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?
28
votes
13answers
2k 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 ...
5
votes
2answers
168 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 $+\infty$?
1
vote
4answers
146 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, ...
2
votes
3answers
118 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 ...
1
vote
5answers
216 views
Evaluate $\lim\limits_{x \to \infty}\left (\sqrt{\frac{x^3}{x-1}}-x\right)$
Evaluate
$$
\lim_{x \to \infty}\left (\sqrt{\frac{x^3}{x-1}}-x\right)
$$
The answer is $\frac{1}{2}$, have no idea how to arrive at that.
0
votes
3answers
126 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
votes
2answers
139 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 ...
1
vote
1answer
43 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?
9
votes
5answers
743 views
7 Drinks - 7 Flavors - Infinite variety?
Me and three friends are trying to find the answer to a question I posed about a self-service drinks machine in our local Burger King:
There is a drinks machine that has 7 varieties of drinks (coke, ...
2
votes
2answers
93 views
Why do geometric sets such as $(\infty, x]$ never have infinity included?
I have a question about the use of infinity and geometric sets. Say I am trying to graph an equation, and the result is all values greater than or equal to, say, $3$. From what I've seen, the proper ...
1
vote
1answer
44 views
$A+\alpha\sim A$ when $\omega\le\alpha<h(A)$
I have been considering about $A+\alpha\sim A$ when $\omega\le\alpha<h(A)$, in which $h(A)$ is the Hartogs number of $A$.
If there is $\alpha$ s.t. $\omega\le\alpha<h(A)$, we can get $\alpha ...
1
vote
2answers
79 views
Is there any Dedekind-infinite set can be split to two smaller Dedekind-infinite sets?
I have been considering about $A+\alpha\sim A$ when $\omega\le\alpha<h(A)$, in which $h(A)$ is the Hartogs number of $A$.
If there is $\alpha$ s.t. $\omega\le\alpha<h(A)$, we can get $\alpha ...
4
votes
1answer
81 views
Is $\aleph_0$ the minimum infinite cardinal number in $ZF$?
$\aleph_0$ is the least infinite cardinal number in ZFC. However, without AC, not every set is well-ordered.
So is it consistent that a set is infinite but not $\ge \aleph_0$? In other words, is it ...
-2
votes
1answer
198 views
Ring polynomials and Zero Divisors
Let R[x] be a polynomial ring. Show that if R is finite and has zero divisors, R[x] has an infinite number of zero divisors.
I'm having trouble wrapping my head around what exactly polynomial rings ...
3
votes
1answer
282 views
Infinite expected value of a random variable
How can a positive random variable $X$ which never takes on the value $+\infty$, have expected value $\mathbb{E}[X] = +\infty$?
1
vote
2answers
57 views
Prove that if a set is Peano finite, then it is Dedekind finite.
I understand that this should be done by induction, but I have very limited knowledge on proof by induction. Could someone explain it in a way which also makes clear exactly what each stage of ...
1
vote
4answers
83 views
Prove sum is bounded
I have the following sum:
$$
\sum\limits_{i=1}^n \binom{i}{i/2}p^\frac{i}{2}(1-p)^\frac{i}{2}
$$
where $p<\frac{1}{2}$
I need to prove that this sum is bounded. i.e. it doesn't go to infinity ...
2
votes
3answers
125 views
Could $\frac x0 = \pm\infty$? [duplicate]
Possible Duplicate:
Is it wrong to tell children that 1/0 = NaN is incorrect, and should be ∞?
I remember that dividing by zero is frowned upon, because it is said that there is no real ...
3
votes
0answers
98 views
Cantor and infinities [closed]
I know we have accepted Cantor's ideas a long time ago and many mathematicians use sets and infinities without ever realizing that thinking about sets and infinities intuitively fails, because there ...
3
votes
1answer
97 views
Rational numbers and series going to infinity
(1) The sum of two rational numbers is a rational number.
(2) The series $\sum\limits_{n=0}^{\infty} \frac{(-1)^{n}}{2n+1} = \frac{1}{1} - \frac{1}{3} + \frac{1}{5} - \cdots = \frac{\pi}{4}$ is ...
-2
votes
1answer
125 views
Why infinite sums of positive real constants definitely yield infinite?
According to the last step in proof of the unmeasurability of Vitali_set, it said that summing infinitely many copies of the constant $\lambda(V)$ yields either zero or infinity, according to whether ...
5
votes
1answer
138 views
Can an infinite cardinal number be a sum of two smaller cardinal number?
Let $\kappa$ be an infinite cardinal number.
My question is whether there are $\lambda$ and $\mu$ such that both $<\kappa$ but $\lambda+\mu=\kappa$?
If AC holds, then the answer is definitely ...
0
votes
1answer
56 views
Identifying an Error in Determining the Convergency of an Infinite Series
Given the infinite series of $(-1)^n/(nln(n))$ for $n = 2,3,4,\ldots$ to infinity, is the series conditionally convergent, absoultely convergent, or divergent?
I took two approaches to solve this ...
1
vote
2answers
104 views
Harmonic Series Paradox
How to resolve the harmonic series paradox presented in this video by James Tanton?
9
votes
4answers
692 views
Two paradoxes: $\pi = 2$ and $\sqrt 2 = 2$ [duplicate]
Possible Duplicate:
Is value of $\pi = 4$?
Can anyone explain how to properly resolve two paradoxes in this YouTube video by James Tanton?
1
vote
4answers
262 views
Integration with infinity and exponential
How is
$$\lim_{T\to\infty}\frac{1}T\int_{-T/2}^{T/2}e^{-2at}dt=\infty\;?$$
however my answer comes zero because putting limit in the expression, we get:
$$\frac1\infty\left(-\frac1{2a}\right) ...
3
votes
4answers
248 views
Does $\log(x)$ stop at a finite value when x is infinite?
Does $\log(x)$ stop at a certain value when x is infinite? Or is it also infinite?
I can see the graph go straighter and straighter in the horizontal direction, and I wonder if it will eventually be ...
1
vote
0answers
65 views
Simplifying this infinite series [duplicate]
Possible Duplicate:
How can I evaluate $\sum_{n=1}^\infty \frac{2n}{3^{n+1}}$
I have an infinite series like so:
$$\sum_{i=0}^\infty (i+1)x^i$$
or basically
$$ 1 + 2x + 3x^2 + 4x^3 +... ...
0
votes
1answer
92 views
Contour Infinites and Vector Spaces
We usually define Hilbert or finite dimensional vector spaces, and even topologies or differential geometry on $\mathbb{R}^n$ , so I wonder what is the implication of doing that on some extended ...
0
votes
0answers
68 views
How to observe infinity?
In my calculus course, there's example stated on the book:
Given that $M$ is an ordered set and the sequence $\{a_n\}\subset M$, prove that there's a (weakly) monotonic subsequence of $\{a_n\}$.
...
1
vote
2answers
91 views
A vertical line in a cartesian coordinate system
Let's say I have points $A(a,a)$ and $B(a,0)$. What is the equation of the line $AB$? If I'm correct the slope is infinite, but it never has a y-intercept. This would give $y=\infty x$, but there are ...
1
vote
1answer
84 views
Why should the set have finite measure in the following proposition?
Here is a proposition in Royden:
Assume $E$ has finite measure. Let $\{f_n\}$ be a sequence of measurable functions on $E$ that converges pointwise a.e. on $E$ to $f$ and $f$ is finite a.e. on $E$. ...
3
votes
1answer
100 views
Prove that a formal language is infinite
I'm having trouble with the following exercise:
Let $\Sigma = \{a,b,c\}$ and $L$ be a formal language, that consists of all words which contain all three letters at least once. Show that $L$ is ...
3
votes
1answer
162 views
“Real” cardinality, say, $\aleph_\pi$?
Is there any meaningful definition to afford for $\aleph_r$ (as in cardinality) where $r\in\mathbb{R}^+$? $r\in\mathbb{C}$? What about $\aleph_{\aleph_0}$? Can we iterate this? ...
1
vote
1answer
74 views
Continuity and pushing a limit inside the function's domain
Consider some right-continuous function $f:\mathbb{R\cup\{-\infty,\infty\}}\to [0,1]$. I have to evaluate (i) $\lim_{b \to 0^+} f(\frac{a}{b})$, and (ii) $\lim_{b \to 0^-} f(\frac{a}{b})$ where $a \in ...
2
votes
3answers
130 views
Find: $\lim_{n\to\infty} r^n$, for $r>1$ and $r<1$
Prove:
$$\lim_{n\to\infty}r^n = +\infty\,, r > 1;$$
$$\lim_{n\to\infty}r^n = 0\,, 0 \le r < 1.$$
I am not quite sure how to prove this, but once someone proves it I will make sure to ask ...
2
votes
1answer
47 views
Is this set finite?
Let's say you are given a function $\mu:S\rightarrow(0,1]$ and you can additionally assume $$\sum_{s\in S}\mu(s)=1$$ Does this imply that $S$ is finite?
0
votes
2answers
183 views
How large is the infinity of real numbers [closed]
Umm ... Can someone disprove my proof that there are aleph-1 number of real numbers? Even comments to make my proof more rigorous are welcome.
https://www.dropbox.com/sh/1fz28jlwrprh4jv/rhA7Ad7OtX
2
votes
2answers
157 views
What is the use of such concepts as potential infinity and actual infinity?
I'm aware of such mathematical concepts as and potential infinity and actual infinity. But I do not understand how those concepts are being used. Are there any applications to such concepts? Are there ...
-1
votes
2answers
210 views
Is there such math concept as potential zero?
It seems that I need to use concept of potential zero in my work and I want to know whether I could reference some other works in order to fully understand what I'm dealing with.
Specifically for my ...
0
votes
0answers
69 views
Size of infinite sets [duplicate]
Possible Duplicate:
Different kinds of infinities?
I heard from a lecture at university, many years ago now, that some groups of infinite sets are bigger than other groups of infinite sets. ...
4
votes
4answers
198 views
The concept of infinity
This evening I had a discussion with a friend of my about a mathematical riddle and the concept of 'infinite'
The riddle
Imagine a hotel with an infinite amount of rooms, and all of the rooms are ...
23
votes
5answers
2k 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 ...
3
votes
4answers
130 views
Limit of difference of two irrational functions
Firstly, this is not a homework. I just want to solve this limit for my own curiosity and self-learning. I have tried to solve this limit for 5-6 hours with no luck. Then I tried to read information ...
1
vote
3answers
109 views
Is infinite a infinite or finite
Long back I have watched a documentary based on a mathematician named Cantor. According to that documentary, Cantor claimed that infinite does not exists, it is only finite. Is that True?
0
votes
1answer
79 views
Normalize infinite range into finite one
First of all; I'm a programmer, not a mathematician so please excuse the informality of my math-vocabulary.
I have a series of slopes, calculated out of random angles (their tangents). These angles ...
4
votes
1answer
174 views
What is the representation for a number that is not quite one?
If: $$0.\overline{9999999} \equiv 1$$
Then how would you represent a value that is infinitesimally close to one, but not quite one?
i would have thought: $$1-\frac 1 \infty $$
But i would take that ...
1
vote
3answers
72 views
What is positive-0 squared minus positive-0?
I've got a basic limit problem that I think I'm solving the right way, but I've run into something that looks confusing enough to make me wonder if I'm doing it right.
$$
\lim_{y\to0} \frac{1}{y^2-y} ...
0
votes
2answers
100 views
Is it possible to iterate through an infinite set?
Is it coherent to suggest that it is possible to iterate, one-by-one, through every single item in an infinite set? Some have suggested that it is possible to iterate (or count) completely through an ...
1
vote
4answers
108 views
Does every sequentially ordered infinite set contain sequentially ordered infinite subsets?
I am not very familiar with mathematical proofs, or the notation involved, so if it is possible to explain in 8th grade English (or thereabouts), I would really appreciate it.
Since I may even be ...
