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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1answer
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absolutely integrability implies function approaches zero at positive infinity

Is the following statement true? $$\text{If function $f$ is absolutely integrable on $[0, \infty)$, this implies } \lim_{x \rightarrow \infty} f (x) = 0.$$ If yes then how would I prove it? Note: I ...
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1answer
23 views

Interchange summation (outer infinite, inner dependent on outer)

For finite summation limits, I believe that the following holds (for some general function $f$): $\sum_{i=2}^n \sum_{j=1}^{i-1} f(i,j) = \sum_{j=1}^{n-1} \sum_{i=j+1}^{n} f(i,j)$ ... (1) However, I'm ...
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0answers
52 views

Why are positive rational numbers countable but real numbers are not? [duplicate]

If we can say that any positive rational number is countable or listable by showing that every positive rational number is the quotient of p/q of two positive ...
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3answers
56 views

Solution of the equation $\cot \theta = 2\cot 2\theta$

I've tried to solve the equation $\cot \theta = 2\cot 2\theta$ with the command 'Reduce' of Mathematica and obtained $\theta = n\pi$ as the solution with n an integer. But $\theta=n\pi$ is clearly a ...
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2answers
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What is the result of a number greater than 2 raised to the power of {Aleph-0}?

So, I know that $2^{\aleph_0} = \beth_1$, right? What about another number, say $10$, raised to the power of $\aleph_0$? Is $10^{\aleph_0} = \beth_1$ also true, or is $10^{\aleph_0} > \beth_1$ ...
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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 ...
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1answer
28 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 ...
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1answer
69 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 ...
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2answers
83 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?
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3answers
435 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 ...
2
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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 ...
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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 ...
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2answers
58 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 ...
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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 $$g(x)=x-1$$ ...
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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 ...
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3answers
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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} ...
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3answers
873 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 ...
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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: ...
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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 ...
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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?
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1answer
170 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. ...
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4answers
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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" ...
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7answers
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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, ...
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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 ...
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1answer
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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.
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5answers
585 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 ...
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1answer
201 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 ...
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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 $A$. ...
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0answers
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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 $n = \infty$?
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1answer
413 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 ...
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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. ...
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1answer
84 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| < ...
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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 ...
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3answers
859 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 ...
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1answer
86 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 ...
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1answer
203 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 ...
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2answers
92 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 ...
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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 ...
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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? ...
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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 ...
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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 ...
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3answers
279 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 ...
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0answers
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Prove the squared vector 2-norm is $\leq$ sum of 1-norm and infinity-norm

How do I prove that $$\|x\|_2^2 \leq \|x\|_1 \|x\|_\infty?$$
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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?
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2answers
72 views

Can I assume the continum hypothesis in a proof

I am showing that the cantor ternary set has the same cardinality as $\mathbb{R}$ I want to use the fact that it is uncountably infinite and a subset of $\mathbb{R}$. ($|N| < |C| \leq \mathbb{R}$) ...
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2answers
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Limit laws when not both limits exist

In the calculus textbooks I've come across, the limit laws are given on the condition that both individual limits exist. Is it safe to weaken that condition by saying that they are valid as long as ...
2
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1answer
67 views

There are more languages than programs?

I am reviewing some Turing machine material...and I come across this the set of all programs are countable (convert them into binary string, each of which represent an integer) whereas the set of ...
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1answer
37 views

What is $1^\omega$?

In Wolfram Mathworld, Ordinal exponentiation $\alpha^\beta$ is defined for limit ordinal $\beta$ as: If $\beta$ is a limit ordinal, then if $\alpha=0$, $\alpha^\beta=0$. If $\alpha\neq 0$ then, ...
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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 ...
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3answers
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probability on countable infinite sets

My question relates to probabilities on countable infinite sets. For example, what is the probability of choosing an even number from the positive integers. Believe it or not I am interested in this ...