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. S

    Calculating the Limit as x Tends to Infinity of a Cube Root Expression

    Homework Statement calculate the limit as x tends to infinity of: \sqrt[3]{x} ((x+1)^{(2/3)}-(x-1)^{(2/3)}) Homework Equations The Attempt at a Solution using the identity: a-b=(a^2-b^2)/(a+b) ; and dividing top and bottom by x, = lim...
  2. H

    Prove n^(1/n) tends to 1 as n tends to infinity

    Homework Statement Need to prove n^(1/n) tend to 1 as n tends to infinty Homework Equations The Attempt at a Solution Have tried comparing to n^(1/n)=(1+h) and using binomial series but no joy..please help
  3. B

    BOOBIES .actually a question on limits approaching infinity

    BOOBIES...actually a question on limits approaching infinity...please help Homework Statement lim x^2-x^4 x>Infinity Homework Equations i made both of them...e^(2lnx)-e^(4lnx)...then get stuck im allowed to use limit laws, and no l'hospitals rule The Attempt at a...
  4. A

    Find a Function f(x) with Limit at Infinity but no Derivative Limit

    Hello I need to give an example of a function f(x) such that limf(x), as x goes to infinity, exists but the limf'(x) does not. Any help? Thx
  5. A

    Find Limit as x approaches infinity

    1. Find the limit as x approaches infinity of (cos(1/x))^(x^2) attempt at a solution I tried using e to change it's form to e^(ln(cos(1/x)*x^2) and taking the limit of the power, the problem is I'm really stuck at this point with the limit as x approaches infinity of ln(cos(1/x))*x^2...
  6. F

    Find the limit of ln(n)/ln(n+1) as n--> infinity

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  7. H

    The limit of a power as it tends to infinity.

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  8. S

    What is the limit of 1/n^2 + 2/n^2 + + n-1/n^2 as n-> infinity

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  9. mbisCool

    Limit as x goes to infinity (algebraic)

    Homework Statement limx->infinity (x+2)/(sqrt(81x^2+15)) Homework Equations The Attempt at a Solution The only thing i could think of doing was rationalizing the denominator to get (x+2)sqrt(81x^2+15) / 81X^2+15 however I am pretty sure this is the wrong route cause there doesn't...
  10. G

    Prove that a sequence which is bounded above cannot tend to infinity

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  11. M

    Understanding the Limit of (1+1/n)^n as n Goes to Infinity

    I'm studying for the GRE that's coming up in a week or two and I came across a problem where the answer given in the book does not make sense to me and I was wondering of someone here could explain it to me. Question: Lim as n goes to infinity of X_(n+1) / X_n Where X_n = n^n / n...
  12. B

    Does Nuclear Reaction Energy Contribute to the Existence of Infinity?

    Is it true that 0.1% of the mass in an uncontrolled nuclear chain reaction gets converted into energy via E=Mc^2? And if this is so, then does the mass technically go to infinity in accordance with the Lorentz transformation before turning into energy?
  13. Loren Booda

    Electron mass density approaches infinity?

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  14. M

    Find a value of constant k so a limit to infinity exists

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  15. D

    Space L Infinity Homework: Open or Closed?

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  16. A

    Finding Solutions to x Goes to Infinity and Beyond

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  17. L

    Finding Limits Approaching Infinity for Improper Integrals

    Homework Statement Lim ((e^{2})-(\frac{t-3}{3}( * e^^{\frac{t}{3}}))) t -> -\infty [sorry for the formatting, I tried my best! that is "The limit as t approaches negative infinity of e squared minus (t-3/3) e to the t/3)]"] Homework Equations I am solving improper...
  18. A

    What Happens When You Calculate One to the Infinity Power?

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  19. K

    What is the limit of the square root of a polynomial as x approaches infinity?

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  20. G

    Point charge attractioning a positive charge from infinity to 30cm

    Homework Statement A point charge of 2.0 10-9 C is located at the origin. How much work is required to bring a positive charge of 4.0 10-9 C from infinity to the location x = 30.0 cm? and it requests the answer in J Homework Equations W=-q2(delta)V DeltaV = Ke q1/d where...
  21. J

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  22. N

    Lim n to infinity for factorial

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  23. D

    Limits As X Approaches Infinity and Negative Infinity

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  24. M

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  25. L

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  26. T

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  27. F

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  28. P

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  29. N

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  30. P

    More on Infinity: Why Does Balloon Analogy Not Describe Universe?

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  31. N

    Is the Wave Equation at Infinity Always Zero?

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  32. D

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  33. G

    Visualizing Point at Infinity - Euclidean Plane

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  34. M

    Exploring Infinity: How Baby Universes Form from Black Holes

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  35. madmike159

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  36. E

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  37. N

    Exploring the Boundless Expanse: Unraveling the Mysteries of Space

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  38. P

    Defining Analyticity at Infinity: How Do You Define and Calculate It?

    How does one define, given a complex function, the following: The function is analytic at infinity. The derivative of the function at infinity. It turns out that it's supposed to be quite common to define these terms, however I have never been shown either of them. I have a few...
  39. G

    Wave Function Zero At Infinity?

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  40. K

    Convergent Sequences on l infinity

    Homework Statement Define R^\infty_f = \{ (t^{(1}),t^{(2}), \ldots, ) |\; t^{(i}) \in \mathbb{R}\; \forall i, \; \exists k_0 \text{ such that } t^{(k})=0 \; \forall k\geq k_0 \} Define l^\infty = \{ (t^{(1}),t^{(2}), \ldots, ) |\; t^{(i}) \in \mathbb{R}\; \forall i, \; \sup_{k\geq 1} |...
  41. A

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    Guys How do you think our reality would differ if the phi ratio were exactly 3 ? Is infinity a possibility? To me It must be, but in he same breath it can't be! "It is impossible to imagine an impossibility" By the way I am a retired Mechanical Engineer (Electricity power...
  42. G

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  43. N

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  44. M

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  45. L

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  46. Beer w/Straw

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  47. C

    Limit as x approaches Infinity

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  48. R

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  49. C

    ZF Axiom of Infinity: Natural Numbers Exist

    In ZF, the axiom of infinity says that the set of natural numbers exists. I was wondering if there was a (finitist?) weakening of ZF that included the axiom "the class of natural numbers exists".
  50. T

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    [SOLVED] Voltage Query Homework Statement An insulating spherical shell with inner radius 25.0 cm and outer radius 60.0 cm carries a charge of + 150.0 \mu C uniformly distributed over its outer surface. Point a is at the center of the shell, point b is on the inner surface and point c is on...
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