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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Looking for some function such that $\lim\limits_{x\to\infty}f(x) \ne \infty$

I am looking for a function $f$ that is differentiable and $f'(x) \ge c \gt 0$ for all $x \in \mathbb{R}$ and $\lim\limits_{x\to\infty}f(x) \ne \infty$? Is there such function, or am I wasting my ...
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3answers
2k views

Is the infinite root of any number equal to $1$?

I was messing around in IRB and I decided to make a $n^{th}$ root function and noticed that for very large roots of numbers, the answer always converges to $1$. It has been a while since I have done ...
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6answers
1k views

Limits of $f(x)=x-x$

It's obvious that $f(x)=x-x=0$. But what exactly happens here? You have a function $f(x)=x-x$ and you have to calculate the limits when $x\to \infty$ This'll be like this: $$\lim\limits_{x\to ...
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1answer
273 views

Equivalence of sequences and subsets of natural numbers

For me, facts like the independence of the continuum hypotheses from ZFC cast a doubt on the "law of the excluded middle". (In this context, the doubt is that there might be no "final set theory" such ...
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3answers
1k views

What did Gauss think about infinity?

I have someone who is begging for a conversation with me about infinity. He thinks that Cantor got it wrong, and suggested to me that Gauss did not really believe in infinity, and would not have ...
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2answers
97 views

Infinite or unknown?

If you have $0$ clients on Monday, and $5$ clients on Tuesday, how many times have the number of clients you had grown from Monday to Tuesday? $A$ - Infinite times $B$ - Unknown $C$ - ...
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1answer
139 views

Can the ongoing need for a meta language be stopped by a loop?

As an afterthought to this question on sets in set theory, and more specifically to the observation that a (first-order) logic requires a meta-language to explain itself (i.e. there is already an ...
3
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1answer
350 views

What does universal quantification mean?

In ZFC, for example, there is no universal set, so what does it mean to write $\forall x (\cdots)$, i.e., for all sets something is true? Does it avoid the problem by quantifying over all elements but ...
0
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1answer
185 views

Evaluation of $\sum_{x=1}^{\infty}x^{-x}$ [duplicate]

Possible Duplicate: “Closed” form for $\sum \frac{1}{n^n}$ Is it possible to evaluate this sum, and if so, how would you do it? This question has been irritating me for a ...
5
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5answers
479 views

Why does it matter to approach zero from the left or right in 1/0?

I was surprised to see that 1/0 is undefined. One answer mentions that $1/0$ can be +$\infty$ or -$\infty$ depending on whether $0$ is approached by the left or the right: ...
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3answers
6k views

Isn't there a bijection between real numbers and natural numbers?

I'm not a math guy and probably this is a stupid question. However, I was browsing Wikipedia out of curiosity and I could not understand this one statement: [...] not all infinite sets have the ...
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1answer
1k views

Why do some divergent series apparently seem to converge (e.g. Grandi's series)?

Grandi's series is defined as: $$\sum_{n=0}^{\infty} (-1)^n = 1 - 1+1-1+\cdots$$ By plainly looking at this series it seems like the value of it is either $1$ or $0$ by doing the following ...
3
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2answers
201 views

Is $\infty$ enough or do I need to write $+\infty$

This is a question of notation. I have seen in many articles that people often denote $+\infty$ when talking about 'positive infinity' of the real numbers. Is that a convention, or it can be written ...
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12answers
17k views

I have learned that 1/0 is infinity, why isn't it minus infinity?

My brother was teaching me the basics of mathematics and we had some confusion about the positive and negative behavior of Zero. After reading a few post on this we came to know that it depends on the ...
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3answers
170 views

Finding a finite integer in infinite space

An adversary selects an integer k from the set of non-negative integers. Does any algorithm exist that, using only tests for equality or inequality (<, =, >), ...
3
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2answers
2k views

There's a real between any two rationals, a rational between any two reals, but more reals than rationals?

The following statements are all true: Between any two rational numbers, there is a real number (for example, their average). Between any two real numbers, there is a rational number (see this proof ...
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2answers
929 views

Infinite Disjunctions and Conjunctions

While doing work on propositional logic (namely, proving the Generalized De Morgan's Laws), I found myself wondering why precisely an infinite conjunction or disjunction are not permitted, due to the ...
2
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5answers
462 views

Is there a scientific name for 0.infinity?

First of all I want to say that when coming to math - I know absolutely nothing - so please forgive me if my question is not "scientifically" correct, if it is not "syntax-correct" - or even too ...
2
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1answer
119 views

Terminology for a property that holds in the finite but not infinite case?

(I apologize if this is a duplicate, but I don't know what terms to search for. Please feel free to close this if this has already been asked.) There are some properties of finite objects that don't ...
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2answers
471 views

Infinite palindromic numbers

Today’s Abstruse Goose comic got me thinking: Does an “infinite palindromic” number (other than the obvious $x\times1.\overline1$) make sense? In any conventional number system, the answer (as far ...
3
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2answers
570 views

Sierpinksi like triangle construction. How to find the number of triangles in each iteration?

So here is the question: If we look at the Sierpinski triangle (left column of attached image) and think about how many triangle's it takes to make the shape at each iteration we can get the sequence ...
3
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1answer
944 views

speed of convergence to infinity

Lets take for example $\lim_{x\rightarrow\infty} \log(x)$, from a mathematical point of view this is $+\infty$, but from a logical point of view it's clear that $x$ converges to $+\infty$ much more ...
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2answers
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Practical Use of Series Expansion at $x=\infty$

Asking WolframAlpha on certain functions, it happens that you get a series expansion at $\infty$. Thinking of the expansion as an approximation of the function in the vincinity of a point $a$, like in ...
2
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2answers
117 views

Subset of infinite set

A point. There is an infinite number of lines intersecting it, but this is less than the number of possible lines. How do we represent this in mathematical notation?
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4answers
2k views

Examples of continuous growth rates greater than exponential

I read on Wikipedia that growth rate of a function can sometimes be greater than exponential. Can you give me some examples of such functions (preferably continuous ones)? Obviously $x^x$ grows ...
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5answers
889 views

Repeating digits in $\pi$

As $\pi$ has infinite digits in its decimal expansion, one could argue that its digits will repeat after a finite number of digits. If so, it is a rational number. What's wrong with this argument?
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1answer
568 views

Am I thinking about infinitesimals correctly?

I was all set to begin Calculus 2 when I thought, "I should have a more intuitive sense of what's happening with differentials before I move on." I want to tell you what I've learned and ask you all ...
2
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1answer
213 views

$\frac{-1}{0}$ is infinity or -infinity? [duplicate]

Possible Duplicate: Division by $0$ I was solving a question for my brother today when i got this doubt, i I arrived at the answer as $\frac{-1}{0}$ Will the answer be infinity since ...
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4answers
1k views

Is a subset of infinity still infinity?

In looking at the post here which really got me thinking what infinity and how it is notated. As the question states, is a subset of infinity still infinity? Also, what area of the Math world covers ...
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1answer
599 views

Is there an absolute notion of the infinite?

Skolem's paradox has been explained by the proposition that the notion of countability is not absolute in first-order logic. Intuitively, that makes sense to me, as a smaller model of ZFC might not be ...
4
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1answer
209 views

Maximizing gambling performance over the long run

Background. We can play a game in which we can put one dollar and get out $X$ dollars, where $X$ is 2 dollars with probability $p>1/2$, or zero dollars with probability $1-p$. We also assume that ...
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2answers
253 views

Resolving a paradox concerning an expected value

We have a coin that has a probability $p>1/2$ of coming up heads (and probability $1-p$ of coming up tails). We now play the following game: We start with a fortune of one dollar. We toss the ...
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3answers
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The Aleph numbers and infinity in calculus.

I have a fairly fundamental question. What is the difference between infinity as shown by the aleph numbers and the infinity we see in algebra and calculus? Are they interchangeable/transposable in ...
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2answers
579 views

What is infinity times the reciprocal of infinity?

I was talking with a friend about interesting properties of numbers and their theoretical contradictions and solutions when we came up with this. What is the answer? So... $x * ∞ = ∞$ and... ...
3
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0answers
197 views

Proof that $ -1 = \infty $ . [duplicate]

Possible Duplicate: Infinity = -1 paradox $1+2+4+8+16+\ldots = \infty$ $LHS=1(1+2+4+8+16+\dots)$ $LHS=(2-1)(1+2+4+8+16+\ldots)$ $LHS=(2+4+8+16+32+\ldots)-(1+2+4+8+16+\ldots)$ ...
0
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2answers
414 views

Set of finite and infinite $0$-$1$ sequences countability [duplicate]

Possible Duplicate: Is the set of all finite sequences of letters of Latin alphabet countable/uncountable? How to prove either? Is the set of all strings with countably infinite length ...
5
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2answers
439 views

How many times more than $0$?

If I have $10$ apples, but you have $5$ apples, then I have $2$ times more apples than you. But what if I have $10$ apples, but you don't have any apples? If you look at the graph ...
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0answers
53 views

Probability in infinitary logic

Let X be a random variable taking the value 0.2 with probability 0.2, 0.4 with probability 0.4 and 0.8 with probability 0.2 and 1.0 with probability 0.2. Using Infinitary logic I can ask the ...
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2answers
96 views

The limit of a rational function

$$ \lim _{x \to -\infty} \frac{3x^2+3x}{2x^2+2}$$ Is a good practice to do this? Change the $ -\infty $ to $ \infty $, and change the sign of the $ x $ variables: $$ \lim _{x \to \infty} ...
2
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4answers
144 views

Sequence of numbers with infinite number of primes

If I have an infinite sequence of positive integers with infinite number of primes and if I have an infinite number of distinct sequences with such properties may I claim that there is an infinite ...
0
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2answers
408 views

Proof of an infinite sum of probabilities [duplicate]

Possible Duplicate: Alternative Expected Value Proof If $X$ is a random variable that takes values in the range $\left \{ 1,2,3,4,5,6,\ldots \right \}$ how can I prove the following ...
0
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1answer
102 views

Divisibility with sums going to infinity

I can't quite wrap my head around this. Given the formula $(1-x)(1+x+x^2+...) = 1$ It seems clear to me why this is true. All the x terms cancel out and we are left with one. And this is clearly ...
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1answer
104 views

Solve an infinite sum

I need to find the sum of this series: $1, 2 \left ( 1 - \frac{1}{\sqrt{15}} \right ), 3 \left ( 1 - \frac{1}{\sqrt{15}} \right ) ^ 2, 4 \left ( 1 - \frac{1}{\sqrt{15}} \right ) ^ 3, 5 \left ( 1 - ...
5
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2answers
287 views

Does an infinite random sequence contain all finite sequences?

If we have a finite alphabet, with each letter having a non-zero probability of being selected, will an infinite sequence of letters selected from that alphabet contain all finite sequences of letters ...
2
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3answers
238 views

What is the definition of $\infty$

I saw in a note that say $\infty$ is not a real number, and there is no interval of the form $(a, \infty]$? So what is the definintion of $\infty$?
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1answer
265 views

Division by zero [duplicate]

Possible Duplicate: Division by $0$ Everyone knows that $(x/y)\times y = x$ So why does $(x/0)\times 0 \ne x$? According to Wolfram Alpha, it is 'indeterminate'. What does this mean? ...
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2answers
349 views

Doubly infinite numbers

Real numbers are sequences of integers which are infinite in one direction. If I have a string which is infinite in both directions, say ...345123985..., then I can form an injection from these ...
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1answer
510 views

Can one compare negative infinity and positive infinity?

I have always wondered, is negative infinity less than positive infinity? Can I compare them?
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3answers
472 views

What is the result of $\infty - 1$?

I was wondering after reading "What is the result of infinity minus infinity", is there any logical result for $\infty - 1$ ?
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3answers
1k views

Rejecting infinity

I've heard about mathematicians who defend a strictly finite conception of mathematics, with no room for infinity. I wonder, how is it possible for these people to do this? Are there any concepts that ...