What is Infinity: Definition and 981 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. docnet

    B Confusion about infinity

    We have come to accept that Infinity times two is infinity. In the sense of 'size' we use to think about everyday numbers, the rules of arithmetic with infinities seem like nonsense. For example, consider the computable number $$0.100100100100100....$$ In the decimal expansion, there are...
  2. Kekkuli

    I Criticism of the doctrine of infinity

    On the one hand, Cantor showed that not all real numbers can be enumerated, while on the other hand he showed that rational numbers can. Cantor demonstrated this with a grid. In the picture below, a natural number (yellow) is assigned to each rational number in order, but since the natural...
  3. N

    B 1 divided by infinity equals zero (always?)

    I understand the mathematics that 1 divided by infinity is virtually zero and so equals zero. I look on the internet and that is the answer that I get. Is this a simplification for early mathematics learning and, if I continue, will I find a more complex answer? The reason that I ask is that I...
  4. Lokalgott

    I Infinity Numbers (Expanded P-Adic Numbers)

    Hello, I'm new here and I'm looking to talk about p-adic numbers. It's not a specific school problem . I'm trying to expand p-adic numbers into something that I call "Infinity Numbers". Is it possible here to discuss topics like that? Kind regards
  5. Jaime Rudas

    I The boundary conditions at infinity

    Section 5 (pg. 29) of the Michel Janssen's paper EINSTEIN’S QUEST FOR GENERAL RELATIVITY, 1907–1920(*) says: 1. From the above I understand that the application of general relativity to an infinite universe was considered problematic. 2. On the other hand, I understand that it is currently...
  6. L

    B Is one out of infinity different from zero?

    In another question I posted on here, I asked about a hypothetical roulette wheel with an infinite amount of spaces. Each number has a one-in-infinity chance of being selected, yet each time the wheel is spun, one number wins those odds. My initial question was how this was possible, and I had...
  7. L

    I How is it possible to win one-in-infinity odds?

    Imagine a roulette wheel with an infinite amount of numbers. Every number on the wheel has a one-in-infinity chance of being selected. Every time the wheel is spun, one number wins those one-in-infinity odds. How is this possible? Isn't one-in-infinity basically zero? It's infinitely far from...
  8. L

    I Taylor expansion of f(x)=arctan(x) at infinity

    I have to write taylor expansion of f(x)=arctan(x) around at x=+∞. My first idea was to set z=1/x and in this case z→0 Thus I can expand f(z)= arctan(1/z) near 0 so I obtain 1/z-1/3(z^3) Then I try to reverse the substitution but this is incorrect .I discovered after that...
  9. T

    I Integrity of matter / space and time

    I have probably a basic question from Space, Time and Matter area. My 11 years old daughter asked me once why we exist physically in a stable form if everything is infinite. We had a conversation about it but then it got me thinking about this and it seems I can't find the answer. There is...
  10. jaketodd

    B Dividing by infinity, exactly, finally!

    Why not use these number systems, in place of the real number system, when these allow us to divide by infinity exactly? According to these, division by infinity equals exactly zero! No need for calculus limits, which only can say it approaches zero when tending towards infinity...
  11. Euge

    POTW Find Limit of $$\frac{x}{e} - \left(\frac{x}{x+1}\right)^x$$ at Infinity

    Find the limit $$\lim_{x\to \infty} x\left[\frac{1}{e} - \left(\frac{x}{x+1}\right)^x\right]$$
  12. Mohmmad Maaitah

    L'Hopital's Rule case: How does x^(-4/3) equal 0 when x approches infinity?

    I'm talking about the x^(-4/3) how does it equal 0 when x approch infinite?? so I can use L'Hopital's Rule
  13. C

    Prove that l^p is a subset of l^q for all p,q from 1 to infinity

    Dear everyone, I am having trouble with this problem. I have convinced myself that the ##a^t-a\leq 0## is true. Now, I am trying to applying this inequality for the finite series and I don't know where to start. After that, proving that the p-norm is less or equal to the q-norm. Thanks...
  14. N

    B Gravity Between Two Stars 45 Billion Light Years Away?

    Hi, mathematically in the F = GMm/r^2 equation r can be very close to infinity (or the size of the universe), but gravitational force always will be some number. But how is that in the real world? Let's say we have a perfectly empty universe but only with two sun-like stars. If they are away...
  15. S

    Is infinity the answer of all question that we cannot answer?

    TL;DR Summary: Is infinity the answer of all question that we can not answer? If I ask a scientist how big our universe is? He will say infinite. If i ask how small anything can be? Will scientist say infinitely small ? As our visual limitation or device limitation we might not be able to see...
  16. S

    Proving limit of f(x), f'(x) and f"(x) as x approaches infinity

    I imagine ##f(x)## has horizontal asymptote at ##x=k##. Since the graph of ##f(x)## will be close to horizontal as ##x \rightarrow \infty##, the slope of the graph will be close to zero so ##\lim_{x \rightarrow \infty} f'(x) = \lim_{x \rightarrow \infty} f^{"} (x) = 0## But how to put it in...
  17. robphy

    Video: A Trip to Infinity (Sept 26, 2022 on Netflix)

    A Trip to Infinity https://www.netflix.com/title/81273453 (Sept 26, 2022 on Netflix) trailer: https://www.imdb.com/title/tt21929356/ lists the cast as: Anthony Aguirre Stephon Alexander Eugenia Cheng Moon Duchin Kenny Easwaran Delilah Gates Rebecca Goldstein Brian Greene Janna Levin...
  18. U

    B About infinity and all possibilities of Multiverse/Omniverse

    If all possibilities happen somewhere, can there be a universe where there are more explanations and content (articles, blogs, videos etc.) in that universe explaining about the multiverse, omniverse, dimensions etc.? And speaking of multiverse, why there is not enough information about...
  19. nomadreid

    "1/2 = infinity, great is the glory.... "

    I am looking for the details of when a famous mathematician in history (Gauss? Euler?) tried to find an infinite sum (integrate?) in two different ways, and got two different answers, one of them one-half and the other one infinity (where maybe a negative was attached to one of them). When he...
  20. wrobel

    I The behavior of a potential-like integral at infinity

    I need a help in the following problem. I feel that the question is stupid. Take a function ##f\in C(\mathbb{R}^3)\cap L^1(\mathbb{R}^3)## and a number ##\alpha\in(0,3)##. Prove that $$\lim_{|x|\to\infty}\int_{\mathbb{R}^3}\frac{f(y)dy}{|x-y|^\alpha}=0.$$ I can prove this fact by the Uniform...
  21. 0

    I A question about limits and infinity

    I hope I can make this question clear enough. When we have a function such as f(x) = 1/x and calculate the side limits at x = 0, the right side goes to positive infinity. The left side goes to negative infinity. In calculus we are pluggin in values closer and closer to zero and seeing what the...
  22. D

    B Why 1 / ∞ = 0 but ∞ * 0 is not equal to 1?

    As we know those relations are true: if a/b = c, then a = b*c and b = a/c Therefore if 1/ ∞ = 0, ∞ * 0 should be equal to 1 and 1/0 = ∞
  23. A

    Compute lim as n tends to infinity of f(xn)

    As n tends to inf, the fraction goes to zero so would the lim just be X?
  24. Frabjous

    Physics problem with Thanos and Infinity Gauntlet

    Apparently there are problems with him snapping his fingers while wearing a glove https://www.sciencenews.org/article/finger-snap-physics-new-high-speed-video-thanos
  25. A

    I The atomic Coulomb potential extends to infinity?

    I'm studying nuclear physics in a text, but at one point that is said: "Both the Coulomb potential that binds the atom and the resulting electronic charge distribution extends to infinity" , I don't understand what is that "resulting electronic charge distribution extends to infinity" what they...
  26. M

    I An infinity of points on two unequal lines- an intuitive explanation?

    I am familiar with Cantor's work on the concept of infinity and his use of the set theory to explain various types of infinities. Having said that my intuition never seems truly grasp/accept it. Is there a way to train my mind to see this seemingly contradictory situation as a fact? This is...
  27. Dale

    I Future null infinity confusion

    So I was trying to get a bit better handle on the definition of the difference between an event horizon and a Killing horizon. Locally they are indistinguishable, and the key difference (to my understanding) is that the event horizon is the last Killing horizon that escapes to future null...
  28. M

    MHB Calculating $2\sqrt{2}$ to Infinity

    Calculate $2\sqrt{2\sqrt[5]{2\sqrt[8]{2\sqrt[11]{2 \cdots}}}}$. I know only that $...=2^{1+{1\over2}+{1\over10}+{1\over80}+{1\over880}+\ldots}$
  29. Jarvis323

    Optical Ideas for Infinity Mirror Geometries

    I'd like to make a large scale infinity dodecahedron, or icosahedron, or something similar. I'm just curious if anyone around here has an idea for a different/more complicated geometry that would work well?
  30. T

    B Difficulty with understanding whether 1/n converges or diverges

    Hi, I have a quick question about whether or not the infinite series of 1/n converges or diverges. My textbook tells me that it diverges, but my textbook also says that by the nth term test if we take the limit from n to infinity of a series, if the limit value is not equal to zero the series...
  31. V

    Determine the limit of 2^x/x^2 as x approaches infinity....

    How would I determine the following limit without substitution of large values of x to see what value is approached by the complex function? ## \lim_{x \rightarrow +\infty} {\dfrac {2^{x}} {x^{2} } } ## where ## x\in \mathbb{R}##
  32. S

    B Integration of tan^2 x from - to + infinity

    ∫tan^2 x ( -infinity to +infinity)
  33. karush

    MHB -b.1.1.2 behavior of y'-2y=-3 as t goes to infinity

    determine the behavior of y as t →∞. If this behavior depends on the initial value of y at t = 0,describe the dependency \begin{array}{lll} \textit{rewrite} &y'-2y=-3\\ \\ u(t) &=\exp\int -2 \, dx=e^{-2t}\\ \\ \textit{product} &(e^{-2t}y)'=-3e^{-2t}\\ \\ \textit{integrate}...
  34. Z

    Ampere's Law & Biot-Savart: Dealing with Infinity

    How to lose surface integral in derivation of ampere law from biot-savart law if current goes to infinity? How does current that goes to infinity obey Helmholtz theorem for vector fields?
  35. S

    I Different kinds of infinity in physics?

    I am reading several books on infinity as first developed by Georg Cantor. Some physicists claim that the multiverse might be infinite. But they don’t seem to mention two of the kinds of infinity which might (exist?). It makes an infinite difference to make a bad pun. The integers define...
  36. D

    B My argument why Hilbert's Hotel is not a veridical Paradox

    Hello there, I had another similar post, where asking for proof for Hilbert’s Hotel. After rethinking this topic, I want to show you a new example. It tries to show why that the sentence, every guest moves into the next room, hides the fact, that we don’t understand what will happen in this...
  37. TopsyKrate

    Gravitation: Object moving from infinity

    Express the condition of 𝜎 where the celestial body B collides against the celestial body ASo this is the original figure of the problem. This is my attempt at a solution Since I need to find σ, I have assumed sigma to be an multiple of the radius ## R ##. So, let ## \sigma = \lambda R ##...
  38. D

    B Hilbert's Hotel: new Guest arrives (Infinite number of Guests)

    Hilberts Hotel has infinity numbers of rooms and in every room is exactly one guest. On Wikipedia Hilberts Hotel gets described as well: Suppose a new guest arrives and wishes to be accommodated in the hotel. We can (simultaneously) move the guest currently in room 1 to room 2, the guest...
  39. LCSphysicist

    Question about potential energy as a mass approaches infinity

    I was thinking, what would be the consequence if we wouldn't adopt the ro in the infinite, and i conclude that it would just irritate the accounts, with one constant more, am i right? Once what matter is the diference between the U, and no the U infact.
  40. F

    Sum of a series from n=1 to infinity of n^2/(2+1/n)^n

    I tried to write it as n^2/2^n (1+1/2n)^n But I am stuck there and don't know what to try next.Thanks for any help in advance!
  41. agnimusayoti

    Limit of a function as n approaches infinity

    If there is no ##(-1)^2## factor, I can find the limit. But, now I have no idea how to find limit for the ##(-1)^\infty##. I thought ##(-1)^\infty## is an indeterminate form. So, how to modify this? Thanks!
  42. agnimusayoti

    Find the limit of this sequence as n approaches infinity (ML Boas)

    First I assume that $$(1+n^2)^\frac{1}{\ln n}=\exp {\ln (1+n^2)^\frac{1}{\ln n}} $$But, $${\ln (1+n^2)^\frac{1}{\ln n}}={\frac{\ln (1+n^2)}{\ln n}}$$ By L'Hopital Rule, I got $$\lim_{n\to\infty} {\frac{\ln (1+n^2)}{\ln n}}=\frac{\lim_{n\to\infty} (\frac{2n}{1+n^2})}{\lim_{n\to\infty} 1/n}$$...
  43. S

    Limit of a fraction as n-> infinity in the numerator and denomominator

    Hello,This is actually a piece of a little bigger problem (convergence of a series) - you can see the ratio test ak+1 / ak That's why the (n) and (n+1) terms I have lim n->∞ of (n√n) / (n+1)√(n+1) ∞/∞ I have tried L'Hopitals rule (requiring multiple times) and I am not seeing an end...
  44. GregoryC

    Is Nothing Really Something? Exploring the Concept of Absolute Nothingness

    Summary:: Would nothing & infinity be considered polar opposites? Neither can be observed and are hypothetical at 2 extremes. "Nothing" is one of the questions much like "infinity", I find myself questioning these supposedly two "real" expressions. Logically they both make sense.
  45. J

    Find the range of y in a DE

    From integration by parts, and using y(10) = 0, I get the equation ##2e^{3t-30} = \frac{|y-2|}{|y+1|}.## Let ##f(t) = 2e^{3t-30}##. Since it's for t>10, f(10) = 2, and we have ##2=\frac{|y-2|}{|y+1|}##. Depending on the sign I choose to use, I get either that y=-4 or y =0. Since ##t: 10...
  46. N

    Can someone help me with this limit where x approaches - infinity?

    I don't know what do do from here other than i can make the 3/e^x a 0 due to the fact its divided by such a large number. What do i do with the e^-3x? Thanks for the help
  47. Physics lover

    Sum of a series that tends to infinity

    I tried by ##S=1+(1/1!)(1/4)+(1.3/2!)(1/4)^2+...## ##S/4=1/4+(1/1!)(1/4)^2+(1.3/2!)(1/4)^3..## And then subtracting the two equations but i arrived at nothing What shall i do further?
  48. S

    Understanding Sum to Infinity in Geometric Progression

    My question is Why is the sum to infinity used as opposed to Sum to n? and How can I deduce that the sum to infinity must be used from the question?Total Distance = h + 2*Sum of Geometric progression (to infinity) h + 2*h/3 / 1-1/3 h + 2h/3 *3/2 = h + h = 2h At first I did sum to infinity...
  49. Troxx

    I Trouble with infinity and complex numbers

    Summary: Trouble with infinity and complex numbers, just curious. I'm not too familiar with set theory ... but <-∞, ∞> contains just real numbers? Does something similar to <-∞, ∞> exist in Complex numbers? My question, is it "wrong"?
  50. S

    I Assigning a value for integrating a divergent oscillatory function to infinity

    There are meaningful ways to assign values to things like 1 - 1 + 1 + ... or 1 - 2 + 3 - 4 + ... In a similar spirit, is it possible to assign a value to the integral of a function like this: ##f(x)=x*sin(x)## or this one: ##g(x)=Re(x^{1+5i})## (Integrals from some value, say zero, up...