Somewhere beyond the numbers lies the concept of Infinity. But what exactly does "infinity" mean? What rules does it obey? What interesting properties does it have?

learn more… | top users | synonyms

6
votes
4answers
186 views

The nature of infinities

I have been thinking about the nature of infinity lately. I have no experience with higher mathematics or theorems regarding infinity, so please forgive me if my ideas on this topic are extremely ...
2
votes
7answers
204 views

How the cardinality of $\mathbb{R^+}$ and $\mathbb{R}$ same?

Let me first confirm you that this question is not a duplicate of either this, this or this or any other similar looking problem. Here in the current problem I'm asking to disprove me(most probably ...
2
votes
1answer
46 views

Could “$\infty$” be understood by taking the reciprocals of the Hyperreal numbers?

When learning mathematics we are told that infinity is undefined. (*) Recently I read about the infinitesimal version of Calculus and how we can in fact treat $dy/dx$ as a fraction under this ...
-6
votes
0answers
81 views

Is $\infty=-\frac12$? [duplicate]

edit I'd like to dispute the dupe-closing: The other question ($1+1+1+\cdots=−\frac12$) asks for a proof of that formula, while this questions asks whether its implication is valid in a more general ...
2
votes
2answers
55 views

How are some infinities larger than other infinities

I heard an expressions, some infinities are larger than others recently, and they stated that it was proved to be so. I haven't been able to find this proof, and ...
1
vote
0answers
72 views

Understanding infinity [on hold]

I want to understand in a greater depth the concept of infinity. Can someone give me any reference/ text from where I can study and understand about the concept of infinity in mathematics? I would be ...
92
votes
17answers
12k views

Is 10 closer to infinity than 1?

This may be considered a philosophy but is the number "10" closer to infinity than the number "1"?
0
votes
0answers
15 views

Request for good resources on 'history of infinity' topics [migrated]

Im writing/starting with my bachelor thesis, the subject is about "infinity": what the hell is it, why do we accept it, but most of all my goal is to give an overview of the history of the ...
19
votes
4answers
629 views

Was there anybody before Cantor who conjectured existence of infinities of different sizes?

Georg Cantor is formally known as the first one who discovered existence of infinities of different sizes. But the history of thinking about the concept of "infinity" in maths and philosophy goes back ...
0
votes
4answers
95 views

What does $[0, \infty]$ mean?

Can we "close" the subset with a bracket on the right of infinity like: $[0, \infty]$? What is the difference to $[0, \infty)$, which is already considered a closed set?
3
votes
1answer
27 views
0
votes
0answers
20 views

is it in fact impossible to construct a machine which can know if a macine ever prints a character?

In $\S\ 8$ of his paper "On computable numbers, with an application to the Entscheidungsproblem" Turing uses his proof that $\mathfrak{D}$ (a machine which given the S.D. of another machine ...
0
votes
1answer
35 views

Proving that if a set $A$ is infinite then necessarily $|A|\geq|\mathbb{N}|$ [duplicate]

A set $A$ is set to be infinite if it is not finite, i.e. if there exists no $n\in\mathbb{N}$ such that $|A|=n$, meaning there exists a bijection $A\leftrightarrow\{1,\dotsc,n\}$. How do I prove that ...
2
votes
7answers
374 views

When does it make sense to say that something is almost infinite?

I remember hearing someone say "almost infinite" on one of the science-esque youtube channels. I can't remember which video exactly, but if I do, I'll include it for reference. As someone who hasn't ...
-1
votes
2answers
161 views

Does infinite time = time with no end = never?

Say an object is to travel from point A to point B, a finite distance of 2 meters. Say the object travels at 1m/s. After 1 meter the speed of the object is halved. After another half of the previous ...
0
votes
4answers
228 views

Is $\infty / \infty = 1$?

Lately, my friend and I were arguing about what $\infty / \infty$ equals. My thinking was that $\infty / \infty = 1$, since no matter how high you go in the numerator, it would have to go equally as ...
0
votes
1answer
57 views

Is$\ +\infty$ greater than any other number (surreal, superreal, hyperreal, …)?

Let$\ \mathbb{A}$ be an arbitrary totally ordered set and consider the largest element of the set of extended real numbers,$\ +\infty$. Can we say that$\ +\infty > \chi $, for *any*$\ \chi \in ...
1
vote
3answers
141 views

Question about $\infty\cdot0$ as not defined number

Let's say we have this: $$ \infty \cdot 0 + \infty$$ since $\infty\cdot0$ is not defined can we do this: $$ \infty \cdot(0 + 1) = \infty \cdot 1 = \infty $$ and therefore can I say $\infty\cdot0 + ...
2
votes
0answers
34 views

Infinity Gradient

I calculate infinity gradient, but I am not sure is this correct.
3
votes
0answers
94 views

Is it a “paradox”, or a flaw in the question?

(Clearly not a pardox per-se but I would like to hear what you think) The basic riddle (not a very interesting one even) goes as follows: A first client comes into a barber shop, takes a hair cut ...
-8
votes
5answers
942 views

How does the number $ e $ relate to the limit of $(1+1/n)^n$ as $n \to \infty$? [closed]

The number $e$ is sometimes defined as $\lim_{n\to\infty}\left(\frac{n+1}{n}\right)^n$. Does this mean we can say that $e = \left(\frac{\infty + 1}{\infty}\right)^{\infty}$? Why or why not? And what ...
1
vote
3answers
51 views

Limit question $\infty^{0}$ type

$$\lim_{x\to\frac{\pi}{2}^-} (\tan x)^{\cos x}$$ I just tried to write $e^{\ln(\tan x^{\cos x})}$ form but I couldn't solve the limit.
4
votes
1answer
273 views

Can a curve be an asymptote?

$f(x)=x^3+\frac{3}{x-1}$ This was the question given to me.I replied that $f(x)$ will have only a single vertical asymptote of $x=1$. My teacher told that there'll be be two asymptotes.One is the ...
1
vote
1answer
63 views

Something I don't understand about Hilbert's grand hotel

So I want to know if Hilbert's hotel "story" holds for this statement: $\wp (\mathbb{N}) \sim \wp (\mathbb{N})\smallsetminus \left \lbrace\emptyset\right\rbrace$ So, If the statement wasn't talking ...
0
votes
0answers
9 views

Integration help with Hermite polynomials or direct integration!

This is my formular: $$ \psi_2=N_2 (4y^2-1) e^{-y^2/2}, $$ where $y=x/a$, $a= \left( \frac{\hbar}{mk} \right)$, $N_2 = \sqrt{\frac{1}{8a\sqrt{\pi}}}$. Here is my integral: $$ <x^2> = ...
2
votes
3answers
134 views

What is the difference between $\omega$ and $\aleph_0$?

The book I'm using says that the cardinality of a set $X$ is the least ordinal $\alpha$ such that $|X| = |\alpha|$. So then $\omega = \aleph_0$, but $\omega + \omega \ne \omega$, while $\aleph_0 + ...
1
vote
2answers
65 views

Uniqueness Proof for solution to $\nabla^2 G(\textbf{r}) = \delta(\textbf{r})$ with $G \rightarrow 0$ when $|\textbf{r}| \rightarrow \infty$

I'm having difficulty understanding the derivation of solution to this equation: $\nabla^2 G (\textbf{r}) = \delta(\textbf{r})$ with $G \rightarrow 0$ when $|\textbf{r}| -> \infty$ in $R^n$ where ...
8
votes
4answers
2k views

how do we assume there is infinity?

Definition of infinite: A set is infinite iff it is equivalent to one of its proper subsets. We know that our universe doesn't contain infinite number of elements (including subatomic particles), so ...
1
vote
2answers
67 views

Limit of $\frac{n!}{(n+1)!}$ as n approaches infinity.

I know that factorials grow faster than any exponential function, but what if you put two factorials up against each other? My problem is finding the limit of: $$\frac{n!}{(n+1)!}$$ as $n$ ...
12
votes
0answers
362 views

“Infinito”, a combinatorial game with infinite width game-tree

I recently designed a combinatorial game (sequential game of perfect information) with an infinite branching factor, that is it has a game-tree of infinite width. I'm wondering how is it possible to ...
2
votes
0answers
33 views

mathematical limit for a ouroboros torus

The other day i was watching an episode of Tom and Jerry in which a similar situation was present toms head comes out of his own mouth. My head hurts when i think how is that even possible so i ...
6
votes
3answers
94 views

Irrational numbers in between $n$ and $n+1$

Is the amount of irrationals numbers in between consecutive integers always the same? is this amount infinite?
2
votes
8answers
570 views

Is 0.9 repeating = 1 disproved by asymptotes?

I'm discussing proofs that 0.9 repeating equals 1 with some friends, and they use asymptotes to disprove this. One says if we had the function $y=x/0.000\ldots1$ (and he's only using that impossible ...
1
vote
3answers
50 views

About the infinite geometrical sequence factored with n [duplicate]

I just came across this thread, and i asked myself: I know that $\sum^\infty_{n=0} x^n = \frac{1}{1-x}$ But what happens when we set up the sum like $$\sum^\infty_{n=0} nx^n = ?$$ There is ...
5
votes
1answer
60 views

Proof that the harmonic series is < $\infty$ for a special set..

In one of my books i found a very interesting task, i am really curios about the solution: Let $M = \{2,3,4,5,6,7,8,9,20,22,...\} \subseteq \mathbb{N}$ be a set that contains all natural numbers, ...
1
vote
1answer
34 views

Question about probability in infinite game of chance

Imagine a game where you start with \$100 and toss a coin repeatedly. If it's heads - you lose \$1, if its tails - you double your money. Game ends when you lose all the money. Given infinite amount ...
1
vote
2answers
56 views

Divergent to $\infty \Rightarrow$ Divergent?

In our lecture, we defined a sequence $\left(a_n\right)_{n\in\mathbb N}$ to be divergent if it does not converge, and additionally to be divergent to $\pm \infty$, iff: $$\forall \epsilon \in \mathbb ...
7
votes
2answers
240 views

New Axioms of Infinity

Axiom of Infinity says there is an inductive set (i.e. a set which includes $\emptyset$ and is closed under successor operator). Formally: $Inf:\exists x~(\emptyset\in x~\wedge~\forall y\in ...
19
votes
2answers
1k views

Are there number systems corresponding to higher cardinalities than the real numbers?

As most of you know, the set $\omega$ with cardinality $\aleph_0$ corresponds to what we normally know as the natural numbers $\mathbb{N}$, and the set $\mathcal{P}(\omega)$ with cardinality ...
0
votes
3answers
51 views

Is a nonzero number infinitely greater than zero? [closed]

So many years ago, I posted this question on Yahoo! Answers, and was not really happy with the response. I ran across it again recently and decided to try and breathe new life into this in the hopes ...
1
vote
2answers
46 views

Finding the sum and nth term of a series

How do you find the value of this series? $$\sum^\infty_{n=2}{\frac{2^n + (-1)^n}{4^n}}$$ I tried writing out the series at $n=2, n=3,$ and $n=4$, and I attempted to look for a pattern with which ...
0
votes
1answer
61 views

Will the Declaration of Independence ever show up in pi? [duplicate]

If pi goes on forever and is completely random, if ascii would be mapped onto pi would you eventually find the Declaration of Independence in it? If so, by what digit of pi can we reasonably expect ...
3
votes
1answer
39 views

Hypercomputation & Higher Dimensional Variants of Conway's Game of Life

Conway's Game of Life is a simple and important mathematical game with some rules of evolution in a two dimensional space. It appears in many subjects in mathematics, artificial intelligence and ...
1
vote
3answers
81 views

Why does Infinity x Zero not Equal One? [duplicate]

Why does Zero Times Infinity not equal One ($0 \times \infty \neq 1$)? If Infinity = $\infty$ and Zero = $\frac{1}{\infty}$ Then Zero Times Infinity = $0 \times \infty = \frac{1}{\infty} \times ...
3
votes
0answers
115 views

Which infinity do we mean by $\infty$ in the symbol $x\rightarrow\infty$? [closed]

In ordinary mathematics we use the "limits" frequently. In principle the notion of "limit" is closely related to the notion of "infinity". Intuitively when we are calculating a "limit" we begin from ...
2
votes
3answers
124 views

Example of set of cardinality $\aleph_2$

I am looking for an example of a set of cardinality $\aleph_2$, such as the continuum is an example for cardinality $\aleph_1$.
1
vote
3answers
63 views

How to prove that $\lim_{x\rightarrow \infty}\dfrac{x^2}{e^x}=0$?

I need to prove that $\lim_{x\rightarrow \infty}\dfrac{x^2}{e^x}=0$.
3
votes
1answer
72 views

Why aren't there $+\infty^{+\infty}$ real numbers?

I was reading this pop math piece on "the different sizes of Infinity." The article explains why the real numbers are uncountably infinite. Taking a real number, my uneducated mathematical mind ...
3
votes
2answers
39 views

Can I say that a fixed constant is less or equal infinity?

Mathematically speaking, given $c\in\mathbb{R}$, can I say that: $c\leq\infty$? E.g., is $10 \leq \infty$ a correct mathematical statement? I know this comparison is true in computer arithmetic, ...
2
votes
2answers
41 views

Difference in treatment of Infinity and Undefined

I understand that $$1)\; \lim_{x\to0}\frac1{x} = +\infty$$ $$2)\; \frac1{0} is\,undefined $$ If both infinity and undefined ...