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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1answer
14 views

Show that the set of all finite subsets of $\mathbb{N}$ whose size is exactly $n$ where $0<n\in\mathbb{N}$ is countable

I got this problem: Show that the set of all finite subsets of $\mathbb{N}$ whose size is exactly $n$ where $0<n\in\mathbb{N}$ is countable. I.e. Show that $|\{P\in\mathbb{P}(\mathbb{N})| ...
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0answers
45 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 ...
0
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1answer
15 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 ...
6
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1answer
63 views

Dominating strategically $\omega_1$ reals

For a given $\kappa > \omega$, define the game $d(\kappa)$: In every stage, player I chooses a sequence of elements of $\omega$, $g_n$ of length $\kappa$, and player II picks a natural number ...
1
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1answer
40 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 ...
2
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1answer
28 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?
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3answers
74 views

Can we simplify analysis by getting rid of the uncountable reals? [duplicate]

Since the entire observable part of the universe can only be in a finite number of physically distinguishable states, it seems rather strange that an efficient formal description of the universe would ...
0
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1answer
35 views

If $A$, $B$, $C$ are any infinite sets then is $|A|=|B|$ and $|A|=|C|$ $\Longleftrightarrow |A|=|B\cup{}C|$?

Suppose we have three sets $A$, $B$, and $C$ that we know are infinite sets, but we do not know anything else about the cardinality of $A$, $B$, and $C$. Is $|A|=|B|$ and $|A|=|C|$ ...
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2answers
88 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, ...
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1answer
41 views

Set $T$ is Countably Infinite [closed]

How can it be shown that $$T = \{\,(i, j, k) \mid i, j, k \in\mathbb N\,\} $$ is countably infinite?
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0answers
18 views

A question on product space [duplicate]

If $|X|=\mathfrak c$, then what is the cardinality of the product space $X^{\omega}$? Thanks very much.
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3answers
65 views

What properties does $A\to B$ satisfy under 1-1 correspondence?

A 1-1 correspondence between two sets $A$ and $B$ is a function $f\colon A \to B$ satisfying what properties? I do know that we say that two sets $A$ and $B$ are equivalent, and we write $A \sim B$ ...
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3answers
27 views

Limit Ordinals as Infinite Ordinals and other questions

I am studying set theory and I am confused in the following: Are limit ordinals the same as infinite ordinals? I would say yes since the least non-zero limit ordinal is $\omega$. Infinite limit ...
0
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3answers
83 views

Proving $a + a = a$ if a statement is true

That's the question : Let $a$ be a cardinality such that this following statment is true : For every $A, C$, if $ A \subseteq C$, $|A| = a$ and $|C| > a$, then $|C \setminus A| > |A|$. ...
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2answers
22 views

What is the cardinality of all binary sequnces (infinte and finite) that the sequnce 01 does not apper in them

As the title suggests, the question is : What is the cardinality of all binary sequnces (infinte and finite) that the sequnce 01 does not apper in them ? I'll tell you where im stuck, let's say f is ...
0
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1answer
22 views

Cardinality and Set Thoery

(|x| = cardinal # of x for clarification) let A,B be two finite sets, show that $|A \cup B| = |A| + |B| -|A\cap B|$ Proof: let $x\in A\cup B$ $x \in (A -A\cap B) + (B- A\cap B) + (A \cap B)$ let ...
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0answers
27 views

Show that same cardinal number defines and equivalence relation

Same cardinal number must satisfy all three properties : symmetry, reflexivity, transitivity Symmetry Suppose bijection $f:A→A$, Then by definition, |A| = |A| Reflexivity Suppose bijections ...
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2answers
21 views

Relation between successor cardinals and power sets

What are the known relation between successor cardinals $\kappa^+$ and power sets $2^\kappa$ (when GCH is not assumed)? For example, is it true that $\kappa^+ \le 2^\kappa \le \kappa^{++}$? In ...
1
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1answer
19 views

Help understanding cardinal multiplication and infinite Cartesian products

The cardinal product of two sets is defined to be the cardinality of the Cartesian product. The Cartesian product is: $$\prod_{\alpha \lt\beta}\kappa_{\alpha}=\{f\mid f\colon\beta\rightarrow ...
3
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1answer
41 views

A generalization of “any countable limit ordinal is the union of a sequence of increasing ordinal”

Using the fact that every countable ordinal is isomorphic to a closed subset of $\mathbb Q$, I find out that any countable limit ordinal is the union of a sequence of increasing ordinal. Now I'm ...
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4answers
198 views

Is $2^{\aleph_0} = \aleph_1$?

I was reading a thread on Examples of Common False Beliefs in Mathematics on MathOverflow, in which a user wrote: $$2^{\aleph_0} = \aleph_1$$ This is a pet peeve of mine, I'm always surprised ...
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1answer
35 views

uncountable repetitions

I have a question (or two) about recursive naming conventions. Consider the following recursive naming sequence: Base step: Let S be any nonempty set. Let x be any arbitrary element of S. Let S* be ...
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1answer
22 views

What is the limit of the cardinality of a set of bins in finite range, as bin width approaches zero?

Let's say that we divide the region $(0,1)$ into $N$ bins of width $1/N$. Of course, it makes sense to take the limit $1/N \rightarrow 0$ in this configuration, because that's simply how we define an ...
3
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1answer
33 views

Is $\operatorname{card}(I)=\operatorname{card}(D)$

When I was answering number of integrable functions is greater than number of differentiable functions I got to wonder if the inequality was strict. So with $\mathcal I$ being the set of integrable ...
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1answer
72 views

Cantor's Diagonal: Why not a 1-2 Correspondence between the Naturals and Reals?

Hopefully I'm following Cantor's Diagonal Argument with a minimum of distortion and omission: We start from an enumeration T of all infinite binary sequences. We then construct a list S of elements ...
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2answers
32 views

Cardinality of $\{ (x, y) \in \mathbb{R}^2 \mid \left| x \right| + \left| y \right| = 1 \}$ and $\{ (x, y) \in \mathbb{R}^2 \mid x^2 + y^2 = 1 \}$

Do $\{ (x, y) \in \mathbb{R}^2 \mid \left| x \right| + \left| y \right| = 1 \}$ and $\{ (x, y) \in \mathbb{R}^2 \mid x^2 + y^2 = 1 \}$ have the same cardinality? One can draw a square in the two ...
0
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1answer
98 views

What's the largest number

Originally this question started as 'what is the largest number' using $\aleph_0$ as a start, and continuing using concepts such as ${\aleph_0}^{\aleph_0}$, and Knuth's Tower notation $\uparrow$, so ...
2
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2answers
52 views

$\forall \alpha \exists \beta: \beta > \alpha$ where $\alpha$ and $\beta$ cardinals

I have to prove ZF $\vdash$ $\forall \alpha \exists \beta:\beta > \alpha$, where $\alpha, \beta-$ cardinal numbers. I can prove it only in ZFC. Let's fix some cardinal number $\alpha$. By ...
0
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1answer
22 views

$\sigma$-algebra with cardinality $\aleph_0$ [duplicate]

Can a $\sigma$-algebra in a set $X$ have cardinality $\aleph_0$, the cardinality of the naturals? I do not have a clue on how to start with this? Can someone please give me a hint?
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1answer
61 views

Why does this author define cardinality indirectly?

I'm studying Enderton's Elements of Set Theory and in the page 129 he defines what it means two sets being equinumerous: After that in the page 136 he defines cardinality: Why doesn't he define ...
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1answer
69 views

Is the cofinality function monotonic?

Is the cofinality function $\operatorname{cf}$ monotonic? I.e., if $\lambda \le \kappa$ for cardinals $\lambda$ and $\kappa$, does it then follow that $\operatorname{cf}(\lambda) \le ...
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0answers
32 views

Mathematics with and without continuum hypothesis

This is a follow-up to a recent question. Are there "interesting" differences between CH-mathematics and (non-CH)-mathematics? Has anybody developed mathematics with c = $\aleph_2$? $\aleph_3$? ... ...
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2answers
53 views

Sets of “Isolated” Cardinals

Let $C\neq\emptyset$ be a set of infinite cardinals with the property that NO member of $C$ occurs as the supremum of strictly smaller members of $C$. So the cardinals in $C$ are sort of "isolated". ...
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0answers
31 views

The cardinality of dense subsets of infinite-dimensional Hilbert spaces

If $H$ is an infinite-dimentional Hilbert space, then does $\dim H$ coincide with the smallest cardinal of a dense subset of $H$?
2
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3answers
283 views

A question about cardinal number.

Let $X$ be a infinite set and $n$ be a positive integer. We denote the cardinal number of $X$ by $|X|$ and denote the family of all subsets of X which contains n elements by $\mathfrak{F}$. Then ...
2
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1answer
20 views

Is ordering of (possibly infinite) sets by cardinality a total ordering?

Given sets $A$ and $B$. Can you show that either there exists an injective map of $A$ into $B$ (that is, a map such that each element of $A$ maps to an element of $B$ and no two elements of $A$ map to ...
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3answers
73 views

Decidability of the cardinality of a set given that the Continuum Hypothesis is independent from ZFC

I'm a Total Amateur (TM), please forgive me if this question makes no sense. The Continuum Hypothesis states that there are no sets with cardinality strictly between that of the integers and the ...
4
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1answer
45 views

Cardinality of sets of reals without choice

Assuming just ZF (no axiom of choice): Does $\aleph_n\leq|\mathbb{R}|$ for all $n<\omega$ imply $\aleph_\omega\leq|\mathbb{R}|$? (with $\kappa\leq|\mathbb{R}|$ meaning that there is a set of reals ...
2
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3answers
83 views

Teaching cardinality

I would like to give a class of 60 minutes to my undergraduate students about cardinality. I would like to begin with the definition of cardinality and end with one or two good application of this ...
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0answers
46 views

How to calculate the dimension of an infinite direct product of copies of a field?

Let $F$ be a field and $I$ an arbitrary infinite index set. I'd like to know how to calculate the dimension of $\prod_{i\in I}F$. By the way, I know $\dim(\prod_{i\in I}F)\geqslant ...
1
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1answer
46 views

Can you say that almost all $\mathcal{C}^\infty$ functions are not polynomials?

This question asked whether there are functions other than the trigonometric ones whose Maclaurin series contains infinitely many terms, i.e. that never become zero under repeated differentiation. ...
4
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2answers
44 views

How can I prove that the cadinality of a set minus a finite number of elements of it is still the same as the original set?

A is a finite subset of S, which is an infinite set. How can I prove that $|S| = |S \setminus A|$? I just finished proving that $|T \cup S|$ where $T$ is infinite and $S$ is countable is $|T|$. They ...
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1answer
34 views

How could I prove that the cardinality of the union of two sets is equal to R? $|T U S| = |T| = |\mathbb{R}|$

I have to prove that $|T \cup S|$ where $T$ is infinite and $S$ is countable, equal to $|T|$, and this is also $|\mathbb{R}|$. How can I approach this? $|T \cup S| = |T| = |\mathbb{R}|$ I tried to ...
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3answers
71 views

What is the cardinality of the set of all functions from $\mathbb{Z} \to \mathbb{Z}$?

How can I approach this? I have to find the cardinality of the set of the functions from $\mathbb{Z} \to \mathbb{Z}$ and I have no idea on how to solve it. Can someone hint me here? The approach ...
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0answers
26 views

What mean $L(\mathbb{R})$ and $L(\mathbb{R})^*$?

I found them relating a cardinality question here. Does it have anything to do with regularity/computability?
2
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1answer
77 views

What axioms are needed in proofs of the independence of the continuum hypothesis?

My understanding is that the proofs that CH and not-CH are consistent with ZFC are both about ZFC and in ZFC. Is it possible to do these proofs about ZFC but in a weaker axiomatic system? (It is also ...
3
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1answer
83 views

Measure of an elementary set in terms of cardinality

In Terry Tao's textbook on measure theory and integration, he notes that, given an elementary set $A$, the length of $A$, denoted $|A|$, may be written discretely as $$|A| = \lim_{n \to ...
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0answers
19 views

Cardinality of symmetric density functions relative to the cardinality of all density functions

Is there anyone who has some idea bout the following question? $X=$(total number of all pairs of probability density functions $(f_0,f_1)$ on the real numbers) and let $Y=$(total number of all ...
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3answers
210 views

Bijection between open and closed interval [duplicate]

I am not sure how to approach the following problem: Show the open interval $(a,b)$ is bijective with the closed interval $[c,d]$. I was thinking of using $a+u$ where $u$ is a really small number ...
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1answer
42 views

cardinality of polynomial

What is the cardinality of the following sets? (Choose from finite, countably infinite, or uncountably infinite.) The set of polynomials of the form $ax+b$ with $a \in\Bbb N$ and $b \in\{0,1\}$ ...