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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59 views

Can Aleph Numbers be multiplied?

i.e., does it make sense to say something like $(2 * \aleph_0) > \aleph_0$ ? The original question I was thinking about is: if A = $\mathbb{Z}$ and B = {the set of even integers} is it correct to ...
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10answers
2k views

Is there a maximum value between open (0,1) set?

This question came up in my interview for a job application(you won't believe it but it was a C# programmer job application). Let's say we have a open set (0,1). Can we say that there is a maximum ...
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4answers
167 views

How much bigger is 1 than 0? [closed]

Bear with me -- nonsense to ensue. How much bigger is one than zero? The obvious answer is one. One is one bigger than zero. The backstory: Greg Fishel on WRAL said something along the lines of ...
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1answer
55 views

How to make a good “infinity plot”?

This is what I mean (note the labels in the x axis): The reason I'm looking at this problem is because I've always felt something was not right with truncated plots (e.g. of the exponential ...
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2answers
32 views

Indeterminate form as a series

We know that $0 \times \infty$ is an indeterminate form. However, is it equivalent to $0 + 0 + 0 + \cdots$? If yes, why we do not consider $\displaystyle \sum_{n = 0}^\infty 0$ an indeterminate form? ...
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1answer
48 views

Is the spacing between the set of all natural number powers bounded?

I was wondering whether the set of all numbers that can be expressed as a natural number to the power of another natural number has "infinitely wide" gaps or if there is some upper bound between the ...
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4answers
138 views

What's wrong with this “proof” that $\infty = -1$?

So I'm not great at math which is why I'm asking this. Someone send me the next math: Sum($1+2+4+8+16+$..)= infinity Which I understand S=sum($1+2+4+8+16+$..) S=1+sum($2+4+8+16+$..) So this ...
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2answers
64 views

why is $\lim_{x\to -\infty} \frac{3x+7}{\sqrt{x^2}}$=-3?

Exercise taken from here: https://mooculus.osu.edu/textbook/mooculus.pdf (page 42, "Exercises for Section 2.2", exercise 4). Why is $\lim_{x\to -\infty} \frac{3x+7}{\sqrt{x^2}}$=-3*? I always find 3 ...
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0answers
44 views

Why can't infinity be a number? [duplicate]

Why can't infinity be a number? All reasons why it can't that I come up with, require that it already isn't one. Yes you are right, by definition $\infty$ is larger than all numbers, take $\infty ...
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2answers
56 views

Sum of alternate terms of Riemann Zeta function

If $\sum\limits_{n=1}^{\infty}\frac{1}{n^{4}}=\frac{\pi^{4}}{90}$ Then find the value of $\sum\limits_{n=1}^{\infty}\frac{1}{(2n-1)^{4}}$ The book I took this problem from makes no mention of Riemann ...
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1answer
52 views

Are these “infinity” sequences true? [duplicate]

For $1\over 3$, you get $0.\overline3$, which is $0.33333...$. The threes go on forever. You can't ask "What happens if it ends in an eight?" because it simply doesn't end. For SSSSS..., what if it ...
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1answer
43 views

Limit of $x^2+3\sin x$ as $x$ goes to negative infinity [closed]

For every $x, x^2+3 \sin x \ge x^2 −3$ and, for every $c \ge 3, c^2 −3 \ge c$ hence, defining $x(c)=−c,$ one gets: $\forall c\ge 3, \exists x(c), \forall x \le x(c), x^2 + 3\sin x \ge c.$ I got this ...
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1answer
16 views

Number of equivalence classes of binary sequences which differ only by finitely many elements.

This question rose up when i was reading a problem the author used to argue against the axiom of choice. Consider the set of all (infinite) sequences of 0's and 1's. Q1) How many such sequences are ...
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3answers
102 views

Is the number of finite strings infinite?

I already asked this question on Stack Overflow and people kept voting me down and telling me it's "more of a maths question" so I will ask the question again: Assuming a finite alphabet, (eg: A,B), ...
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2answers
60 views

Calculating $ \lim_{n\to \infty} (1+\sin({1}/{n}))^{n}$ without L'Hopital or series expansions [duplicate]

I am trying to calculate the following limit, without using the L'Hopital rule or series expansions: lim (1+sin(1/n))^(n), n->infinity I now that it is the ...
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0answers
27 views

Logarithmic Series [duplicate]

I was doing a bit of math when I came across logarithmic series. I have no idea from where they come from. They seem so unrelated, that I have no intuition behind them at all. So, can anyone prove ...
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5answers
105 views

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
133 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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0answers
31 views

Infinity and the complex infinity

What's the difference between infinity and the complex infinity, and why is $\tan 90^{\circ}$, according to Wolfram Alpha, equal to the complex infinity and not undefined? Please see the following ...
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2answers
71 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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2answers
51 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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0answers
29 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
73 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
77 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
112 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
100 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, ...
6
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1answer
84 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
70 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
74 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
104 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
107 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
119 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
239 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
65 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
98 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
23 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
39 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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0answers
39 views

Infinity Gradient

I calculate infinity gradient, but I am not sure is this correct.
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0answers
100 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
52 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
13 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> = ...
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1answer
65 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
141 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
69 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
66 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 ...
2
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0answers
36 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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3answers
98 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
51 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 ...