What is Infinity: Definition and 982 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. 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...
  2. L

    A Construct BMS Coordinates near Null Infinity

    Let us consider Ashtekar's definition of asymptotic flatness at null infinity: I want to see how to construct the so-called Bondi coordinates ##(u,r,x^A)## in a neighborhood of ##\mathcal{I}^+## out of this definition. In fact, a distinct approach to asymptotic flatness already starts with...
  3. B

    MHB Proving a limit to infinity using epsilon-delta

    lim 2x + 3 = ∞. x→∞ Pretty intuitive when considering the graph of the function. But how would I show this using the epsilon-delta definition?Thanks!
  4. K

    Infinity question about known numbers?

    1) Can I say that humans in history named, recognized countable set of numbers (1,2..Pi,sqrt(2) ... etc) the count of this set is named aleph-0 ? 2) Do mathematicians investigate the "other" continuum set of numbers which is infinity bigger ? Do we suspect, knows anything about that set like...
  5. Physics lover

    A trignometric limit going to infinity

    I wrote cos(pi(n^2+n)^(1/2)) as cot(pi(n^2+n)^(1/2))/cosec(pi(n^2+n)^(1/2)) and as we know cot(npi)=infinity and cosec(npi)=infinity , so i applied L'Hospital.After i differentiated i again got the same form but this time cosec/cot which is again infinity/infinity.But if i differentiate it i...
  6. karush

    MHB 16.1 Show that e^{2x}, sin(2x) is linearly independent on + infinity -infinity

    16.1 Show that $e^{2x}$, sin(2x) are linearly independent on $(-\infty,+\infty)$ https://www.physicsforums.com/attachments/9064 that was the example but... \begin{align*} w(e^x,\cos x)&=\left|\begin{array}{rr}e^x&\cos{x} \\ e^x&-\cos{x} \\ \end{array}\right|\\ &=??\\ &=?? \end{align*}
  7. babaliaris

    Torque with Infinity length can lift anything?

    I was always wondering if you can lift anything (no matter how heavy it is) if you just use a really long pipe. Or does torque increases in a way like ##e^x## , ##a^x## and after some point it barely increases? Also if this can be explained mathematically, I would love to see it.
  8. P

    I Infinity: Is it Completed or Contradictory?

    If actual infinity represents a completed set of infinite data points, wouldn't that be a contradiction of terms?
  9. SamRoss

    I Why is the Laplace transform unchanged when t is replaced with -t?

    In Mathematical Methods in the Physical Sciences by Mary Boas, the author defines the Laplace transform as... $${L(f)=}\int_0^\infty{f(t)}e^{-pt}{dt=F(p)}$$ The author then states that "...since we integrate from 0 to ##\infty##, ##{L(f)}## is the same no matter how ##{f(t)}## is defined for...
  10. P

    I Exploring the Science of Infinity: Is It Possible?

    Einstein mentioned that our universe if a finite spherical universe inside an infinite space. If this said infinite space is, as said, infinite, then infinity is possible? How do you explain the science of infinity?
  11. D

    B Can black holes truly be infinite?

    Good morning. Not sure if I am doing this right, but I just wanted to ask a question.
  12. Mlesnita Daniel

    B Could singularities be just rips in the space-time fabric?

    First off, this is just an assumption. My knowledge of the field is extremely limited and I beg you to come and correct my mistakes, so I can learn. So, I guess we all know how that space-time fabric is bended by gravity. When a star dies, all of the atoms are brought extremely close...
  13. navneet9431

    B 1 to the power of infinity, why is it indeterminate?

    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 to 1? So why would the limit not exist?
  14. A

    A Uncertainty principle, removing infinity in the Fourier Transform

    I have come across a paper where it is stated that if the infinity assumption in the FT is removed, the uncertainty doesn't hold. Is this a sensible argument? Thank you.
  15. N

    I Collimating an extended source to infinity?

    I need to collimate an extended source (for example a cell phone) to infinity or as far as possible. I show an illustration below The collimator lens is about 2cm from the eye and the extended source is about 60cm from the eye. My understanding is that a collimator lens with a focal length...
  16. PainterGuy

    B Do the fields extend to infinity in a solenoid and for a wire?

    Hi, I wanted to clarify a point about the magnetic fields of a solenoid and wire. Do the fields extend to infinity? In my opinion, they don't but they can assuming the current also goes to infinity. They don't extend to infinity for a limited amount of current because they need to follow a...
  17. H

    MHB Calculus: Understanding Infinity in Functions

    When have a function and I know by investigation of that it getting "bigger and bigger" or getting "smaller and smaller", how could I know that in infinity it continue by that way always?
  18. A

    MHB Problem about limit to infinity

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  19. Adgorn

    Convergence of Roots at Infinity

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  20. Klystron

    Exploring Life at Infinity Bar

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  21. Y

    Help Taking the Limit as K goes to infinity

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  22. D

    I Infinity x 0: Is the Answer Zero?

    Hi. Is infinity multiplied by zero defined and is it zero ? And is infinity multiplied by any positive number , infinity ?
  23. Max Loo Pin Mok

    I How do we integrate x^2/(e^x - 1) from xg to infinity?

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  24. QuantumJames

    A How to take Pz in quasiPDF to infinity to get Light-cone PDF

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  25. M

    Proving Negative Infinity Divergence of (5-n^2)/(3n+1)

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  26. F

    I Infinity: The Limit Concept and Cantor Transfinites

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  27. H

    MHB Actual infinity vs. potentially infinity - Math philosophy

    what the differences between actual infinity to potentially infinity?
  28. G

    I Can Turing Machines Understand Infinity?

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  29. Mr Davis 97

    Showing that partial sums diverge to infinity

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  30. Y

    MHB Improper integral from 1 to infinity

    Hello everyone, I am stuck on this homework problem. I got up to (ln (b / (b+1) - ln 1 / (1+1) ) but I'm not sure how to go to the red boxed step where they have (1 - 1 / (b+1) ) if anyone can figure it out Id really appreciate it. thank you very much.
  31. Mukund

    B Is There a Model That Includes Infinite Space and Finite Matter in Our Universe?

    We have a Universe that can be seen by Hubble maximum and we can imagine millions time more than that but if there is no boundary, there may be another millions of universe. Suppose we gather this all millions of universe and say this is a giant universe of universes and thus matter do not stop...
  32. C

    Solve a limit with a nth root, with n -> infinity

    Homework Statement Solve the ##\lim_{n \rightarrow +\infty} \sqrt [n] \frac {n²+1} {n⁷-2} ## 3. The attempt of a solution: First I thought about using L'Hopital's rule, but the nth root makes it useless. Then I thought about to eliminate the root multiplying it by something that is one, but...
  33. Wrichik Basu

    B Does undefined always mean infinity?

    While using L' Hospital's rule in evaluating limits, one comes across limits of the following type: $$\lim_{x \to 0} x \ln x$$ Such limits are generally evaluated by taking ##x## to the denominator and make it ##x^{-1}##. In such a case, an indeterminate form ##\frac{\infty}{\infty}## comes...
  34. R

    I Understanding the concept of infinity

    In Hilbert infinity hotel, all the rooms were occupied. Then how did the occupant were able to shift to their adjoining room?? Here I understand, by full mean, ALL the infinite room has a corresponding occupant. I also understand some infinity number are greater because it can be proove when...
  35. A

    I Quantum Field Theory -- Does electron have infinity size?

    Art Hobson said that quanta propagate in space to infinity. (sorry can not give a link)
  36. bland

    I Does the concept of infinite parallel universes justify absurdities?

    I know this has been discussed ad infinitum (pun intended) however I have recently been informed by Max Tegmark's book Our Mathematical Universe, that apparently the multiverse concept is becoming mainstream. In fact he mentioned at a recent quantum conference that a show of hands revealed no...
  37. Auto-Didact

    Is 'Avengers: Infinity War' worth the hype?

    First off please keep this thread SPOILER FREE, at least for the time being seeing the film has just released, spoilers go in spoiler tags. Carrying on. Just saw it. Absolutely loved it. Highly recommend dropping whatever you are doing now and to immediately go see it in IMAX asap. Also, and...
  38. N

    Velocity of a charged particle as it approaches infinity

    Homework Statement Homework Equations F = k(q1q1/r^2) K = (mv^2)/2 The Attempt at a Solution I got number 18 easy enough, number 19 seems simple but I'm not getting the right answer. I'm calculating Force exerted by each charge on the new charge using F = k(q1q1/r^2) for the three charges...
  39. F

    I Why doesn't a photon's mass increase to infinity?

    Pretty self explanatory really. If a photon has a mass (1.67 * 10^-27 kg), and it travels at the speed of light, why does it's mass not increase to infinity?
  40. Sunanda

    What is the potential of plate B with respect to infinity?

    I have two isolated plates A and B, kept parallel to each other. Now I give charge +Q to the plate A, it will redistribute itself as +Q/2 on the outer plate A and + Q/2 on the inner plate A. Right? Now this will induce charge -Q/2 on the inner plate B and +Q/2 charge on the outer plate B...
  41. P

    MHB Is infinity / infinity equal to 1?

    Dear Members, I tried to prove this indeterminate form of infinity / infinity as 1. I could come up a reasonable approach with Gamma and Product functions. I posted my proof as video in Youtube. Here is the URL for the video. I would like to receive feedback and challenges on where my...
  42. D

    B Is infinity truly infinite if it has something else in it?

    Is infinity truly infinite if it has something else in it? Put differently, say there's an infinite volume of water that has some rocks in it, is the volume of water truly infinite? Though there's a place where there's no water?
  43. shintashi

    A How is Inaccessible Cardinal Written?

    I'm writing some notes on set theory, Aleph Null, etc., and was wondering if there's a Notation or Symbol that abbreviates this (inaccessible/strong/uncountable etc. cardinals). I'm not sure if I've seen notation before but it seems like symbols resembling Theta and phi have been used.
  44. R

    Limit of x^α.sin²(x)/(x+1) as x approaches infinity is?

    Homework Statement : Find [/B] limx->∞(xα(sin2x!)/(x+1) α∈(0,1) Options are: a)0 b)1 c)inifinity d)does not existHomework Equations : -[/B]The Attempt at a Solution : limx->∞(xαsin2x!)/(x+1)[/B] Dividing the numerator and denominator by xα, we have:limx->∞sin2x!)/(x1-α+x-α) clearly x−αis tending...
  45. A

    Finding the limit at infinity

    Homework Statement Finding the value of the limit: $$\lim_{t\to +\infty} t+\frac{1-\sqrt{1+a^2t^2}}{a}$$ ##a## is just a costant The Attempt at a Solution At first sight I had thought that the limit was ##\infty## but then I realized that there is an indeterminate form ##\infty - \infty##. I...
  46. Chris Miller

    B Can Infinity Be Even? | Exploring the Possibility

    Since every point on a circle has exactly one other point (opposite its diameter) whose tangent is parallel, can it be said (proven?) that a circle is composed of an even number of points? It's messing with my head to think of infinity as even. I realize one-to-one mappings in infinite sets...
  47. Arman777

    Insights Intro to Big Bang and Infinity Concepts - Comments

    Greg Bernhardt submitted a new PF Insights post Intro to Big Bang and Infinity Concepts Continue reading the Original PF Insights Post.
  48. H

    Find limit x to infinity from f(x) contains squareroot of x

    Homework Statement ##\lim x \to \infty \frac{\sqrt{x+1} - \sqrt{x}}{\sqrt{3x + 5} - \sqrt{3x + 1}}## Homework EquationsThe Attempt at a Solution ##\lim x \to \infty {\sqrt{x+1} - \sqrt{x}} * \lim x \to \infty \frac{1}{\sqrt{3x + 5} - \sqrt{3x + 1}}## ##\lim x \to \infty \frac{(x+1) -...
  49. F

    Limit proof as x approaches infinity

    Homework Statement Verify the following assertions: a) ##x^2 + \sqrt{x} = O(x^2)## 2. Homework Equations If the limit as x approaches ##\infty## of ##\frac {f(x)}{g(x)}## exists (and is finite), then ##f(x) = O(g(x))##. The Attempt at a Solution Let ##\epsilon > 0##. We solve for ##\delta##...
  50. 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...
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