean below $\omega_\omega$) are regular, namely only $\omega,\omega_1,\omega_2$. ...
1
vote
0answers
25 views

Why Is This Step Needed in Proving Bernstein-Schroeder?

Link to Original Text My question is in the lemma: If $f: A \rightarrow B$ is injective, where $B \subset A$, then there is a bijection between $A$ and $B$. The author commented that, with $Y = ...
0
votes
2answers
50 views

What is the cofinality of $2^{\aleph_\omega}$

There is a similar question in this site but I am not satisfied with the answer, which is basically the same as the proof in the mentioned textbook. The book(Karel Hrbacek&Thomas Jech, ...
1
vote
4answers
66 views

When does the cardinality disappear?

Pardon me, if this question sounds stupid. I am learning real analysis on my own and stumbled on this contradiction while reading this -- http://math.kennesaw.edu/~plaval/math4381/setseq.pdf. I ...
-2
votes
0answers
35 views

Cardinality of union of 2 sets

Given infinite sets $A, B$ and $C$ having the same cardinality, prove that $ |A \cup B| = |C| $ I can do it if their cardinality is the same as that of the set of natural numbers or if it is the ...
2
votes
1answer
51 views

Vaught's two cardinal theorem using Vaught pairs

I've been reading David Marker's Introduction to Model Theory, and found Vaught's two cardinal theorem (4.3.34): if a theory $T$ has a $(\kappa,\lambda)$-model, where $\kappa > \lambda \geq ...
0
votes
2answers
50 views
3
votes
1answer
53 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
1answer
40 views

Proving $|\mathbb{R}-\mathbb{S}|=2^{\aleph_0}$ when $\mathbb{S}\subset R$ is countable [duplicate]

I wish to prove that $|\mathbb{R}-\mathbb{S}|=2^{\aleph_0}$ when $\mathbb{S}\subset \mathbb{R}$ is countable. I want to say that $|\mathbb{R}-\mathbb{S}|= |\mathbb{R}|-|\mathbb{S}|$ but we haven't ...
1
vote
2answers
38 views

Cardinality of all subsets of cardinality $\aleph_0$ or $\aleph$ in $\mathbb{R}$

What is the cardinality of all subsets of cardinality $\aleph_0$ in $\mathbb{R}$? And of all subsets of cardinality $\aleph$ in $\mathbb{R}$? Since both are subsets of $P(\mathbb{R})$ , I conclude ...
6
votes
4answers
1k 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?
3
votes
1answer
50 views

cardinality of the set of real functions that approaches $0$ when $x\longrightarrow \infty$

I know that this set is a subset of the set of all real functions. Hence its cardinality is less than or equal to $\aleph ^\aleph=2^\aleph$. The question is how do I prove the second direction? ...
0
votes
0answers
22 views

set difference between $\mathbb{R}$ and a countable set [duplicate]

How does one show that for a countable set $S$: $|\mathbb{R}\setminus S|=|\mathbb{R}|$ (I'm not familiar with the axiom of choise)
1
vote
2answers
44 views

What is the cardinality of $P(\mathbb{R})\setminus P(\mathbb{Q})$?

I'm a beginner in elementary set theory and I'm looking for a simple way (I can use facts from cardinal arithmetic) to show that: $|P(\mathbb{R})\setminus P(\mathbb{Q})|=|P(\mathbb{R})|$ I ...
2
votes
3answers
337 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 ...
4
votes
1answer
58 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
30 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 ...
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 ...
7
votes
1answer
104 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
53 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
310 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
43 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
147 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
757 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 ...
5
votes
1answer
72 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
73 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
36 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 ...
14
votes
0answers
139 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
34 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?