What is Infinity: Definition and 984 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. T

    What will a voltmeter read if it is connected between c and infinity?

    [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...
  2. J

    Can the Value of e Be Found Using the Taylor Series?

    Why does \lim(1+\frac{1}{x})^x = e x->\infty
  3. M

    Evaluating the Limit of a Function at Infinity: [x+1-ln(x+1)]

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  4. G

    Divergent series and the limit of the nth term as n approaches infinity

    I'm looking for help with my conceptual understanding of part of the following: 1) If a series is convergent it's nth term approaches 0 as n approaches infinity This makes perfect sense to me. 2) If the nth term of a series does not approach 0 as n approaches infinity, the series is...
  5. A

    Do Zero and infinity break laws?

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  6. K

    Cardinality of infinity (3)

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  7. K

    How Is Cardinality of Infinite Sets Defined in Mathematics?

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  8. K

    How Many Constructible Points Exist on the X-Axis?

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  9. M

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  10. S

    Finding the Limit of F(s) as s Goes to Infinity: Exploring Exponential Order

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  11. U

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  12. pellman

    Limits, infinity, and cardinality (oh, and integrals too)

    These are some related questions in my mind, though I am rather confused about them. 1. What does \infty at the "end" of the real number line have to do with \aleph_0, the cardinality of the integers, and C, the cardinality of the continuum? Is \infty equal to one or the other (if such a...
  13. J

    Is the derivative of infinity zero?

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  14. D

    Fnd the limit of sqrt(x^2 + 4x) - x as x goes to infinity

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  15. T

    Limit of Arcsin Approaching Infinity

    Homework Statement limit approaching infinity: (arcsin(x))/(x) = 0 Question is: Why? The 'Sandwich Theorem' 0=[(arcsinx)/x]=0 gives this solution, but looking at the graph of (arcsinx)/x , this appears impossible. Homework Equations lim x->OO [arcsin(x)] - {DNE) lim x->OO...
  16. M

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  17. D

    Limits of polynomials at infinity

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  18. E

    Finding Limits at Infinity: x^4 + x^5

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  19. G

    Limiting x^2-x as x Approaches Infinity

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  20. J

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  21. K

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  22. P

    Find the imit of 2x + 1 - sqrt(4x^2 + 5) as x--> infinity

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  23. D

    What is the Limit as n Approaches Infinity of a Rational Function?

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  24. H

    Infinity subtracted from infinity is undefined.

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  25. H

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  26. P

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  27. P

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  28. M

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  29. M

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  30. S

    ∑ C An =C ∑ An (n from 1 to infinity) ... why?

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  31. S

    Lim of sqrt(x) * sine(1/sqrt(x)): Solving for x --> infinity

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  32. R

    Help Needed: Analyzing Limit as x Approaches Positive Infinity

    The problem is The limit as x approaches pos infinity ln(square root of x + 5) divided by ln(x) In the numerator only x is under the square root. I'm having trouble getting to this answer. If someone can take a look I would really appreciate it.
  33. M

    Find the Limit as n Approaches Infinity

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  34. J

    Is There an Infinite Number of Universes in Infinity Itself?

    Here's an idea that I have been working on for the past few months. After weeks of writing out idea after idea about the universe and other universes I have reason to believe that there is an infinite number of universes contained in infinity itself. Here's the idea broken up: There is one...
  35. K

    Infinity in Mathematics (Calculus and Series)

    Could some one please explain to me 2 things 1) I have seen integrals that are between 0 and ∞ and also between -∞ and ∞. What does this mean 2) I have also seen sigma series (∑) between n=1 and ∞. What doe this mean Thanks heaps
  36. M

    Circle & Infinity: Why Is a Line?

    Why is it that a circle with an infinitly large radius is a line?
  37. C

    What is the Method for Determining Infinity Limits in a Rational Function?

    ok I am confused when x->negative infinity or positive infinity. for example lim (5x^3+27)/(20x^2 + 10x + 9) x-> negative infinty heres what i think, i want to know if i have the right idea or not. - so since the top exponent is larger then the denominator the lim DNE and so i...
  38. L

    Can Two Infinitely Sized Objects Exist in an Infinite Space?

    Spare me the ridicule for asking a stupid question, I'm just your average guy trying to comprehend his environment. I do my best to understand the origins and workings of the universe but no matter what path I take I will end up facing an infinity of one form or other. Ok here goes. I have...
  39. J

    Approaching Infinity: Simplifying Radical Expressions

    Homework Statement \lim_{\substack{x\rightarrow \infty}}f(x)=\sqrt{3x^2+8x+6}-\sqrt{3x^2+3x+1} Homework Equations The Attempt at a Solution I truly have no idea how to solve this. I know I need to get x in some rational form like 5/x but I'm not sure how to do this with the...
  40. J

    Find all the different limits in + infinity

    Homework Statement f(x) = sqroot(ax^2 + 1 ) - x find all the different limits in + infinity that f can have with all the different "a" values Homework Equations The Attempt at a Solution i don't know if i did something wrong here : - if a = 0 then f = 1 - x, limit : -...
  41. Z

    How Big is Infinity? | Endless Possibilities

    How big is Infinity ? like eletron is considered the samllest thing ever, wouldn't that be the lowest value of negative inifinty in teh size of things but overall how big or small is infinity ?
  42. J

    Can Infinity Minus X Ever Be Larger?

    I understand that infinity - X = infinity. However, what I don't understand is this. If I have: infinity - 1 = infinity, AND infinity - 5 = infinity, is the first infinity larger than the second? If so, how can that be? Because they are infinite, how can one really be larger than the...
  43. M

    Infinity: A Concept Beyond Measure?

    If infinity were to be possible, it would seem that nothing in it's surrounding environments could be segmented or finite. How do we record segments of something that is infinite like for example time? It's almost as if infinity is really points of end, causing new beginnings. Thus causing an...
  44. R

    Exploring the Concepts of Zero & Infinity

    We know that anything divided by zero is 'undefined' or equal to infinity. Is it not possible to define in anyway such indeterminate quantities? The concept of zero basically refers to 'nothingness' or 'void', but that indeed has utmost importance in writing numbers. If you consider a general...
  45. I

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    Can someone give me a hint on how to evaluate the following limit? \stackrel{lim}{T\rightarrow\infty} (Texp(c/T) - T) I tried multiplying the numerator and denominator by the conjugate (because that sometimes helps) and got: (T^2exp(2c/T) - T^2) / (Texp(c/T) + T) But I'm not sure what I...
  46. phoenixthoth

    What is (the nature of) infinity?

    Perhaps some consensus can be arrived at in regard to what infinity is. After that, perhaps its nature can then be discussed. One approach to defining infinity is to first define what finite means and then say something is infinite if it is not finite. Rather than define infinity by what it...
  47. K

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    Let f be a differentiable complex valued function on R. If f is square integrable, then it is not the case that f(x) must approach zero at infinity. counterexample: f(x)=x^2 exp(-x^8 sin^2(20x)). If I also require that the derivative of f be square integrable, is that enough to guarantee that...
  48. B

    Finding the Average Number of Packets Needed to Complete the Word SPARK

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  49. U

    What Are the Different Orders of Infinity and How Do They Compare?

    I wasn't sure where to post this, so I'll post it here. Depending on your answers, I may have a few more questions. What's greater: \infty^2 or 2^\infty? Why?
  50. C

    Black holes and entropy: energy with respect to infinity

    G'day! In the paper "Black holes and entropy" (JD Bekenstein, Phys Rev D 7 2333, 1973), in the section on Geroch's perpetual motion* machine, I'm trying to understand why they can state "its energy as measured from infinity vanishes"? What they mean is that the work extracted by lowering...
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