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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111 views

Infinite shots fired in a lattice forest

A hunter is standing in the center of an infinite 2D forest. There are point trees at all the integer lattice points. The hunter fires a gun with a bullet of zero width in a random direction. He ...
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3answers
299 views

How big is the size of all infinities?

"Not only infinite - it's "so big" that there is no infinite set so large as the collection of all types of infinity..." What does exactly mean? How many infinities are there? I've heard there are ...
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1answer
79 views

What are the factors of $\aleph_0$?

Extend the system of positive natural numbers with $\aleph_0$. Then we have: $$\aleph_0 = \aleph_0\cdot n,\quad \forall n \in \mathbb{N}^+$$ Does it make sense to talk about factors of $\aleph_0$? ...
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5answers
2k views

Which infinity is meant in limits?

For example, when we write $\lim_{x\rightarrow \infty} f(x)$ - which infinity is meant and why? Countable? If uncountable - which and why?
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4answers
963 views

infinity times infinitesimal - what happens?

So what happens if we multiply infinite number by. Infinitesimal number? Like $dx \times \infty$ where $dx$ is treated as in one-dimensional integration. Also, can we divide infinite number by ...
6
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3answers
221 views

Use of infinity as an “idealistic approximation”

There have been several recent posts about the work of N. J. Wildberger, a finitist who seems to think that mathematics should only focus on things that have some sort of "real world" connection, ...
2
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6answers
336 views

prove that , $2+2+2+2+2+ \cdots= 1+1+1+1+1+\cdots$

how can we prove that ?? i think they are equal but a friend say that they are not equal my argument is $$1+1+1+1+1 + \cdots = \infty$$ $$2+2+2+2+2+\cdots = (1+1) + (1+1) + \cdots = (1+1+1+1+1 ...
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4answers
1k views

Non-existence of irrational numbers?

I realize the title of my question will probably cause the raising of some eyebrows, so let me explain. Not sure whether to file this under "math" or "philosophy". This also might be able to be ...
4
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3answers
354 views

Why the need of Axiom of Countable Choice?

Two theorems: $(1)$ Countable Union of Countable Sets is Countable $(2)$ Cartesian Product of Countable Sets is Countable Linked are the formal proofs on Proofwiki. I do not understand why they ...
6
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2answers
324 views

Is the proper class of all ordinals equivalent to the potential infinity of pre-Cantor times?

My understanding is that the class of all ordinals is, by definition a proper class. This in the end is done to avoid a paradox: the collection of all sets would be paradoxical if you allow it to be a ...
0
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3answers
317 views

Why I can't calculate $0*log(0)$ but can $log(0^0)$

I got this doubt after some difficult in programming. In a part of code, i had to calculate: $$ x = 0 * Log(0) \\ x = 0*-Inf $$ and got $x = NaN$ (in R and Matlab). So I changed my computations to ...
12
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0answers
400 views

“Infinito”, a combinatorial game with infinite width game-tree

I recently designed a combinatorial game (sequential game of perfect information) with an infinite branching factor, that is it has a game-tree of infinite width. I'm wondering how is it possible to ...
2
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0answers
82 views

Paradox of Infinity? [duplicate]

If a series such as '$a$' below adds to infinity: $a = 1 + 2 + 4 + 8 + 16 + \cdots\to \infty$ Multiplying '$a$' by $2$ yields: $2a = 2 + 4 + 8 + 16 + \cdots\to \infty$ However when I subtract ...
0
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2answers
136 views

Infinity/exponential problems.

I want to evaluate $$\int_{0}^{\infty} e^{(i\alpha-1)x}\,\mathrm dx,$$ where $i$ is the imaginary number. $$\left [ \frac{e^{(i\alpha-1)x}}{i\alpha-1}\right]_0^{\infty}$$ At this point, I beleive, ...
0
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2answers
190 views

Understanding limits at infinity with regard to the definition of a limit

This is sort of a follow up to my previous question Say you have $$ \lim_{x\to +\infty} f(x) $$ where $f : \mathbb{R} \to \mathbb{R} , x \in \mathbb{R}$ What exactly does this mean? From the last ...
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2answers
224 views

Creating the set of natural numbers

I am not a mathematician but an engineer, so I can read some basics of the language proofs are written in. Second I am bad in probability and infinity and my question covers both. So I like to ...
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1answer
55 views

Statements true for all n Vs. statements true as n->infty

Let P be a statement. What are the necessary and sufficient conditions for the following statement to be true? (P is true $\forall n \in \Bbb N$)$\implies$(P is true as n$\to \infty$) As background ...
2
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1answer
134 views

Random infinite binary sequence

What I mean by random infinite binary sequence is an infinite sequence of $0$'s and $1$'s with probability of occurrence in this sequence equal to $1/2$ (all digits being equally likely). How is it ...
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4answers
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1 to the power of infinity, why is it indeterminate? [duplicate]

I've been taught that $1^\infty$ is undetermined case. Why is it so? Isn't $1*1*1...=1$ whatever times you would multiply it? So if you take a limit, say $\lim_{n\to\infty} 1^n$, doesn't it converge ...
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0answers
144 views

Relationship between ordinals and rank of well founded relations on $\mathbb N$

I want to understand the relation between ordinals and well founded relations on $\mathbb N$. I found a nice starting point here cut-the-knot/ordinals. Ordinals start like this 0={}, 1={0}, 2={0,1}, ...
2
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1answer
200 views

What is the probability of guessing the right number $n$ from all numbers $\mathbb{N}$?

As we all know, $\mathbb{N}$ contains infinitely many numbers. What is the probability of guessing the right number $n \in \mathbb{N}$, i. e. what is $\frac{1}{\infty}$? It is clear that there is a ...
2
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1answer
2k views

Infinity = Undefined?

Let's start with the equation $y = |1/(x-1)|$. 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 ...
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0answers
122 views

maximize an objective function with an infinite component

Suppose I have the following maximization problem: $\log\det(\alpha K_p)-c\alpha$ with respect to $\alpha$ with $c$ being a constant and $m$ being the dimension of $K_p$. Here, one of the eigenvalues ...
2
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2answers
105 views

Is there an infinite sequence AB, BC, CD, DX, …, YZ

Is it possible to construct an infinite set of ordered pairs of form S = {(A, B), (B, C), (C, D), (D, x), ..., (y, Z)}? Every element (B, C...) must appear only once as the first object in one of the ...
4
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1answer
186 views

How to prove that $0.01001100011100001111…$ is not periodic decimal number?

I have the following decimal number: $0.01001100011100001111...$ Notice how whenever we have one 0, we also have one 1, two 0's, two 1's, etc. How do you continue it to infinity and prove that this ...
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1answer
106 views
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2answers
169 views

Elaboration of infinite, finite and enumerable definition

I am starting to learn some of the basic concepts of math. The concept I am learning now is infinite, finite, and denumerable. I am having trouble understanding the book's definiton. I am hoping if ...
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3answers
282 views

Random Point on Infinite Line Paradox

I've invented a paradox, or at least I think I have. Here is how it goes: On an infinite line, a point is placed at random. You start at point 0 on the line, and your job is to find the point, but ...
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3answers
154 views

Is ∞ considered defined?

$\infty$ (Infinity) is not a number, but infinity is considered to be defined, right? There are expressions in mathematics such as: $\frac x0,0^0, \frac\infty\infty,$ which are not defined because ...
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3answers
2k views

Does the concept of infinity have any practical applications?

I know what you're thinking: "of course it has, for example, it can be used to tell you how many times you can go around a circle". But that isn't really true, now is it? You'd be dead or the world ...
3
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1answer
87 views

Are distinctions in definitions of “finite” material in, eg, topology or measure theory?

There are several definitions of "finite", like Dedekind's and Tarski's (Thanks to A.K. for point out the latter - first time I've heard of it): From the Wikipedia entry on Finite Set: (Richard ...
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1answer
62 views

Please help me understand this formula for Fourier analysis

I'm a programmer with a poor knowledge of math. Could anyone tell me how to read the infinity above the sigma, and the n=1 below the sigma?
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5answers
2k views

How many different sizes of infinity are there?

It's pretty straightforward to say that there is an infinite number of different sizes of infinity, but then I thought, "What size of infinity is that?" My thoughts are that the number of unique ...
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1answer
114 views

Finding an element of the intersection of an infinite sequence of “compatible” sets of infinite sequences

Let $A$ be a set. Let $A^\omega$ denote the set of infinite sequences of members of $A$ (i.e., functions from $\omega$ to $A$). Define $\omega_n = \omega \setminus \{n\}$. Let $A^\omega_n$ denote the ...
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1answer
55 views

Comparing Improper Integrals Involving Infinity

From my current understanding: $K>J$ and $L>K$ , therefore $L>K>J$. How can I compare the first integral $I$ ?
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1answer
878 views

Comparison Theorem for Integral Calculus

I have narrowed it down to C, E, and F, since we know that $1/x^{1/5}$ is always greater than the original function for all $x\geq 1$. However, the second set of conditions is more difficult to ...
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0answers
102 views

Hadamard regularization isn't working out

As part of an exercise in a grad course on "mathematical methods" (always such a helpful name), I've been asked to evaluate $I=\int_0^{1/2}{(x^2-x+c)^{-2}dx}$ as a Hadamard finite part integral for $0 ...
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7answers
344 views

A proof of a property of limits

Today during lecture my lecturer showed us this property, but provided no proof. If $$\lim_{n\to\infty} {d_{n+1}\over d_n} >1$$ then $$\lim_{n\to\infty}d_{n}=\infty $$ Is this property legit? ...
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2answers
127 views

Why is this proof false?

I know this proof is false, but I don't know why. I need your help. The false proof says that it is possible to create a bijection between a subset of the rational numbers and the Power set of ...
0
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1answer
129 views

Infinite versus unendlich and double-negation

The German term for infinite is unendlich, which transliterates as non-ending, or non-finite. This is just word-play but from a constructive point of view, is the shift from a negative to a positive ...
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4answers
289 views

Is the powerset of every Dedekind-finite set Dedekind-finite?

Is the powerset of every Dedekind-finite set Dedekind-finite? I think this statement can be written in $\textbf{Set}$: If every mono (=injection) $f: A \to A$ is iso (=bijection), then every mono ...
2
votes
2answers
428 views

Infinite sum of floor functions

I need to compute this (convergent) sum $$\sum_{j=0}^\infty\left(j-2^k\left\lfloor\frac{j}{2^k}\right\rfloor\right)(1-\alpha)^j\alpha$$ But I have no idea how to get rid of the floor thing. I thought ...
2
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3answers
166 views

Finite sequences of unbounded value

If I have a finite sequence of expressions $a_1+a_2+a_3+....a_k=\infty$, does that imply that at least one such $a_j=\infty$? I know that if it didn't it would make the sum not equal to infinity, but ...
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12answers
3k views

What exactly is infinity?

On Wolfram|Alpha, I was bored and asked for $\frac{\infty}{\infty}$ and the result was (indeterminate). Another two that give the same result are $\infty ^ 0$ and ...
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2answers
204 views

Is it viable to ask in an infinite set about the Cardinality?

Can you ask given an infinite set about its cardinality? Does an infinite set have a cardinality? So, for example, what would be the cardinality of $+\infty$?
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4answers
561 views

Does such a natural number exist, that it would be divisible by every other natural number

I've got to prove (or disprove) the following statement: $\exists x \in \mathbb{N} \; \forall y \in \mathbb{N}: y \mid x$, which translates into "It exists such $x$ from the set of natural numbers, ...
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3answers
776 views

Number of points on line segment

I know the line segment have a infinite number of points, but i know that exist different kinds of infinity ( $\aleph_0 $). My question is there same number of points on segment of line and entire ...
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3answers
175 views

What does the notion of different sizes of infinity really mean?

I have heard that there are infinities of various sizes. I was wondering what that actually means-how do we compare their cardinalities? I have just started real analysis and I am slowly coming to ...
3
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2answers
423 views

Multiples of numbers up to infinity

A question my wife and I were chatting about last night. Are there more multiples of 3 than there are of 17, if we count from 0 to infinity One point of view was since there are infinite multiples of ...
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1answer
71 views

Finite sums of infinite value

If a sum of a finte number of terms is infinite, does that imply that at least one term in the finite sum is also infinite?