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. C

    Number of vertices of an n-gon as n tends to infinity?

    The following is a problem that I’m sure is considered “basic” for mathematicians. I would therefore be gracious if somebody could, at the least, point me in the right direction to some reference. Since I’m not a mathematician, the simpler the better. :redface: In short, my question is: what...
  2. M

    Beyond Absolute Infinity, then neither Beyond nor not beyond

    What is beyond Cantor absolute infinity? Then can you imagine what is neither beyond nor not beyond, which is we can't even say that 'the Being is beyond the universe or beyond infinity' to that particular Being? And what is neither singular/oneness nor multiplicity?
  3. A

    Sum of a geometric series up to infinity

    Homework Statement A geometric series had first term 54 and 4th term 2. (i) What is the common ratio? (ii) Find the sum to infinity of the series. (iii) After how many terms is the sum of the series greater than 99% of the sum to infinity? Homework Equations N/A The Attempt at a...
  4. dkotschessaa

    Cantor, infinity and cosmology

    I should start a new thread for my questions rather than hijack others... Posted this on another thread but it didn't get any response, so bear with me if you've seen it. Has anybody looked into Cantor's works on infinity and seen how they relate to the question of an infinite universe...
  5. J

    On the infinity of the Universe

    Okay, now this question has been asked over and over, and all that stuff, so I am not going to ask whether the universe is infinite or not. Actually, my physical intuition says that it probably is, and so do two very intelligent people I admire, namely Eliezer Yudkowsky (a mere AI programmer...
  6. F

    Fourier transform, complex exponential and infinity

    I'm taking the Fourier transform of a signal. This integral has bounds from -∞ to ∞, but since the signal is 0 for negative t, the bounds become 0 to ∞ doing the integration, the antiderivative I get is et*(-3-jω+2j) where j is sqrt(-1) Now I have to evaluate this at t=infinity (since it is a...
  7. K

    Integrate exp(-(z-ia)^2) from z = - infinity to z = infinity

    Homework Statement Prove that \int^{∞}_{-∞} exp(-(z-ia)2)dz = √∏ for all real a. Homework Equations The Attempt at a Solution If I use the substitution x = z-ia then dz = dx and if I use the limits x = -∞ to x = ∞ I get the correct answer. However, I do not know how to justify leaving the...
  8. K

    Limits at infinity, lim xF(x) = L then lim (f(x)=0

    Homework Statement show that if F:(a,∞) -->R is such that lim xF(x) = L, x --> ∞, where L is in R, then lim F(x) = 0, x --> ∞. Homework Equations The Attempt at a Solution Let F:(a,∞) →R is such that lim xF(x) = L, x → infinity, where L is in R. Then there exists an α> 0 where given ε...
  9. I

    Intersection of Rationals and (0 to Infinity)?

    Homework Statement Let A = [Q\bigcap(0,\infty)] \bigcup {-1} \bigcup(-3, -2] Homework Equations So A = (0,\infty) \bigcup{-1} \bigcup(-3,-2] The Attempt at a Solution I understand that the Rational numbers are cardinally equivalent to (0,\infty), but why isn't...
  10. C

    Limits to infinity with natural logs

    Homework Statement I'm having trouble calculating these limits. Homework Equations none The Attempt at a Solution 1. lim...((lnx)^5)/(x)= x->infinity (((lnx)^5)/x)/(x/x)= (((lnx)^5)/x)/1= How do I calculate from here? 2.lim ...(14lnx^2)/(6lnx^3)=...
  11. C

    Calculating limits to infinity

    Homework Statement Can someone tell if if these look right? Homework Equations none The Attempt at a Solution 1. lim (lnx)^5/x = x->infinity 5lnx/x = (5lnx/x)/(x/x)= (5lnx/x)/1 = 0/1 = 0 2. lim sinx/x^2= x->infinity (sinx/x^2)/(x^2/x^2)=...
  12. A

    Why doesnt net torque cause angular velocity to increase upto infinity?

    If i have a shaft & I'm applying a driving torque D at one end & at other end there is resisting torque R due to bearing friction, etc. Then if D>R, i have net torque = moment of inertia times angular acceleration. Since the ang. accln is constant with respect to time, will the ang. speed of...
  13. C

    Limit of ln as x goes to infinity

    Homework Statement lim (lnx)^2/x x-->infinity Homework Equations none The Attempt at a Solution =5lnx/x * (1/lnx)/(1/lnx) =5/(x/lnx) How do I calculate x/lnx?
  14. H

    Proving that 'volume' and 'surface' of hypersphere go to 0 as n -> infinity?

    Homework Statement I'm supposed to find the equations of a hypersphere in n-dimensions (meaning the set of points within the radius R), as well as of its surface (the set of points at exactly radius R). I've already found the equations, and now need to show that both go to zero as n goes to...
  15. R

    Improper integral from -infinity to infinity

    What is the improper integral of sin[x]/(1+x^2) from -infinity to infinity equal to? It doesn't seem to be integrable in the real number system. Is it 0, because sin[x] is an odd function and 1+x^2 is an even function, and an odd function divided by an even function is equal to an odd...
  16. V

    Does it matter if a charge (unknown) placed at infinity

    Homework Statement Does it matter if a charge (unknown) placed at infinity w.r.t to one known charge be called positive or negative? Homework Equations - The Attempt at a Solution -
  17. alexmahone

    MHB Approaching Infinity - Finding the Limit of a Sequence

    Find $\displaystyle\lim_{n\to\infty}\frac{\sqrt[n]{n}}{\sqrt[n+1]{n+1}}$.
  18. L

    Why can't the wavefunction equal infinity?

    I see why having mutiple points of infinity in the wavefunction would be bad. But what about one point being infinity and everywhere else being zero? Is this the only case where the wavefunction could have an infinite value? Would this be a case where the expectation value of whatever...
  19. D

    C is infinity thus electrons can be in two places at once

    My thought experiment is this: The slit experiment showing photons are particles and waves leaves out a constant that is wrong. c in a vacuum is 187kph because vacuum is a medium. c is infinite in true nothing. Our sample time of an electron Thus gives probabilities not definates...
  20. R

    Infinity Biscuit Dilemma: Do You Have More Biscuits or Jars?

    Thread title says it all. Lets say you have an indestructible jar. The jar contains two biscuits. Now imagine you had an infinite number of these jars, each containing two biscuits. Would you have more biscuits than jars?
  21. A

    Integral of x*delta((x/y)-t) dx from 0 to infinity

    Homework Statement ∫x*delta((x/y)-t) dx from 0 to infinity Homework Equations ∫x*delta((x/y)-t) dx from 0 to infinity = ty*|y|*θ(ty) The Attempt at a Solution Okay, so using the transformation of variables technique via the Jacobian, I see where the |y| comes from. However, using...
  22. C

    Self inductance of a very thin conductor, is infinity?

    Hi, I was reading some text and came across this problem, the problem is also mentioned in this link from Wiki: http://en.wikipedia.org/wiki/Inductance#Self-inductance They said that it is because the 1/R now becomes infinite, this is what I'm confused about. From my understand, there would be...
  23. anemone

    MHB The Limit of (sinx+1)/(x) - 1 or Infinity?

    Someone told me that the limit as x goes to zero of (sinx+1)/(x) is 1 but I think it's quite obvious that its limit is infinity. I hope you can tell me which is the correct answer? 1 or infinity? Many thanks in advance for replying.
  24. A

    Electric Potential Energy: Work Required to Move 3 Charges Out to Infinity

    Homework Statement Three charges are distributed as follows: http://tinypic.com/r/1fviq0/5 How much work must an external force do to move them infinitely far from each other? Homework Equations W = -\DeltaU = (kq_{1}q_{2})/r The Attempt at a Solution So what I did was find...
  25. N

    Infinity & Particles: Can They Really Travel at the Speed of Light?

    I just started to study and happen to stumble to more questions then answers. I thought that photons would travel with the speed of light, and the problem is that in calculus we learned that 0 and ∞ are not achievable numbers. And for a particle to achieve the speed of light, would it not...
  26. L

    Why does Lim as x approaches infinity of x/(x-9) = 1?

    Homework Statement Limit as x approaches infinity of x/x-9 Homework Equations None The Attempt at a Solution I know the indeterminate form infinity/infinity happens. I don't know how to fix it, but I'm assuming it's quite simple...
  27. D

    Define the potential to be zero at infinity

    Homework Statement A solid insulating sphere of radius a = 5.6 cm is fixed at the origin of a co-ordinate system as shown. The sphere is uniformly charged with a charge density ρ = -159 μC/m3. Concentric with the sphere is an uncharged spherical conducting shell of inner radius b = 10.7 cm, and...
  28. D

    MHB Solving Limit at Infinity Problem: x^(2/3) / (log^2(x))

    Hello, I am working on some limit problems and ran into one in which I am lost on how to proceed with: The problem is: lim x -> infinity x^(2/3) x/(log^2(x)) I have a basic understanding of L'Hopital's Rule and attempted to apply it, but just ended up with a confusing mess. I'm assuming I'm...
  29. J

    Voltage not zero at infinity

    Homework Statement A point charge q = -8 µC is surrounded by two thick, conducting spherical shells of inner and outer radii a1 = 0.2 m, a2 = 0.3 m, a3 = 0.8 m, and a4 = 0.9 m respectively. The inner shell is uncharged; the outer shell has a net charge Q = -8 µC. At this point in the...
  30. C

    Are there Electrostatic Fields at infinity?

    Homework Statement In the region x ≥ 0 there is an electrostatic potential V(x)=xexp(-λx) where λ>0 What is the electrostatic field at x→∞ Homework Equations The Attempt at a Solution My understanding is there is no fields at infinity because there is no electrostatic potential...
  31. K

    Can't figure out how to evaluate a sequence as it goes to infinity.

    Homework Statement An = (((-1)^(n-1))n)/(n^2 + 1) I need to know if it converges or diverges and if it converges the limit. Homework Equations The Attempt at a Solution I know it converges to 0. But I don't know how to show it when evaluating. I tried evaluation An| in the...
  32. S

    [Electromagnetism] How much work is required in moving Q3 to infinity?

    Homework Statement How much work is required in moving Q3 to infinity while Q1 and Q2 remain in their positions? Q3-------a-------| |-----------------| b Q1-------------Q2 a = 16.0 cm b = 6.0 cm Q1 = 5.70 μC Q2 = -5.70 μC Q3 = 1.8 μC Homework Equations W=ΔPE PE=kQ1Q2/r...
  33. K

    (Comparison Theorem) Why is x/(x^3+1) convergent on interval 0 to infinity?

    ∫0->∞ x/(x^3 + 1) dx. Use comparison theorem to determine whether the integral is convergent of divergent. Homework Equations None. The Attempt at a Solution ∫0->∞ x/(x^3 + 1) dx = ∫0->∞ x/(x^3) dx = ∫0->∞ 1/(x^2) dx From my class I learned that ∫1->∞ 1/(x^2) dx , is convergent But...
  34. R

    Time to Travel to One Infinity and Back?

    Homework Statement Consider the ODE x' = x2 + ε, where ε is a small number. Find the time T = T(ε) it takes for the solution to travel from x = -∞ to x = ∞. Let T1 = T1(ε) be the time it takes the particle to travel to x = -1 to x = 1. Show that T/T1 → 1 as ε → 1. Homework Equations...
  35. F

    Singularity, Infinity, Edge, Expansion

    First post in the Physic forum, the thread title are the words that really get on my nerves. I have the following questions. a) Cosmologists say the universe started at a singularity, there for there is a point where the universe started to expand from, if you could look at the singularity...
  36. Q

    Limits with sinx where x tends to infinity help.

    Homework Statement http://prikachi.com/images/26/4273026B.gif Homework Equations The Attempt at a Solution I'm having problems in solving the limit which is shown on the gif file . As you see I use L'Hopital's rule 2 times and I get to a point when I should divide -sinx with sinx...
  37. G

    What is the definition of analytic at infinity in complex analysis?

    Hi, I'm reading "Spectral Theory of Linear Operators" by John Dowson. I've seen the phrase "analytic at infinity" popping up very early in the book, but no definition is given. I wonder if anyone could tell me what the definition is or where I might find the definition and perhaps a few basic...
  38. R

    Speed of light becomes Infinity and Zero

    Speed of light becomes Infinity and Zero! Hi Guys, I am just new to SR & GR. I learned about Lorentz contraction and time dilation, but they don't make sense when i consider the following situation... Your(Moving non inertial frame) Time runs slow relative to a stationary observer when your...
  39. V

    Proof of relativistic energy/mass to grow to infinity for v approaching c?

    By SR, relativistic energy and mass are proportional to the Lorentz factor γ, therefore, grow to infinity for v\rightarrowc. This relationship for the relativistic mass has been confirmed by the Kaufmann experiment and its successors, via measuring the deflection of high velocity electrons by an...
  40. L

    Voltage difference = Infinity?

    Voltage difference = Infinity? So if there was a Q Coulomb point charge and a -Q Coulomb point charge with X meters of separation, and I wanted to find the voltage difference between those two charges... How would I do it?Since V = kq/r + kq/r in this case, wouldn't I have to divide by 0? V =...
  41. U

    The Nature of Infinity in Mathematics and Reality

    Hello all! In the past few months I've stumbled upon an issue that has played games with my mind. I feel I need some help to solve this, as I've tried various other sources and remain without answers. Firstly, I was confronted with a mathematical proof which states that 0.9~ (to infinity)...
  42. J

    Hypergeometric equation at z = infinity

    Homework Statement Show that by letting z = \zeta^-1 and u = \zeta^{\alpha}v(\zeta) that the differential equation, z(1-z)\frac{d^{2}u(z)}{d^{2}z}+{\gamma - (\alpha+\beta+1)z}\frac{du(z)}{dz}-\alpha \beta u(z) = 0 can be reduced to \zeta(1-\zeta)\frac{d^{2}v(\zeta)}{d\zeta^{2}} +...
  43. R

    Asymptotic series as x approaches infinity

    Homework Statement Find the first 3 (non-zero) terms of the asymptotic series as x→∞ for ln(ex+1) Homework Equations ln(1+ε) ~ ε-ε2/2+ε3/3-O(ε4) (x→0) The Attempt at a Solution I found x+ln(1+e-x) (x→∞) = ln(ex+1) (x→∞) ε=e-x which allows me to use the Maclaurin series...
  44. M

    Is Infinity possible without duplication?

    Would an infinite existent necessitate that duplication exist? For instance, in a model that mandates a multiverse system on an infinite scale, does that mean that there exists more than one of each possible object? I guess I'm asking a more fundamental question than that.
  45. F

    Determining convergence of (-1)^n/ln(n) as n goes to infinity

    Homework Statement Determine whether (-1)^n/ln(n!) is divergent, conditionally convergent, or absolutely convergent. Homework Equations None, really? :SThe Attempt at a Solution Okay, so I know the series converges by the Alternating Series test - terms are positive, decreasing, going to...
  46. M

    Infinite Possibilities: Exploring the Different Types of Infinity in Mathematics

    How many kinds of infinity does math have? in my point of view "THERE ARE INFINITY KINDS OF INFINITY" What's your point?
  47. P

    Understanding Limits: Does it Exist or Go to Infinity?

    How can you tell if a limit does not exit or that it goes to infinity? examplelim\underbrace{x\rightarrow}_{0}(\frac{\sqrt{x+1}}{x}) The x goes to 0The book says the limit is \infty but if you take the left side limit you get -\infty and if you take the right side of the limit you \infty. So...
  48. R

    Why wave function should goto 0 f faster than 1/sqrt(x) at infinity

    Can anybody please explain the reason why a normalizable wave function ψ(x) → 0 faster than 1/√x as x → ∞. I can understand the reason why ∫ψψ*dx < ∞ But do not understand how quadratic integrability implies that. I would be very thankful to anybody who can give me some idea.
  49. R

    Zero to infinity can we assume same for universe ?

    Zero to infinity...can we assume same for universe ?? Hi, I am new to this forum, Basically I am not a scientist(not at all), like others in the forum, I belong to computer and Network field, However the science fantasies and attracts me very much just like others, ever since I was a kid , I...
  50. K

    Apostol: infinity as finite point

    I found a torrent online of Apostol's "Mathematical Analysis" 1st edition and I think I found a typo, or whoever scanned the book cut off the edge a bit... Apostol writes that the extended real number system R* is denoted by [-∞, +∞] while the regular real number system R is denoted by (-∞...
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