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?

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If $0{.}9\ldots$ is $1$, what does that make $0{.}3\ldots$?

So I recently learned that $0{.}9$ repeating is equal to $1$: $$ x = 0{.}9\ldots\\ 10x = 9{.}9\ldots\\ 9x = 10x - x = 9{.}9\ldots - 0{.}9\ldots = 9\\ x = 9x/9 = 9/9 = 1\\ x = 1 $$ Or a simpler ...
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4answers
172 views

Number of iterations to reach cosine's fixed point

I was messing around with my calculator the other day when I saw something interesting happen. Whenever I repetitively took the cosine of any number, it always ended up on a particular number ...
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2answers
83 views

'Smaller than infinity' notation

I've been coming across some papers (written in the 1960s - 1970s) that use the following peculiar statement: Let use denote by $H$ the space of all grid-functions $w_r$ for which: $$ ...
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63 views

Ice cream issue in Lem's 'Extraordinary Hotel'

Could you clarify the ice cream issue mentioned at the end of the story The Extraordinary Hotel (pages 189-190 here)?
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37 views

Is it true that the slope of a vertical line times the slope of a horizontal like don't equal $-1$, even though they're perpendicular?

I know that the slopes of two lines that are perpendicular have a value of $-1$ when multiplied because they're opposite reciprocals (e.g. $5$ and $-{1\over 5}$), but what if there's a horizontal and ...
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2answers
105 views

What is the one point compactification of the reals?

In several of my questions this theorem has come up. What is the one-point compactification of the reals? Does it have to do with limits and dividing by $0$? I vaguely remember something about a ...
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2answers
81 views

Is there any unreachable result?

I hope that this question is reasonable and make sense because I am not sure. Every theorem's proof is consisting of finite logical steps. Can a proof of the theorem require infinitely many ...
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4answers
123 views

Does the commutative property of addition hold when we're dealing with infinity? [closed]

I was wondering, if I evaluated some kind of algebraic expression and I got the following: $-\infty+\infty$. Is infinity commutative like it is with real numbers? Could I say that $$-\infty+\infty = ...
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1answer
109 views

Is $\mathbb{N}$ a well-founded set?

I was reading about Von Neumann's construction of $\mathbb{N}$, I understood that $\mathbb{N}=\{\emptyset, \{\emptyset\}, \{\emptyset, \{\emptyset\}\},...\} $. I see that, with this construction, ...
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1answer
94 views

The game with countable amount of steps

Here is a cute problem. The angel and the devil play a game. Firstly the angel has an empty box and the devil has a box which contains all numbers from $\mathbb{N}$ (one copy of every natural ...
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1answer
85 views

Is there a proof that zero multiplied by infinity = a real number [duplicate]

Someone told me that $0\times \infty = 1$. I am baffled by this because I thought you cannot multiply by infinity because it isn't a real number. If you can, is it possible to explain how and give ...
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2answers
89 views

A peculiar observation about infinity.

Let ${\sqrt2^\sqrt2}^{\sqrt2^...}=y$. Then $\sqrt 2^y=y$ $\implies \sqrt 2=y^{1/y}$ $\implies \sqrt 2 =1$ $\implies 2 =1$ !! but how come that be. Can anyone explain this and point out what is ...
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5answers
113 views

Analysis: Prove divergence of sequence $(n!)^{\frac2n}$

I am trying to prove that the sequence $$a_n = (n!)^{\frac2n}$$ tends to infinity as $ n \to \infty $. I've tried different methods but I haven't really got anywhere. Any solutions/hints?
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3answers
131 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 ...
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3answers
180 views

Understanding infinity

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 ...
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7answers
276 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 ...
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2answers
102 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 ...
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1answer
29 views

proving a limit of a series with a sum [duplicate]

I just can't find a way to prove it.
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4answers
100 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?
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0answers
25 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 ...
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1answer
41 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 ...
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68 views

Infinity Gradient

I calculate infinity gradient, but I am not sure is this correct.
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130 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 ...
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3answers
143 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 + ...
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3answers
53 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.
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0answers
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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> = ...
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1answer
89 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 ...
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3answers
152 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 + ...
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2answers
74 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 $n$ ...
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2answers
69 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 ...
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0answers
59 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 ...
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107 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?
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3answers
52 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 ...
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1answer
65 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 ...
5
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1answer
89 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, ...
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2answers
63 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 ...
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2answers
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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 ...
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2answers
51 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 ...
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1answer
102 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
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1answer
54 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 ...
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3answers
147 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 ...
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3answers
201 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 $\aleph_1$.
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3answers
75 views

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

I need to prove that $\lim_{x\rightarrow \infty}\dfrac{x^2}{e^x}=0$.
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1answer
85 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 ...
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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, ...
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2answers
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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 ...
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Zero and infinity

Introduction [can be skipped without loss of generality]. This question was closed and, recently, deleted, perhaps for good reason. It did have an answer with 10 upvotes, and another (mine) with 15 ...
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1answer
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1/∞ is 0 or infinitesimal?

Since ∞>0 , so 1/∞>0, thus I think 1/∞ should be infinitesimal, but the calculus book says $\displaystyle \lim_{x \to \infty} \frac{1}{x}= 0$ So is 1/∞ 0 or infinitesimal ? P.S.I mean 1/∞ and ...
19
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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 ...
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1answer
43 views

Dividing a number into infinite pieces

Last day in physics teacher said that any number divided into infinitely many pieces is zero.It got me thinking in kind of weird direction so here is what I was thinking about and how I tried to ...