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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0
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2answers
66 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 ...
1
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1answer
49 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
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1answer
35 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 perspective,...
1
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3answers
115 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} \left((...
8
votes
3answers
976 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 ...
1
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2answers
65 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
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1answer
43 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 \...
2
votes
2answers
91 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?
0
votes
1answer
199 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. ...
36
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" ...
10
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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
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2answers
53 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
36 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.
0
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5answers
618 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 ...
12
votes
1answer
447 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
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0answers
62 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
37 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
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1answer
424 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
288 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
87 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| < 2^{\...
0
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2answers
58 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 ...
27
votes
3answers
950 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 ...
1
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1answer
90 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 ...
0
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1answer
209 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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votes
2answers
102 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 ...
0
votes
1answer
41 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 ...
4
votes
3answers
100 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? ...
1
vote
3answers
39 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
24 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 ...
0
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3answers
296 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 ...
2
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1answer
59 views

Prove the squared vector 2-norm is $\leq$ sum of 1-norm and infinity-norm

How do I prove that $$\|x\|_2^2 \leq \|x\|_1 \|x\|_\infty?$$
2
votes
2answers
35 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?
2
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2answers
79 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 $\mathbb{R}$. ($|N| < |C| \leq \mathbb{R}$) ...
-1
votes
2answers
82 views

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
votes
1answer
78 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 ...
3
votes
1answer
45 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, $\...
1
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3answers
74 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 ...
0
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3answers
84 views

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 ...
0
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1answer
57 views

Thomson's Lamp Question

The Thomson's Lamp paradox: A mad scientist owns a desk lamp. It begins in the toggled on position. The scientist toggles the lamp off after one minute, then on after another half-minute. After a ...
0
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3answers
64 views

calculate the limit of the following function.

$$\lim_{x \to 0} \frac{5x - e^{2x} + 1}{3x + 3e^{4x} - 3}$$ For some reason I can't find the trick to solve this. Tried a lot but it always come to a place where its $\frac{0}{0}$ or $\frac{\infty}{\...
0
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0answers
49 views

evaluate $\int_0^\infty \frac{(ln(x))^2 }{1+x^2}dx$ [duplicate]

I am attempting to evaluate the following integral: $$\int_0^\infty \frac{(ln(x))^2 }{1+x^2}dx$$ Using the substitution $x=e^u$ and $dx=e^u du$, I get: $$\int_{-\infty}^\infty \frac{u^2}{e^{-u} + e^u}...
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1answer
35 views

Zero-infinity hypothesis [duplicate]

math.stackexchange community. I have joined to inquire on a hypothesis a friend of mine has recently proposed. Please note: before posting this, I have repetitively told him that his logic is flawed ...
0
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2answers
35 views

size of infinite strings and infinite alphabets

Please forgive the lack of formal vocabulary. Which set has a larger cardinality? A) a set of all possible countably infinite strings with a finite alphabet of symbols. B) a set of all possible ...
0
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2answers
55 views

The intersection of an infinite number of subspaces is a subspace

Let $V$ be a finite dimensional vectorspace over a field $\mathbb{ F}$. It's easy to show that if $U$ and $V$ are subspaces of $V$ then $U \cap V$ is a subspace. But what if there are an infinite ...
0
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0answers
51 views

integral vs. residue at infinity

I have an issue with residues at infinity. I am computing the integral $\displaystyle{\int_{C_3^+(0)} \dfrac{e^{3z}}{z^2(z^2+2z+2)} dz} $ Since all three poles ($0$ of order 2, $1\pm i$ of order 1) ...
5
votes
4answers
125 views

Is $\aleph_1\cdot\aleph_1=\aleph_1$?

I'm currently trying to understand the basic notions concerning infinity. I think I understand that $\aleph_0\cdot\aleph_0=\aleph_0$ but how about $\aleph_1$? Is $\aleph_1\cdot\aleph_1=\aleph_1$ i.e. ...
4
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2answers
444 views

Compute a limit or prove that it does not exist

Do the following limits exist? Compute them or prove that they do not exist. (a) $\lim_{x\to 1}\frac{x^2-x}{2x^2-x-1}$ (b) $\lim_{x\to 1}\frac{|x-1|}{2x^2-x-1}$ For (a) it's pretty easy to see that ...
2
votes
2answers
40 views

Proving the following limit statements

I need to prove those If $f(x)\ge 0$ and $\lim_{x\to x_0}f(x)=L$, then $\lim_{x\to x_0}\sqrt{f(x)}=\sqrt{L}$. If $\lim_{x\to x_0}f(x)=L$, then $\lim_{x\to x_0}|f(x)|=|L|$. If $f(x)\ge g(x)$ for ...
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0answers
38 views

Friend B and C have eaten zero apples. How many more apples has C eaten?

Friend $A$ claims that he has eaten $1$ apple today. Friend $B$ responds. Congrats, I have eaten $0$ apples, so that is $\infty$ more apples than me. Friend $C$ says, but I have also eaten $0$ ...
1
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1answer
27 views

Probability of selecting a number in a repeating decimal series

For example in a infinitely repeating series such as $\frac{110}{111}=0.\overline{990}$, what would be the probability of selecting a 0 in the series generated by the infinitely repeating decimals? I ...