What is Infinity: Definition and 983 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. binbagsss

    Order of zero of modular form from its expansion at infinity

    1. Homework Statement order of zero of a modular form ? 2. Homework Equations 3. The Attempt at a Solution Apologies if this is a stupid question but I'm pretty confused. So, a modular form ##f(t) \in M_k ## is usually given by it's expansion about ##\infty## expressed in the variable...
  2. D

    I Infinity & Limits: Why Does Energy Not Become Infinite?

    Hi I'm confused about something from quantum mechanics but it concerns infinity and limits. For an infinite well the energy levels vary as n2 and for an harmonic oscillator the energy levels vary as n with n taking integer values in both cases with no upper bound. In both cases there are...
  3. R

    MHB Slope of y=-7 as x increases from 8 to infinity

    The function that passes through (8,3) and every time x increases by 1 unit, the fuction decreases by 7 units
  4. S

    I HUD; virtual image at infinity

    Hi, I am trying to create an HUD display - basically create a virtual image at infinity and look at it. I have two plano-convex lenses; the distance between them is the sum of their focal lengths; my object is therefore imaged at infinity. Here is the schematics that I'm pretty much trying to...
  5. Kara386

    I Calculate Time at Infinity for GR Observer: A Photon's Journey

    We were shown the answer to this question as a worked example: A photon is emitted from a radius ##r_2## and travels radially inward to ##r_1## until it's reflected by a fixed mirror and travels back to ##r_2##. Calculate the time taken for the photon to travel in and back, according to a...
  6. F

    Can a set include negative infinity and be bounded below

    Homework Statement Prove that {##x \epsilon \mathbb{R} : x^2 \ge 1##} is "not" bounded below. EDIT: I Looked closely and realized there is a "not" that we all had to write in...sorry if you lost some time.. Homework Equations Defintion: We say a nonempty subset ##A## of ##\mathbb{R}## is...
  7. Mr Indeterminate

    I Where has this proof gone wrong? ∞= 1/0

    Now I expect that most of you on this forum would be familiar with the equality between point nine reoccurring and one: 0.999...=1 If your not familiar please review https://en.wikipedia.org/wiki/0.999... Now this equality can be used to imply something else, which is rather heterodox...
  8. M

    Python Integrating to Infinity Numerically

    Hi PF! I am trying to integrate functions over an infinite domain. One example is $$\int_0^\infty \frac{e^{-x}}{\sqrt{x}}\,dx$$ I know the substitution ##u = \sqrt{x}## reduces this problem to integrating ##\exp(-x^2)##, but if I want to integrate the function as is, how would I do this? I've...
  9. L

    I The Expanding Universe: A Scientist's Perspective on Infinite vs Finite Space

    I have read some of the other posts about this topic but am still left unsatisfied. Could just be me. :cool: Did the universe, one minute after the big bang, consist of a finite volume of spacetime? If so, then is it not logically inconsistent that the universe can possibly be infinite now...
  10. S

    Limit as x approaches infinity.

    Homework Statement Compute Limit as x--> infinity of (logx)(log(logx)) / x The Attempt at a Solution Graphically, I see that the answer is perhaps zero, but I am not sure how to approach this algebraically. I worked at this for a couple hours, trying L'Hospital's rule but that did not really...
  11. Math Amateur

    MHB ZFC .... Axioms of Foundation .... and Infinity ....

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  12. Math Amateur

    I ZFC .... Axioms of Foundation .... and Infinity ....

    I am reading D. J. H. Garling's book: "A Course in Mathematical Analysis: Volume 1: Foundations and Elementary Real Analysis" ... ... I am at present focused on Part 1: Prologue: The Foundations of Analysis ... Chapter 1: The Axioms of Set Theory ... I need help with an aspect of the proof of...
  13. Ethan Singer

    I (Conceptual) Infinity energy via quantum tunneling & Nuclear Fusion?

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  14. Derek Hart

    B Proof that (x^n)/n has a limit of 0 at infinity

    I understand that the standard proof is a bit different from my own, but I want to know if my reasoning is valid. PROOF: Firstly, I assume that x is positive. I then consider p = inf{n∈ℕ : n>x} . In other words, I choose "p" to be the smallest natural number greater than x. If we choose n>p...
  15. Saracen Rue

    B Integrating to infinity issue

    Say we have two functions; ##f\left(x\right)=\frac{1}{e^x-1}## and ##g\left(x\right)=\ln \left(\frac{1}{x}+1\right)##. Let us find the limit of both functions as x approaches infinity; ##\lim_{x \rightarrow \infty} {f(x)} = \frac{1}{e^\infty-1} = \frac{1}{\infty} = 0## Therefore as ##x...
  16. manjuvenamma

    I Is it possible to find the limit of (1+1/x)^x as x approaches -infinity?

    Is it possible to find the limit of (1+1/x)^x as x approaches minus infinity using only the fact that it is e if x approaches infinity?
  17. H

    I Why probability current = 0 at infinity? Why must wavefunction be continuous?

    Q1. Why is the probability current ##j(x,t)=0## at ##x=\pm\infty##? (See first line of last paragraph below.) My attempt at explaining is as follows: For square-integrable functions, at ##x=\pm\infty##, ##\psi=0## and hence ##\psi^*=0##, while ##\frac{\partial\psi}{\partial x}## and hence...
  18. david2

    I What are the implications of infinity?

    hi, say you have an infinite distance. is half of that distance also infinite? or 1/10000000000000 of that distance. and so on and so on. i suspect it must be because if you say that half of infinity is for example 10000 km then infinty must be 20000 km, which it is not. so, if this...
  19. yecko

    Sum to Infinity: Proving Divergence

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  20. T

    Value of alternating sum at infinity

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  21. Snen

    Limit of x^cos(1/x) as x approaches 0+ | Calculus Homework Solution

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  22. S

    A Gauge transformations at infinity

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  23. S

    Evaluating limit at infinity by Maclaurin series

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  24. binbagsss

    Elliptic function - does limit at infinity exist?

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  25. Y

    MHB Solving the Infinity Hotel Problem: 161 Guests

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  26. M

    I Find t to Satisfy e^iat=e^ia_0, e^ibt=e^ib_0, e^ibct=e^ic_0

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  27. W

    I Carolo Rovelli on infinity in LQC

    I just read Carlo Rovelli's new book about loop quantum gravity. https://www.amazon.com/dp/0241257964/?tag=pfamazon01-20 Its a great book and i really enjoyed it. However there is a claim in that seems a little odd to me. In LQC singularities are resolved and replaced by bounces. So the the...
  28. A

    How to interpret the infinity of Hilbert Space?

    This is basically just a comprehension question, but what makes elements of the Hilbert space exist in infinite dimensions? I understand that the number of base vectors to write out an element, like a wavefunction, are infinite: \begin{equation*} \psi(x) = \int c_s u_s (x) ds = \sum_k^{\infty}...
  29. S

    Differentiate Infinity: What Is It?

    What is the differential of infinity?
  30. N

    I By Continuity definition 1/0 is infinity

    lim 1/x as x->0 is infinity, but the function taking it to infinity is continuous, but for continuous functions f(a)= lim f(x) as x->a, so by defininition 1/0 is infinity, what is wrong with this logic?
  31. S

    B Is infinity an imaginary number?

    Is infinity a imaginary number?
  32. Thalooth Bin Khalid

    I Is it really possible that relativistic mass tends to reach infinity?

    I have seen at many places that if ever matter travels more faster than light, it's relativistic mass will reach nearly infinity. Some says it's the inertia, so very high energy is required to accelerate. But since it is traveling with the velocity above 3×10^8 m/s, i believe that the high...
  33. K

    B Is Infinity x 0 Equal to Zero?

    What is infinity times zero? Isn't it zero? I mean, infinity times zero is the same as zero an infinite amount of times, and adding zero infinitely would give zero because even though zero is always being added, this process is the same as nothing ever being added to zero.
  34. BL4CKB0X97

    B Is Infinity Cyclical or Linear?

    In what may be an obvious observation in all accounts. I realized I did not have as good a grasp on infinity as I thought I had. For infinity to be infinite in must go on for ever(obviously, this I get), however, it must reach a point on where it begins to return to numbers it has already used...
  35. franktherabbit

    Find the limit of the expression

    Homework Statement $$\lim_{x\to\infty} \left(\frac{n^2+2n+1}{n^2+2n+3}\right)^{\frac{2n^2}{n+1}}$$ Homework Equations 3. The Attempt at a Solution [/B] I tried ##\lim_{x\to\infty} \left(\frac{n^2+2n+3-2}{n^2+2n+3}\right)^{\frac{2n^2}{n+1}}=## ##\lim_{x\to\infty}...
  36. Nader AbdlGhani

    B Understanding the Concept of Infinity as a Reference in Physics: Explained

    I'm facing a problem in my physics course which is accepting that infinity can be a reference point in both Electrostatics (calculating the voltage of a point) and Matter Properties (calculating the gravitational potential energy), how come we use a reference point which we don't know where it...
  37. ChrisisC

    B Trouble with Infinity: Understanding Density at a Point

    How is it possible that there is a infinite amount of density at a point? I understand how number can be infinite but how does something tangible like matter reach infinity?
  38. F

    I Where do we use infinity in physics, and why?

    Hello! I'm doing a school project, where I am writing about Infinity in Math and Physics. I've got the math part settled, but it's the physics part that has begun to bother me. One part of my task is to write about some of the mathematical expressions in physics, where we use infinity - but the...
  39. P

    B Curve where both x and y approach infinity

    What is the equation for a curve where x approaches infinity as y approaches infinity?
  40. B3NR4Y

    Finding the Magnetic Field Due to a Wire stretching from Infinity to 0

    Homework Statement An intensity current I descends down the z-axis from z = \infty to z = 0, where it spreads out in an isotropic way on the plane z = 0. Compute the magnetic field. Homework Equations The only relevant equation I can think of is Ampere's Law, \oint_\gamma \vec{B} \cdot...
  41. U

    MHB Find the limit as x goes to infinity

    Hi, I am having trouble with these kind of questions where we have to use L'Hospital's Rule. I took the ln of the function to get the x out of the exponent, and then followed the Rule by taking the derivative of the top and bottom (using a shortcut we learned: lim x --> infty f(x)g(x) = lim x...
  42. binbagsss

    A (sum to infinity of divisor function)^{2} -- simplify the expression

    Hi, I have ## 120 \sum \limits_1^\infty (\sigma_{3}(n))^{2} ## , where ## \sigma_{3}(n) ## is a divisor function. And I want to show that this can be written as ##120\sum \limits_{k=1}^{n-1} \sigma_{3}(k) \sigma_{3}(n-k) ## I'm pretty stuck on ideas starting of to be honest, since the sum is...
  43. Math Amateur

    MHB Axiom of Infinity and Garling, Theorem 1.7.4 - the successor set Z^+

    I am reading D. J. H. Garling: "A Course in Mathematical Analysis: Volume I Foundations and Elementary Real Analysis ... ... I am currently focused on Garling's Section 1.7 The Foundation Axiom and the Axiom of Infinity ... ... I need some help with Theorem 1.7.4 ... and in particular with...
  44. Math Amateur

    I Axiom of Infinity & Garling, Th. 1.7.4 & the successor set

    I am reading D. J. H. Garling: "A Course in Mathematical Analysis: Volume I Foundations and Elementary Real Analysis ... ... I am currently focused on Garling's Section 1.7 The Foundation Axiom and the Axiom of Infinity ... ... I need some help with Theorem 1.7.4 ... and in particular with the...
  45. CaptainJonathanNorth

    B Is Infinity a Prime Number: The Confusing Concept of Infinity Explained

    Infinity is both a number and a concept. I asked my 10 year old niece what kind of number infinity might be and she said, "It's a composite number." But I want to think about weather infinity is a prime number? Clearly if you divide infinity by any number, you get infinity. Also if you divide...
  46. M

    A Laurent series at infinity

    Question 1: Find the Laurent series of \cos{\frac{1}{z}} at the singularity z = 0. The answer is often given as, \cos\frac{1}{z} = 1 - \frac{1}{2z^2} + \frac{1}{24z^4} - ... Which is the MacLaurin series for \cos{u} with u = \frac{1}{z}. The MacLaurin series is the Taylor series when u_0 = 0...
  47. RicardoMP

    I Square integrable wave functions vanishing at infinity

    Hi! For the probability interpretation of wave functions to work, the latter have to be square integrable and therefore, they vanish at infinity. I'm reading Gasiorowicz's Quantum Physics and, as you can see in the attached image of the page, he works his way to find the momentum operator. My...
  48. A

    B What are surreal numbers and how do they work?

    Hey guys! I have heard of this concept in various places and sort of understands what it attempts to do. Can anybody please explain it to me in more detail like how it works, how to notate it, and how to expand it to infinities and infinitesimals. Thanks in advance! Aakash Lakshmanan xphysx.com...
  49. Battlemage!

    Lim n^n/n as n approaches infinity

    Homework Statement \lim_{n\rightarrow ∞}\frac{n^n}{n!} Homework Equations n! = (1)⋅(2)⋅(3)⋅...⋅(n-1)⋅n The Attempt at a Solution \lim_{n\rightarrow ∞}\frac{n^n}{n!} \lim_{n\rightarrow ∞}\frac{n^n}{(1)⋅(2)⋅...⋅(n-1)⋅n} I then factor n out of the denominator n times, or rather, nn, leaving...
  50. Radu Mitroi

    B What percentage of the universe do we know

    Hello. First of all, I must say that I'm new to this forum, so I apologize if I'm posting in the wrong section. I'm a 17 year old with not that much knowledge about physics, so if what I'm talking about makes no sense or is completely stupid, just let me know. A couple of days ago I asked...
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