What is Infinity: Definition and 984 Discussions

Infinity represents something that is boundless or endless, or else something that is larger than any real or natural number. It is often denoted by the infinity symbol shown here.
Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions among philosophers. In the 17th century, with the introduction of the infinity symbol and the infinitesimal calculus, mathematicians began to work with infinite series and what some mathematicians (including l'Hôpital and Bernoulli) regarded as infinitely small quantities, but infinity continued to be associated with endless processes. As mathematicians struggled with the foundation of calculus, it remained unclear whether infinity could be considered as a number or magnitude and, if so, how this could be done. At the end of the 19th century, Georg Cantor enlarged the mathematical study of infinity by studying infinite sets and infinite numbers, showing that they can be of various sizes. For example, if a line is viewed as the set of all of its points, their infinite number (i.e., the cardinality of the line) is larger than the number of integers. In this usage, infinity is a mathematical concept, and infinite mathematical objects can be studied, manipulated, and used just like any other mathematical object.
The mathematical concept of infinity refines and extends the old philosophical concept, in particular by introducing infinitely many different sizes of infinite sets. Among the axioms of Zermelo–Fraenkel set theory, on which most of modern mathematics can be developed, is the axiom of infinity, which guarantees the existence of infinite sets. The mathematical concept of infinity and the manipulation of infinite sets are used everywhere in mathematics, even in areas such as combinatorics that may seem to have nothing to do with them. For example, Wiles's proof of Fermat's Last Theorem implicitly relies on the existence of very large infinite sets for solving a long-standing problem that is stated in terms of elementary arithmetic.
In physics and cosmology, whether the Universe is infinite is an open question.

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  1. R

    Infinity: Beyond the Beyond the Beyond

    Infinity : Beyond the Beyond the Beyond by Lillian Lieber, and Hugh Lieber I am not sure how many of you on this forum are familiar with this book but I have a copy of it and it seems very interesting but very strange. I want to know if its worth a read. I enjoy the talk of SAM and I want...
  2. A

    Question about the multiverse and infinity

    I've wanted to ask this for a while since it has been confusing me for too long now... if our universe is infinite and nothing can possibly exist outside of it (other than a putative God although no one has an idea how that could be, since Kant proved quite a while ago that existence itself is...
  3. T

    What is the induction at infinity ?

    What is the induction at infinity ? I told teacher it to be 0 but she disagreed. Using the formula for calculating induction and substituting big distances into the formula gives me also 0. Is my teacher stupid ?
  4. Loren Booda

    Oo = -oo? positive infinity be equivalent to negative infinity?

    In what regard might positive infinity be equivalent to negative infinity?
  5. S

    What Is Infinity? Explained for Beginners

    Hello, Please excuse me if I may sound an ignorant compared to all of you, but I was always stupid and never paid attention to any of my physics classes and now I feel so stupid for missing on the most important subject of our lives and our universe. I was discussing a notion with a friend of...
  6. J

    + infinity and - infinity join ?

    hey in advanced maths , does +infinity and -infinity join at some point ? a bit like if the axis was a cilinder
  7. P

    A sinusoid integrated from -infinity to infinity

    I had a sort of odd question on my homework, Sin(x)^3 dx, integrated over all reals (from negative infinity to infinity). The problem also gives this morsel of ambiguity: "Hint: think before integrating. this is easy" Now my initial guess because of the antisymmetry of the...
  8. W

    Solving infinity limit, something weird.

    Homework Statement This question asks you to evaluate the limit of the following function lim sqrt(9x^2-3x) -3x x-> infinity I did it using 2 methods: 1) lim sqrt(9x^2-3x) -3x x-> infinity = lim x(sqrt(9-3/x)) -3x...
  9. Holocene

    Cardinal Numbers and the Concept of Infinity in Mathematics

    Does subtracting infinity from infinity leave you with zero? Or could you subtract infinity from infinity and still have infinity?
  10. K

    Algebra & Infinity - Solve Complex Math Problems

    *edit* nm sorry didn't see this has been covered before.
  11. A

    Real's Union Infinity: Exploring the Unknown

    Hey, Just a tiny question, what is the space called: Reals union infinity?
  12. M

    How do I calculate the distance of the ∞-norm between two vectors in lR^3?

    Alright, so if I want to find the distance of the ∞-norm between two vectors in lR^3, then would I take the max of the vectors first and then subtract, or should I subtract the vectors and then take the max? I think that the vectors are subtracted, and then the norm is taken, but I just want to...
  13. M

    Solving for the Limit of a_p as p Goes to Infinity

    help! limit find the limit as p goes to infinity a_p = sqrt(p^2+p)-p really don't know how to solve this... i know the limit is 1/2, but i need to prove that 1/2 is really the limit!1
  14. 2

    Understanding infinity in terms of geometry and motion maybe

    I was thinking about how math was taught to me as a kid and how it was taught in symbols, not shapes. I am visual thinker, not a symbolic thinker, I see pictures and reflections of the world in my mind as a "movie" or virtual world I guess you could say. I can create very complex shapes...
  15. P

    Infinity intensity of light-Possible?

    Dear Friends, My longtime pending doubt here...! When we focus a mirror on the wall we get bright spot of the light. Ok. Now say, there is cube 6" x 6" x 6" whose inner walls are of mirror surfaces and opaque surfaces are the outer surfaces of the cube. In the center of the cube, in...
  16. P

    Will intensity of this light become infinity?

    Dear Friends, My longtime pending doubt here...! When we focus a mirror on the wall we get bright spot of the light. Ok. Now say, there is cube 6" x 6" x 6" whose inner walls are of mirror surfaces and opaque surfaces are the outer surfaces of the cube. In the center of the cube, in...
  17. D

    What is the limit as x approaches negative infinity for x + sqrt(x^2+3)?

    Homework Statement limit as x-> -oo of x+(x^2+3)^(1/2) (square root) Homework Equations n/a The Attempt at a Solution i first multiplied the top and bottom (which is just 1) by the conjugate to get: -3 -------------- [x - (x^2+3)^(1/2)] then i divided by x on top and...
  18. R

    What is the Process to Find the Limit at Infinity of (x+(x^2+12x)^1/2)?

    Find the limit as x-> -infinity for (x+(x^2+12x)^1/2) so first of all..i multiply and divide by the conjugent then i get... -12x/(x-(x^2+12x)^1/2) i divide by x in both the nummerator and denominator to get ... -12/1-(1+12/x)^1/2 so the 12/x goes to 0 and the squroot of 1 is 1 so it...
  19. D

    Proving a^n/p(n) Tends to Infinity with n

    i've to show (as part of a bigger assignment) that a^n/p(n), where p is any polynomial and a>1, tends to infinity as n does. I've proved that: a^n/n^k does so, but I'm not sure how to extend this to a complete polynomial such as (c1)n+(c2)n^2+(c3)n^3... thanks for any help NB: edited
  20. P

    Calculating Pole Order: Is Infinity Possible?

    I'm calculating the order of a pole of some function and I'm wondering: is it possible that a pole has order infinity?
  21. S

    Taking the Limit As N -> Infinity

    Taking the Limit As N --> Infinity \mathop {\lim }\limits_{n \to \infty } \frac{{n^n x}}{{(n + 1)^n }} Does this limit exist? Somehow it's supposed to come down to x/e
  22. S

    Calculating Limits Approaching Infinity: An Example

    I don't remember any rules on how to calculate limits approaching infinity. for example, can someone please explain the following limit to me (-1/2) lim (t->infinity) 1/(t^2 + 2) + 1/4 = 0 + 1/4 = 1/4 Thanks!
  23. mattmns

    Exploring Infinity in Computer Mapping: A Problem

    This issue of infinity (undefined?) keeps coming up in the following problems. For example, the following question: Computer the image of the sector 0 \leq r \leq 1, 0 \leq \theta \leq \pi, under the map ln(z). ------------- So I first graphed this thing in the x,y (z-plane) and obviously...
  24. S

    Calculate the integral of x^2 e^(-x^2) from -infinity to + infinity

    \frac{1}{\sqrt{2 \pi \sigma^2}} \int_{-\infty}^{\infty} x^2 e^{\frac{-x^2}{2 \sigma^2}} dx I think just slving this would be fine too \int_{-\infty}^{\infty} x^2 e^{-x^2} dx what is the trick to solving this?/ Cnat integrate by parts because that would yeild Erf function which i have no...
  25. O

    How Do You Solve the Sum of 3^k/k! from 0 to Infinity?

    I've been staring at this stupid problem for a while, and finally gave up and came here. The question is to evaluate the sum of 3k/k! from 0 to infinity. Basically, I'm looking for a starting spot, since I have none. The closest thing to a strategy I've come up with is plugging it into the...
  26. G

    Do Probabilities Always Approach Extremes at Infinity?

    Is it just me or do all high probabilities dwindle to nothing as time approaches infinity and all small probabilities increase to 1 as time approaches infinity?
  27. kreil

    How to show that lim 2^n/n = 0 with n--> infinity

    Using the definition of a limit, show that \lim_{n \rightarrow \infty} \frac{2^n}{n!}=0 If someone could get me started that would be great. thanks josh
  28. L

    Infinity Paradox or just confusion

    Given infinite time, can one do infinite tasks an infinite amount of times? How many times can one do infinite tasks in infinite time? One would think the ansewr to that question could not be >1, for that would mean that the tasks must have an end and all of them can eventually be...
  29. S

    By photon's POV, would light travel infinity in no time?

    According to photon's standpoint, since light travels any distance in 0 time, then doesn't it mean that it can travel infinity in 0 time, but for my mind, that doesn't really made sense b/c I thought nothing could reach infinity, but apparently, light can.
  30. S

    Infinity: Limitless or Limited? Exploring the Concept of Infinity in Mathematics

    Does infinity have direction? Dimensions? As we all know it does, then doesn't it mean that infinity is actually limited? Just like values b/w 1 and 2 x values are infinity but have a total sum.
  31. K

    Indeterminate: 0 times infinity, limits

    I have this problem, with three functions: f(x)=5x(x-2)(1/(x-2)) g(x)=5x(x-2)(1/(x-2)^2) h(x) 5x(x-2)^2(1/(x-2) The problem says that as x approaches 2, the functions take the form 0 times infinity, and asks you to "show" this. I tried plugging in 2 for x, but I got 0 times 1/0 (either...
  32. G

    Limits, infinity, and my calculator

    I understand that the limit of sec^2(x) as x approaches pi/2 is infinity (increasing without bound), and I understand the meaning of this in terms of the epsilon-delta definition of an infinite limit. I also understand why the limit of sec(x) as x approaches pi/2 doesn't exist. What I'm a...
  33. E

    Sizes of Infinity: Countable vs Uncountable

    So we talked about this in class but there's nothing in the textbook. Basically I want to make sure I get this: Reals are uncountable, while natural numbers, integers, and rationals are countable. So really the cardinality: |Z| = |R| = |Q| < |R| then there are only 2 sizes of infinity...
  34. D

    Behaviour of Algorithms at Time at Infinity

    Where might I learn more about the behaviour of algorithms given an infinite amount of time, if such a question makes sense?
  35. H

    Is the Limit at x=0 of a Reflected Symmetric Function Negative Infinity?

    Lets say you have a function that is constant except at interval [-2,2] where it drains down to infinity. The whole function is reflected symetrically at x=0. Is the limit of this function, as x approaches 0, negative infinity?
  36. A

    A -ve number greater than infinity?

    Please follow the following arguments. 5/3=1.66 5/2=2.5 5/1=5 5/0.5=10 ... ... 5/0=infinity and then 5/(-1)= -5 What you see? As the denominator is...
  37. E

    Infinity geometric series question

    Hi there everyone! Have a quick question for you. The question is: The sum to infinity of a geometric series is 9/2 The second term of the series is -2 Find the value of r, the common ratio of the series. I understand that we have to use the sum to infinity of a geometric series...
  38. J

    Infinity x Zero: James' Math Problem

    I am a high school student, and I am doing extension mathematics, and me and the rest of the class always get into big arguments about number planes. I say that on a basic 2d numberline with a parabola or hyperbola, when x = infinity, y = 1/infinity, which I think is zero, but they think...
  39. U

    Derivative of cos\frac{1}{x}: Is it 0 or Infinity?

    What is the derivative of cos\frac{1}{x}? Also, would cos\frac{1}{x} equal 0 or infinity?
  40. D

    Complex Infinity: Exploring Directional Infinity in Complex Variables

    I'm trying to learn some elementary complex variables, and I was reading this book on it when I came upon this Consider the function f(x)=\left\{\begin{array}{cc}0\Leftrightarrow Im(x)\neq 0\\e^x\Leftrightarrow Im(x)=0\end{array}\right. Wouldn't it make more sense if we had a concept of...
  41. V

    Alternate Treatment of Infinity

    Infinity has been something that has been talked about a lot. A lot of Questions are being posted on this forum and elsewhere about the paradoxes involving infinity. I want to know if anybody knows some alternate treatment of infinity.
  42. Mk

    Zero Times Infinity: Exploring Indeterminate Forms

    0 * infinity What is up? I read that it this expression is called an "indeterminate form." Why isn't zero multiplied by infinity equal to zero? Even if infinity is really big, if there are zero amounts of infinity, that would make zero. How about 1infinity? 1? Guess not. infinity / infinity...
  43. T

    Ampere's Law, must the current extend to infinity?

    Ampere's law states that, the closed integral of B over the loop enclosed it equals uI, where u = permeability of the material and I = current "passing" through the loop. I feel confused, because, should the current be extended to infinity? I mean, when we have an infinite wire of current...
  44. quasar987

    Understanding Holomorphic Functions at Infinity: Exploring Complex Analysis

    Hi, could someone explain what it means for a function to be holomorphic on \mathbb{C}\cup \{\infty\}? More precisely, what does it mean for it to be holomorphic at \infty. Thx.
  45. C

    Why is -1/(0^2) = -Infinity? Exploring the Difference from 1/0

    Why does my calculator tell me -1/(0^2) = -infinity. How is this different from 1/0?
  46. S

    Cantor's Infinity and the Concept of Sets: Understanding the Comparison of Sizes

    I have tried to understand Cantors ideas of infinity, but they still don't make sense to me. If you use the mathematical concept of sets to investigate something like this - a 'value' that can't be written because it has no end - then surely the size of the set is what you are evaluating ? How...
  47. M

    Driven RL circuit, why at infinity is this the current 2, and yet at 0+, its 0?

    Hello everyone I'm confused...i was just doing RL and RC circuits without a driven source, they made sense after doing them a few 100 times. Now I'm at Driven RL circuits. My professor said: iL(infinity) = 2mA iL(0+) = 0. From my understanding, i thought at infinity, the switch has been...
  48. T

    Is Infinity a Point in $\mathbb{R}$?”

    We defined the definition of a closed set to be: "F\subset\mathbb{R} is closed if the limit of any convergent sequence in F is an element of F." Now we have also defined that a sequence may "converge to infinity". Is infinity considered a point in N?
  49. H

    Average value of a function sent to infinity

    I was wondering if it would be possible to find the average value of a function with the only condition that x is element of R. For example, could we say that f(x)=4 has an average value of 4 since no matter what values we give for a or b in the integral from a to b of f(x)/(b-a) (b is not equal...
  50. George Jones

    Infinity in Finite Proper Time

    Working on pervect's "messy unsolved" problem has led me to an interesting result. Let \left( x , t \right) be a global inertial coordinate system for Minkowski spacetime. Consider the worldline given by t \left( \tau \right) = \frac{\tau^3}{3} - \frac{1}{4 \tau} x \left( \tau \right) =...
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