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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7
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3answers
3k views

Does 0% chance mean impossible? [duplicate]

Suppose we pick a random real number between 0 and 1 and call it $x$. There are $2^{\aleph_0}$ possible values, so the chance of picking any specific number (such as $x$) in that range is 0. But in ...
3
votes
1answer
60 views

Circle is similar to a polygon with infinite number of sides

It is know from the time of Euclid, that a circle is similar to a polygon with infinite number of sides. But this ^^ is informal. Do you know any formalization where it appears that a circle is a ...
1
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3answers
409 views

Why is the Axiom of Infinity necessary?

I am having trouble seeing why the Axiom of Infinity is necessary to construct an infinite set. According to a professor of who's mine teaching a class on "infinity," the Peano axioms are only ...
4
votes
2answers
79 views

Why was $\aleph$ (aleph) chosen for infinities?

Why did Cantor choose a letter from the Hebrew alphabet to represent infinities, rather than using some Greek letter?
1
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1answer
26 views

What happens to Chebyshev polynomials integration when n=1

The integration of Chebyshev polynomials of the first kind has the following value, $$\int T_{n}(x) \, dx = \frac{1}{2} \, \left( \frac{T_{n+1}(x)}{n+1} - \frac{T_{n-1}(x)}{n-1} \right)$$ what happens ...
-1
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0answers
54 views

Proof of 1^2 + 2^2 + 3^2 + 4+2 +… infinity = 0 [on hold]

Leonhard Euler proved that the sum of the series $$ 1^n + 2^n + 3^n + 4^n + ... ∞ = 0 $$ where n is an even number. I know that this is a diverging series, but the infinity part of it somehow makes ...
0
votes
1answer
27 views

Is the Probability of Selecting 3 Random and Colinear Points nil?

Recently, the mathematics YouTube channel released a video titled "Triangles have a Magic Highway - Numberphile". In the video, at 6:40, the expert being videoed says that the probability of any three ...
31
votes
8answers
4k views

Is the set of all valid C++ programs countably infinite?

I have heard that the set of valid programs in a certain programming language is countably infinite. For instance, the set of all valid C++ programs is countably infinite. I don't understand why ...
0
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5answers
580 views

Is arithmetic with infinite numbers fictitious?

In 1933 Skolem constructed models for arithmetic containing infinite numbers. In a 1977 article Stillwell emphasized the constructive nature of Skolem's approach; see here. Is this at odds with ...
2
votes
1answer
51 views

Cardinality of polynomials with real coefficients

What is the cardinality of the set of all polynomials with real coefficients? I know the power set of R is "more infinite" than R, so to speak, but I'm unsure of how to prove that there does or does ...
1
vote
1answer
658 views

A formal proof that a sum of infinite series is a series of a sum?

I feel confused when dealing with ininities of any kind. E.g. the next equation is confusing me. $$\displaystyle\sum^\infty_{n} (f_1(n) + f_2(n)) = \displaystyle\sum^\infty_{n_1=1} f_1(n_1) + ...
0
votes
2answers
57 views

Different infinity, same limit?

I heard that there are different ranks of infinity, like $\aleph_0, \aleph_1, \aleph_2$, etc, my question is, the base of natural log, i.e. '$e$' is defined by a limit of taking $n\rightarrow$infinity ...
3
votes
1answer
44 views

What is the derivative of $\int_{-10}^{-3} e^{\tan(t)} \,dt$ with respect to x?

We were learning about the Fundamental Theorem of Calculus today in my high school and the above integral came up as an example of an integral with a "constant" value. At first I accepted that the ...
1
vote
1answer
48 views

What is an infinite gap minus another infinite gap?

I was asked this question on a quiz I received a few days ago and I was kind of confused on what the answer would be. Here it is, Set up and find the area between $$f(x)=x^2-x$$ and $$g(x)=x-1$$ ...
0
votes
1answer
34 views

How to disprove that $\text{ span }\{x_1,…,x_k\}=\text{ span }\{y_1,…,y_l\}$ if $x_i\in \text{ span }\{y_1,…y_l\}\ \forall i=1,…,k$?

If $y_1,...,y_l$ are vectors in vector space V and $x_i\in \text{ span }\{y_1,...y_l\}\ \forall i=1,...,k$, how to disprove that span$\{x_1,...,x_k\}=\text{ span }\{y_1,...,y_1\}$. In my ...
1
vote
3answers
42 views

How to solve this limit involving cube root and infinity?

How can I solve this limit? I know the answer is $2/3$. I tried factorisation, but solving the complicated denominator using L'Hopital's Rule returns a wrong answer, $0$. $$ \lim_{x\to\infty} ...
1
vote
2answers
639 views

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 ...
8
votes
3answers
868 views

The smallest infinity and the axiom of choice

The short version of this question is: which (natural) axiom should be added to ZF so that the statement "$\aleph_{0}$ is the smallest infinity" becomes true? A set $A$ is called infinite if it can ...
4
votes
1answer
709 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 ...
35
votes
4answers
3k views

What's between the finite and the infinite?

I'm wondering if there are any non-standard theories (built upon ZFC with some axioms weakened or replaced) that make formal sense of hypothetical set-like objects whose "cardinality" is "in between" ...
1
vote
2answers
56 views

How can a bijection be made from $\mathbb{N}$ to $\mathbb{Q}$ using diagonalization?

I'm studying Cantor's diagonalization, but something seems unclear to me. There is this table for diagonalization: ...
1
vote
1answer
36 views

Limit of a difference

Let $\lim_{n \to \infty} f_n(x) = f(x)$. Now consider $$\lim_{n \to \infty} (f_n(x) - f(x))$$ Usually I would say that $$\lim_{n \to \infty} (f_n(x) - f(x)) = \lim_{n \to \infty} f_n(x) - \lim_{n \to ...
10
votes
7answers
1k views

How can a Cauchy sequence converge to an irrational number?

I am a physics major and would like to clear a confusion regarding complete metric spaces. I am quoting the definition of a Cauchy sequence from wikipedia below Formally, given a metric space $(X, ...
0
votes
1answer
169 views

Does Pi contain itself? [duplicate]

Alright, recently there was a question on 9gag whether the digits of $\pi$ may contain $\pi$ itself here's the original. One user had - in my opinion - a really plausible answer: Here's his answer. ...
2
votes
2answers
86 views

More numbers between $2$ and $4$ than between $2$ and $3$? (I am not a mathematician.) [duplicate]

Between $2$ and $3$ there are infinite numbers and between $2$ and $4$ there are infinite numbers. So which "infinity" is greater?
1
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5answers
187 views

Find the limit of $x +\sqrt{x^2 + 8x}$ as $x\to-\infty$

$$\lim_{x\to -\infty} x +\sqrt{x^2 + 8x}$$ I multiplied it by the conjugate: $\frac{-8x}{x - \sqrt{{x^2} + 8x}}$ I can simplify further and get: $\frac{-8}{1-\sqrt{1+\frac{8}{x}}}$ I think there ...
1
vote
3answers
434 views

Why does Wolfram Alpha say that $n/0$ is complex infinity?

I typed a number divided by 0 on Wolfram Alpha and thought that it would say "undefined". However, when I pressed enter it told me that the answer is complex infinity. I have always been taught ...
0
votes
2answers
48 views

How come it be $\frac{3}{2}A$ and not only $A$?

OK I admit I was too lazy to type this question so I took a screenshot , I got it from the site @brilliant.org where it asked in terms of $A$ what would be the 2nd summation equation ? The explained ...
-1
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1answer
32 views

Find the ratio of a geometric sequence such that its sum is $4$ times the first term

How to find the sum to infinity: the sum to infinity of a geometric progression is 4 times the first term. Find the common ratio.
27
votes
3answers
851 views

What is the largest set for which its set of self bijections is countable?

I recently came across a problem which required some knowledge about the self bijections of $\mathbb{N}$, and after looking up how to construct some different bijections I came across the result that ...
12
votes
1answer
184 views

Countable-infinity-to-one function

Are there continuous functions $f:I\to I$ such that $f^{-1}(\{x\})$ is countably infinite for every $x$? Here, $I=[0,1]$. The question "Infinity-to-one function" answers is similar but without the ...
0
votes
0answers
55 views

If we think of infinity as a number, how does it affect the compactness/completeness of a metric space?

I was recently reviewing some notes regarding compactness, in which the sequential definition is given i.e. "$A$ is compact if any sequence in $A$ has a subsequence which converges to a limit in $A$. ...
0
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0answers
32 views

If I can prove f(n) = g(n+1) by induction when n is finite, Can I prove f(n) = g(n) by taking n = $\infty$

I have to prove f(n) = g(n) when $n = \infty$. Now I can prove f(n) = g(n+1) by induction when n is finite. Can I say $f(n) = g(n)$ by taking $n = \infty$?
10
votes
1answer
410 views

Infinity-to-one function

Are there continuous functions $f:I\to S^2$ such that $f^{-1}(\{x\})$ is infinite for every $x\in S^2$? Here, $I=[0,1]$ and $S^2$ is the unit sphere. I have no idea how to do this. Note: This is ...
2
votes
2answers
271 views

Where is the flaw in my Continuum Hypothesis Proof?

I am not a mathematician, but rather a computer engineer with a curious mind. The continuum hypothesis (CH) has gripped my attention today, and I even asked a question about it earlier today. ...
2
votes
1answer
83 views

Is this interpretation of the continuum hypothesis correct?

I am not a mathematician, but rather a computer engineer with a curious mind. The continuum hypothesis states (I believe) that there does not exist a set $S$ such that $\aleph_0 < |S| < ...
0
votes
1answer
201 views

Can I have something larger than infinite? [duplicate]

My question is "Can I have something larger than infinite?" Sometimes, we add infinite numbers into our set of numbers by simply extending our set and adding infinite numbers to it. But can't you ...
0
votes
2answers
52 views

Unique infinite subsets of the integers

Edit: Great points on the comments. There is no unique set of unique infinite subsets of the integers. Is this a better question? What is the largest possible cardinality of a set which is a set of ...
3
votes
1answer
4k views

Is Infinity =Undefined?

Let's start with the equation $$y =\frac 1{(x-1)}$$. The positive and negative limit of $x$ at $1$ both approach $+∞$, but at $x = 1$, $y$ is undefined. I know this is because the denominator of the ...
1
vote
1answer
85 views

Limit to infinity and infinite logarithms?

When trying to evaluate$$\ln(\ln(\ln(\ln(\cdots\ln(x)\cdots))))$$I noticed that the answer was bound to be complex for any $x$. Plugging in a very, very large real number in for $x$ will eventually ...
1
vote
7answers
242 views

Does infinite equal infinite?

I have a question. Let $x$ be infinite. $$2x=\infty\times2, \quad 2x=\infty$$ So actually, does $2x=x$?
-2
votes
2answers
91 views

How small is infinite? [closed]

There are a lot of posts concerning how big infinite is, but I wonder how small infinite is. One can clearly see (ignoring a few things) that$$\frac{\infty}2=\infty$$Which means that no matter how ...
4
votes
3answers
89 views

The cardinality of Indra's net?

This question has been asked before, but the title of the post was so general that it received no sufficient answer. What is the cardinality of the set of jewels and reflected jewels in Indra's Net? ...
0
votes
3answers
277 views

Concept behind the limit to infinity?

I can across transfinite numbers and came up with a thought. What if$$\lim_{x\to\infty}f(x)=f(T)$$where $T$ was a transfinite number? Generally, in calculus, I have noted that it is two different ...
0
votes
1answer
35 views

A box comprised of infinite number of small similar boxes.

On Wikipedia, I read, "A box can be thought of 'small boxes' infinitely repeating in all three dimensional directions" I don't understand what does Wikipedia wants to say with a box containing ...
1
vote
3answers
38 views

Classify the type of discontinuity at $x_0 = 0$

for (a) I think it is essential because the right side goes to infinity. for (b) I think it is removable because the function is not defined in $0,$ same goes for (c) I am really not sure about ...
0
votes
0answers
23 views

What is the origin of the distinction between assignable and inassignable number?

Leibniz described his infinitesimals as being inassignable numbers in a number of texts, e.g., in his Cum Produisset that was analyzed in detail by H. Bos in a seminal text dating from the 1970s. The ...
2
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0answers
23 views
1
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3answers
61 views

Proof using formal definition: Infinite limit

I was wondering how get the proof of this limit: $$\lim\limits_{x\to -\infty}\dfrac{{x^2} - x + 1}{x + 4} = -\infty$$ The problem is that I don't know what to do for find the appropriated values to ...
2
votes
2answers
33 views

values that can be attained by random variables

Can a discrete random variables takes the values $+ \infty$ and $- \infty$ ? Can someone explain to me this with an example?