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Questions tagged [infinity]

Use this tag for questions on the concept of infinity. Don't use this tag merely because $\infty$ appears somewhere in the question.

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Is selecting a random person from an infinite population of people an invalid premise to begin with?

This was initially sparked by a hypothetical question: There are two scenarios. In the first, an infinite number of people are living in a completely blissful paradise, but every day a person is ...
mthulq's user avatar
  • 192
0 votes
1 answer
86 views

Is a finite resource that will never run out considered infinite? [closed]

I am a regular on many SE sites, although this is my first (but hopefully not last) question on Math SE. My area of expertise is in digital technology- and although I do know more mathematical ...
security_paranoid's user avatar
1 vote
1 answer
55 views

Brun's theorem and the twin prime conjecture

According to the following extract taken from Wikipedia, almost all prime numbers are isolated given Brun's theorem. Doesn't that mean that there is only a finite number of twin prime numbers (they ...
David's user avatar
  • 37
0 votes
0 answers
57 views

What is the meaning of "infinity" in the Continuum Hypothesis

Quoting from this article How the continuum hypothesis could have been a fundamental axiom by Joel David Hamkins: The continuum hypothesis (CH) is the assertion that the cardinality of the set of ...
zeynel's user avatar
  • 437
0 votes
0 answers
47 views

Hyperreal valued integrals

Consider a set of real numbers $S=\{c_{\vec{x}}\}_{\vec{x}\in\mathbb{R}^3}$. That is a number for every point in 3d space. Consider the product: $$X=\prod_{\vec{x}\in\mathbb{R}^3}c_{\vec{x}}$$ In many ...
Joeseph123's user avatar
2 votes
2 answers
91 views

If I have a sequence $a_0, a_1, a_2, \cdots$ , then is expressing the limit of this sequence as $a_\omega$ sensible?

If I have a sequence created by some rule which comes to a limit , then I can express it as $a_0, a_1,a_2,\cdots$. If I said $\lim_{n \to \infty} a_n = a_{\omega} $ , is that a sensible thing to do ? ...
Q the Platypus's user avatar
1 vote
0 answers
79 views

Is there a name for the set of lines with slopes $1$, $2$, $3$, $\ldots$?

I've recently stumbled across a pattern involving these infinite sets of lines that meet at a single point and have slopes 1, 2, 3, 4... Is there a name for these lines and have they been studied ...
JonasPK's user avatar
  • 31
1 vote
1 answer
74 views

Let $B\subsetneqq A\subset\mathbb{N}$. If there exists a bijection from $A$ to $B$, then there exists a bijection from $A$ to $\mathbb{N}$.

I am reading "Logic in Mathematics and Set Theory" (in Japanese) by Kazuyuki Tanaka and Toshio Suzuki. This book is not a rigorous book. Verify the following fact holds. Let $B\subsetneqq A\...
佐武五郎's user avatar
  • 1,138
-2 votes
1 answer
67 views

𝖈 × 0 > 0 when 𝖈 is the cardinality of real numbers in the real number line [closed]

Is there a flaw in the reasoning below that 𝖈 × 0 > 0 when 𝖈 is the cardinality of real numbers in the real number line? ...
user1286486's user avatar
2 votes
1 answer
174 views

Can we say that $1^\infty$ is indeterminate because we can't classify $\infty$ as even or odd?

While studying indeterminate forms I like many others had used this post to understand why $1^\infty$ is indeterminate. Today I thought of a new and perhaps simpler argument to explain why this is so ...
Madly_Maths's user avatar
0 votes
0 answers
28 views

Where can I find information about such a relationship about the limits of functions that Wolfram suggests?

Where can I find information about such a relationship about the limits of functions that Wolfram suggests 1? I'm interested about whether it is formal method? And whether exist aritmetic for limits f....
linek's user avatar
  • 1
1 vote
2 answers
96 views

why is $L=\{\{x\mid x<q(i)\}\mid i\in\mathbb{N}\}$ not the set of all Dedekind cuts?

Let the set $L$ be definded as $$L=\{\{x\mid x<q(i)\}\mid i\in\mathbb{N}\},$$ where $q(i)$ is some bijection from $\mathbb{N}$ to $\mathbb{Q}$. Clearly, every member of $L$ is neither an empty set ...
Mohamed Mostafa's user avatar
0 votes
2 answers
69 views

What is the definition of an infinite sequence?

Usually when an infinite sequence is described in simple words, someone writes something like this: $(1,2,3,...)$. So it's clear to everyone that the first element is $1$, the second is $2$, the third ...
user3635700's user avatar
1 vote
1 answer
110 views

What is $\log_2{\aleph_0}$?

I understand that $\aleph_0$ is the cardinality of the natural numbers, as well as any set A, for which there’s a way to both match every element to of A to the natural numbers, and match every ...
Anders Gustafson's user avatar
0 votes
1 answer
86 views

An infinite nested radical [closed]

Can anyone help me in finding a closed form of the infinite nested radical here $$\left({\sqrt {4+\sqrt {4+\sqrt {4-\sqrt {4+\sqrt {4+\sqrt {4- ......\infty}}}}}}}\right)$$ The signs are as "+,+,-...
Rieman Tieman's user avatar
2 votes
1 answer
156 views

Is this a valid basis for a transfinite number system?

I've been curious about transfinite number systems including infinite ordinals, hyperreals, and surreal numbers. The hyperreals in particular seem particularly appealing for introducing a hierarchy of ...
Aidan Simmons's user avatar
2 votes
1 answer
94 views

How to evaluate $\lim\limits_{x\to 0} \frac{d}{dx} \frac{e^{-ax}-e^{-bx}}{x} $ using integral?

I tried to find this limit $$ \Omega = \lim_{x\to 0} \frac{d}{dx} \frac{e^{-ax}-e^{-bx}}{x} $$ without using series or finding derivative of that function So I tried to use the integral $$ \int_0^\...
Faoler's user avatar
  • 1,637
1 vote
1 answer
127 views

Nature of infinity [closed]

If there are an infinite number of whole numbers, and an infinite number of decimals in between any two whole numbers, and an infinite number of decimals in between any two decimals, does that mean ...
Blazers's user avatar
  • 35
0 votes
1 answer
76 views

Does the following conditions ensure the infinite set B is enumerable?

Let $B$ be an infinite set. Suppose we take input set as $B$ and output set as $\mathbb{N}$ (1) $∀\ b∈ B, ∃!\ n∈ \mathbb{N}, \text{output $(b)$ is $n$}$ This ensures $f:B→N$ exist. (2) $∀\ n∈ \mathbb{...
lorilori's user avatar
  • 556
-2 votes
1 answer
148 views

can there be an infinite set S whose elements all contain S? [closed]

Just as the title says: Can there be an infinite set S whose elements all contain S? If so, how is it called? An example of a finite set could be the set of all Humans, where each human h knows about ...
arod's user avatar
  • 123
0 votes
3 answers
241 views

Allegedly: the existence of a natural number and successors does not imply, without the Axiom of Infinity, the existence of an infinite set.

The Claim: From a conversation on Twitter, from someone whom I shall keep anonymous (pronouns he/him though), it was claimed: [T]he existence of natural numbers and the fact that given a natural ...
Shaun's user avatar
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2 votes
0 answers
72 views

Dedekind infinite set implies axiom of infinity in Kripke-Platek set theory

I'm referring to this version of Kripke-Platek set theory minus axiom of infinity. Let $S$ be a Dedekind infinite set, and $f\colon S \to S$ be an injection with $f(S) \neq S$. A proof I know picks $s ...
Ris's user avatar
  • 1,292
0 votes
0 answers
65 views

Why doesent the integers have twice as large cardinality as the natural numbers? [duplicate]

I just started learning about set theory and the countability of sets. And there is something my brain does not quite understand about infinities. It might be a dumb question, but here it goes: You ...
volticus's user avatar
0 votes
0 answers
34 views

An asymptotic approximation of stirling numbers of the second kind

How can one prove that an asymptotic approximation of stirling numbers of the second kind S(n,k) is k^n/k!, k is fixed and n goes to infinity .I found this in wikipedia
summer time's user avatar
0 votes
1 answer
30 views

In wheel theory, does $1/0$ have a one-to-one mapping to any set? Or is the concept of a one-to-one mapping outside of its definition?

As I understand it, infinities typically have a one-to-one mapping with known sets. The denumerable infinity has a one-to-one mapping with integers, rational numbers, and natural numbers. The ...
Larry Freeman's user avatar
0 votes
0 answers
93 views

As I understand it, it is possible to define division by zero in a consistent way. The problem is that such a definition is worthless.

I was reading this answer to the question of the division by zero which proposes the idea of "warped" numbers. So, am I correct, the following rules could be a definition of division by zero:...
Larry Freeman's user avatar
1 vote
1 answer
50 views

Limit of $∞.0$ form of an integral and Riemann sum

I was pondering how to solve a limit of the kind $$\lim_{n \to ∞} n^k \left(\dfrac{1}{n}\sum_{k=0}^{n-1} f \left(\dfrac{k}{n} \right) -\int_0^1 f(x)dx \right)$$ where k is chosen such that the order ...
Cognoscenti's user avatar
1 vote
1 answer
116 views

$\omega$-th or $(\omega + 1)$-th when putting odd numbers after even numbers?

The following clip is taken from Chapter 4 - Cantor: Detour through Infinity (Davis, 2018, p. 56)[2]. When putting odd numbers after even numbers, what should the index for the first odd number ($1$ ...
Yif's user avatar
  • 103
0 votes
0 answers
54 views

How to solve the limit of the following trigonometric functions?

How can i calculate limits of $\mathop {\lim }\limits_{\varepsilon \to 0} \mathop {\lim }\limits_{x \to + \infty } \cos \left[ {\left( {a + i\varepsilon } \right)x} \right] = a$, $\mathop {\lim }\...
Tom's user avatar
  • 1
2 votes
1 answer
43 views

How to compare $\infty$ with another $\infty$ in the proof of sub-additivity property.

When reading a proof of the sub-additivity property of the outer measure on $\mathbb{R}$, that is: If $\lbrace A_n \rbrace_{n=1}$ in $\mathscr{P}(\mathbb{R})$ then $$ m^*\left( \bigcup_{n=1}^{\infty}\ ...
Tran Khanh's user avatar
1 vote
2 answers
165 views

Basis for an infinite dimensional vector space

After learning about finite dimensional vector spaces, the time came for learning about infinite dimensional ones. However, these seem much less intuitive to me, and proof of that is the question I ...
user3141592's user avatar
  • 1,919
0 votes
1 answer
72 views

If I had a finite amount of red balls in a infinite sea of blue balls, what is the probability of grabbing a red ball? [duplicate]

If I had a finite amount of red balls in a infinite sea of blue balls, what is the probability of grabbing a red ball? is it 0 because $\frac{n}{\infty} = 0$? But I can't accept that because if every ...
Dave's user avatar
  • 101
2 votes
1 answer
105 views

Taking the limit beyond infinity, with the ordinals

Imagine a function $f:X\to X$ and $x\in X$ (keeping $f$ and $X$ vague on purpose) and let's define $u_1 = f (x)$ $u_2=f^2(x)=f(f(x))$ $u_n=f^n(x)$ Let's also assume that $\forall n, u_{n-1} \neq u_n $ ...
KiwiKiwi's user avatar
  • 169
0 votes
2 answers
81 views

$\aleph_0^c=2^c$

Problem: Prove that $\aleph_0^c=2^c$ My attempt: $\aleph_0^c=\aleph_0^{2^{\aleph_0}}=\aleph_0^{\aleph_1}=\aleph_2$ Then $2^{2^{\aleph_0}}=2^{\aleph_1}=\aleph_2$, so $\aleph_0^c=2^c$. But my intuition ...
MrMustache's user avatar
0 votes
1 answer
45 views

Infinite Series of infinite cardinals in ZFC

$\sum_{n=0}^\omega 2^{\aleph_n}=2^{\aleph_\omega}$ Is this true? And is there a way in ZFC to let $\infty$ range over ALL infinite ordinals (not a concrete one as in the example above) ? $\sum_{n=0}^\...
Michael Lombardini's user avatar
-2 votes
1 answer
73 views

A question about the number of infinities. [closed]

if there are infinite whole numbers, and there are infinite decimals between 0 and 1, and there are infinite decimals between 0.1 and 0.12, and there are infinite decimals between 0.1111111 and 0....
new guy's user avatar
0 votes
1 answer
56 views

Is $\frac{1}{\sqrt{\mid x \mid}}$ integrable in interval -1 to 1?

I know that I can't integrate some unlimited functions while my interval of integration contains the singularity point. For example, $\int_{-1}^{1}\frac{1}{x}dx$ is undefined, it makes sense to me ...
TY FIRE's user avatar
  • 17
3 votes
2 answers
111 views

An alternative way of looking at countable/uncountable infinities

Consider the decimal expansion of a rational number. This will either terminate, or repeat forever a finite number of its final digits. Thus, any rational number can be expressed with a finite amount ...
user19642323's user avatar
1 vote
0 answers
31 views

Ordering and dividing orders of infinity.

I read there are an infinite number of orders of infinity. Can they all be ordered, or are there different orders we can identify where we do not know which has the greater cardinality? Is the ratio ...
Roz's user avatar
  • 21
3 votes
1 answer
97 views

If $+\infty\leq\mu(A)$ then $\mu(A)=+\infty$

Let $(X,\mathbb{X},\mu)$ be a positive measure space. Let $A$ belong to $\mathbb{X}$. If $+\infty\leq\mu(A)$ then $\mu(A)=+\infty$. Is this statement correct? My answer is yes. I believe this is very ...
Jose Esparrago's user avatar
1 vote
1 answer
71 views

why $\lim_{M\to\infty} e^{-kM}=0$

why $\lim_{M\to\infty} e^{-kM} = 0$ if $k>0$ I know that $e^{-\infty} = 0$ but why $e^{-k\infty} = 0$ I was told I could not do operations with infinity
samsamradas's user avatar
1 vote
1 answer
94 views

Find the flaw in this mapping between the naturals and reals

I was studying Cantor's diagonal argument etc. I was testing the ideas and I thought of the following mapping between the naturals and the reals and I need some help to find the flaw in it. For ...
Antonis Karvelas's user avatar
0 votes
6 answers
244 views

Evaluation of a limit at infinity [closed]

Evaluate $$\lim_{x \to +\infty} [\sqrt{x}-\ln(x^2+1)]$$ I tried to multiply both numerator and denominator by conjugate and tried applying L'hopital but the calculations become way too complex. Is ...
a_i_r's user avatar
  • 689
0 votes
0 answers
95 views

Does the Power of a Point Theorem apply to a point at infinity?

It is common to extend the Euclidean plane with points at infinity, one for each direction. The the theorems of Euclidean geometry generalize amazingly well to this extended plane. The Power of a ...
SRobertJames's user avatar
  • 4,450
-3 votes
2 answers
269 views

Could a sequence in the Collatz conjecture actually increase without bound?

If my understanding is correct, than the Collatz conjecture could only be false if there is at least two closed cycle in it or if there is a number which increases without bound. $3x-1$ We know that ...
RBen's user avatar
  • 15
2 votes
4 answers
412 views

How to interpret what a set is to see how it could be infinite?

Currently, 'infinite set' sounds oxymoronic to me, so my question is how to interpret what a set is such that it is consonant with it being infinite. I understand that we take it as axiomatic that ...
Princess Mia's user avatar
  • 3,019
1 vote
1 answer
70 views

A basic question about 0 the one point compactification of $\mathbb R$

It is possible to turn the real line into a compact space by adding a single "point at infinity" which we will denote by $\infty$. The resulting compactification is homeomorphic to a circle ...
boaz's user avatar
  • 4,807
0 votes
3 answers
73 views

An apparent paradox: a different solution when transforming a linear equation

If we have $y=a(y-x)+b$, and we set $a=1$, the solution of that equation will be $x=b$. However, if we rewrite the equation as $y=\frac{a}{a-1}x-\frac{b}{a-1}$, and again we set $a=1$, we would get ...
Ommo's user avatar
  • 349
1 vote
2 answers
114 views

Connecting elements in the set of Natural numbers

Lets say we "connect" two numbers in the set of Natural numbers in a way, that we draw an arrow between them. In each step we draw an arrow from a number to the number which is next to it. ...
RBen's user avatar
  • 15
0 votes
1 answer
226 views

Infinite sequence of random variables

Suppose that we have an infinite sequence of i.i.d. binary random variables $(X_n)_n$ with $X_n\in\{0,1\}$ for all $n$ (with both options having probability 0.5). Now the outcome of all random ...
Riemann's user avatar
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