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. T

    Lim x to 0 (1/sinx - 1/x ) is equals to infinity?

    may i know is it lim x to 0 (1/sinx - 1/x ) is equals to infinity??
  2. H

    Average of Infinity: Is It Possible?

    Hello I was watching lectures on Quantum Computing another day and I came up with a problem. I talked to my school teachers about it but everyone seemed to dodge my question except for one, my math team teacher who told me that without assuming anything it cannot be solved. Or we can take the...
  3. Gamma

    Is the limit also infinity?

    Hi, What is the limit of ln(\frac{1+1/ \sqrt(1+1/R)}{1-1/ \sqrt(1+1/R}) as R reaches infinity. (latex did not show it very well. But the numerator is same as denominator except there is a + sign instead of a - sign. As R reaches infinity the argument of the ln reaches...
  4. Y

    Is the Sun truly the center of the universe?

    i think maybe this question was posted in this furoms for so many times but i want to ask it again in my own version.. d' question goes like ths.. is there such thing as infinity? i mean for example, infinite time or infinite quantity of space or matter? if so, then how could it be...
  5. U

    Exploring Infinity Singularities: Why Don't They Suck in Everything?

    I have an interest in physics, I have not studied it at any high levels (as u will soon find out :rofl: ), i just like to read and think about it, but I have a question, please don't blind me with maths! If a singularity at a black hole has infinity density, I'm guessing this means it has...
  6. E

    Limit with n tending to infinity

    let,s suppose that we have the limit with n tending to infinity: \frac{f(n)}{g(n)}=1 then i suppose that for n tending to infinity we should get: f(n)\rightarrow{g(n)} or what is the same the function f(n) diverges as g(n) as an special case: \pi(n)\rightarrow{n/ln(n)} where Pi is the...
  7. dextercioby

    Exploring the Limit of Cosine as x Goes to Infinity

    of doing this \lim_{x\rightarrow +\infty }\cos^{x}\frac{\pi}{x} than using \cos x \simeq 1- \frac{x^{2}}{2} \mbox {when "x" goes to zero} ? Daniel.
  8. S

    How to find limit of sin^2(x)/x^2 as x--> infinity

    Hello Everyone, I'm doing some practice problems and I am stuck on the following limits. 1) Limit as x approaches infinity of (sin^2(x))/x^2 Now for this one I thought that the answer would just be 1 seeing as the limit of sinx/x is 1, however the answer is apparently 0 and I can't see the...
  9. L

    Vatican to Host Debate On Infinity

    I heard about this debate and wanted to share with you what information I have gathered so far...maybe you'all can find more out about it. I'll try and keep you posted, and thought maybe this would be a great subject to keep posted on!? For now though, I just wanted to let you know about it...
  10. G

    Work needed to move charge from infinity to center of sphere

    Hey, I recenty took an exam and the professor asked us to turn in a corrected version of the exam based on the comments he made to us (I guess people didn't do as well as he expected). On one of the problems it asks: How much work does it take for you to move a charge Q from infinity to the...
  11. M

    Input impedances for both inverting and non-inverting amplifiers infinity?

    are the input impedances for both inverting and non-inverting amplifiers infinity?
  12. K

    Summation of n^2k. k = 1 to infinity

    Is the summation of k = 1 to infinity for n2k equal to n2 / (1 - n2)?
  13. happyg1

    Proving "No Convergent Subsequence -> |s| Diverges to Infinity"

    Hi, Here is the question: Prove that if the sequence {s} has no convergent subsequence then {|s|} diverges to infinity. To me, this seems so easy, but I'm having a really hard time putting it down in a rigorous manner. My thoughts are: every convergent sequence has a convergent...
  14. Z

    A sub-thread to Yomammas Infinity issue

    I`m relativly new to Physics but have heard of a technic used called renomalisation, the cancelling of infinties. If infinity is not a number how does this work? (my appologies to the mods if i`ve posted this in the wrong place)
  15. kakarukeys

    Exploring the Relationship between Infinity and Zero in Feynman Path Integrals

    :grumpy: :yuck: Hello there, please tell me whether e^{iS} = 0 when S = +\infty S is the action of a particle = \int L dt e.g. when the particle goes to infinity and comes back, kinetic energy blows up, then S blows up.
  16. Math Is Hard

    Is Infinity a Fundamental Property of the Physical Universe?

    I can sort of deal with it conceptually, and even work with it mathematically - but are there any examples of it as a property of something in the physical universe? (I'm probably missing something obvious..)
  17. T

    Can We Really Grasp the Concept of Infinity in the Universe?

    I gave roughly an hour's thought to infinity, and came up with these statements and questions; If infinity exists, where would it go? The only infinity that would make sense is the size of the universe, because you couldn't have an infinite amount of any matter without it taking up literally...
  18. P

    Is \Re^\infty Used for Sets with Zero Elements?

    Is it correct that \[ \Re ^\infty = \left\{ {\xi \in \Re ^N \left| {\exists n \in N:\forall k > n\,\xi _k = 0} \right.} \right\} \] , N denoting the natural numbers? If not, is there another symbol for the left hand side of the equation? Is this also used with any other set containing a...
  19. G

    Understanding the Relationship Between 0 and Infinity

    With the real number line. 0 is before -1 and 1. Both zero, and infinity are undefined. How does 0 touch infinity. Both are the same value, and start at 0, so for there to be infinity either infinity starts from infinity, or infinity starts from 0. In fact infinity has both values. It...
  20. B

    Exploring Infinity: Is 1 Still Infinity?

    Is infinity - 1 still infinity. Also if 1\3 = .333... then wouldent 3\3not only = 1 but .999... as well?
  21. B

    Is zero times infinity equal to negative one?

    Zero x infinity = -1 ?! when two lines meet perpendicularly, we hv the formula v m1 x m2 = -1 if a horinzontal line meet v a vertical line,by using same formula ,we hv m1 x m2 = -1 zero x infinity = -1 so..does it make sense??! :eek:
  22. Z

    What is the Definition of Infinity?

    can i ask how do we define infinity? Is it a number that cannot be represented mathematically(like something divided by 0) or is it a number too large that our human mind can't think of? what do statements like 'The universe is infinite' mean? Is it too large to be represented or there are...
  23. S

    Calculating an Integral with the Limit of Infinity

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  24. M

    What is sin x when x tends to infinity?

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  25. Orion1

    Does infinity - infinity equal 0

    Is \infty - \infty = 0?
  26. D

    H and Infinity: What Happens to Time and Distance at the Event Horizon?

    Another question from someone who doesn't know much about physics. If black holes don't let light out, then it has to stop somewhere, right? There has to be a point where the light is not moving at c. It's been proven that gravity can "bend" light waves, so why could it not stop them? Also, if...
  27. L

    Understanding the Revised Infinity Theory: New Rules and Applications

    For those who were on the last thread concerning this, I have started a new one over since the last one is dead. I have thought on the rules for a while, and now have made a few new rules and changed the otherones. For those who have not read the last thread, what I am trying to do is create...
  28. C

    Is Infinity Just an Inverse Zero?

    Hey, I'm new here. I wanted to ask a question about infinity - I've been playing with something in my mind. If you define infinity as a number that is greater than all numbers except itself, and negative infinity as a number that is less than all numbers except itself... Couldn't both...
  29. R

    If f(c) = infinity and c is in [a,b]

    if f(c) = infinity and c is in [a,b] is it equivalent to saying f is not cont. at c ? because infinity is undefined? Thanks k.cv
  30. L

    The Infinity Theory: Proving X=0 Equation True

    First, i would have put this in the theory development, but i can't. And since this is pure mathmatical, but i do not know what type, I have put it here. Second, you need to keep an open mind and do not call me crazy. Third, the person who came up with imaginary numbers was probally...
  31. O

    Is renormalization in quantum field theory a valid approach?

    I had a dream this morning about a Stargate type device, and i remembered that a singularity is infinity. And i got to thinking that a black hole is supposed to be a singularity. So is a black hole a physics representation of infinity? and what does that mean?
  32. C

    Theory of Infinity: Exploring Boundless Possibilities

    This is the theory I made, saying that every moment is infidential. Lets just say 1 second can be dividied into 100milliseconds, and 100milliseconds can be divided into the next smallest unit and so, this process is infidential... Same with digits. Lets take an interger 1. We can divide it...
  33. C

    Sum to Infinity: Solving the Limit of a Series

    Hello! Here is the problem I am attempting: Sum to infinity: \frac{x}{1*2} + \frac{x^2}{2*3} + \frac{x^3}{3*4} + ... Here is what I get: S = \frac{x^n}{(n)(n+1)} \lim_{n \to \infty} \frac{x^n}{(n)(n+1)}} Now what? Should I do partial fractions to split the equation?
  34. R

    Integration of X^2 exp(-aX^2) from 0 to infinity

    Hey can someone tell me the value of the integral of X^2 exp (-aX^2) dx from zero to infinity. I have the general solution from a table of integrals but since the upper limit is infinity, I can't really plug these numbers in. Can't find it in a table of integrals anywhere? If anyone has the...
  35. S

    Integration by parts when a limit is infinity.

    I'm having a tough time trying to do integration by parts with one of my limits being infinity. My Integral looks like: \int_0^\infty x^z e^{-x} dx with z = \frac{-1}{\pi} Now if I let u = e^{-x} and dv = x^z dx, I will have: du = -e^{-x} dx and v = \frac{1}{z + 1} x^{z + 1} and...
  36. C

    Medical Limits of infinity , The simulaion of human brain

    Human brain is an observer , and the problem of this idea is observing himself the brain will observe an observer observe an observer observe an observer ... to infinity. the real problem appears to people who work in computional simulation of brain ( infinite loop ) can this cycle have any...
  37. I

    Can there be a largest prime within the concept of infinity?

    Is infinity even or odd? If it's even (or both), then it would mean there's a finitely largest prime. By coarsely applying limit concepts, and lim(x->inf)x/2 does not yield a remainder.
  38. B

    Can Infinity Multiply by Zero to Create Something Out of Nothing?

    I am a Civil Engineer so I have some college education in mathematics and Calculus. But I am also an old fart so it has been quite a few years since my head had to use this stuff seriously (my work never required it). I tell you this so that you know I am a partial dummy and not a total dummy...
  39. A

    Find the limit of xsin(pi*x) at infinity

    Since the function f(x)=xsin[(pi)x] oscillates, shouldn't the limit as x -> infinity not exist? i was told that it is positive infinity.
  40. C

    Infinity: Square of Infinity + 2 Infinity

    x + x = 2x infinity + infinity = 2 infinity (or) infinity x-x=0 infinity * infinity = square of infinity (or) infinity :tongue: :devil: :blushing: :approve:
  41. P

    Infinity & Zero: Debate Over Math Definitions

    Hi I was having a debate with some guys, and anyhow they were saying 0 divided by infinity is 0, and I said that it was undefined. Whose correct? Also, they said infinity divided by a number, say 2, is not infinity, that that is undefined, and I said the number is still infinity. I don;t know...
  42. D

    Can infinities have different sizes?

    I've had a discussion with two friends about infinity. I am of the opinion that infinities come in different sizes and some infinities can be bigger than others. My argument goes somewhat like this: Between the numbers 1 and 2 there are infinite real numbers. Between 1 and infinity there are...
  43. D

    What is the meaning of Infinity?

    I have a few questions regarding infinity. Its pretty long so I thought of putting it up on my site instead of here, sorry about that. The URL is: http://deydas.com/blog/archives/4-What-is-infinity.html . Can anybody answer my question please?! Thank you.
  44. V

    Exploring the Meaning of 'Almost Infinity

    well, yesterday i was talking to my friend, and during the conversation he used the term "almost infinity". though he intended to refer to something very large, but he got me thinking. what is "almost infinity"? or let me rephrase it in a beter way. is it correct to use the term "almost...
  45. K

    Prove (lim n -> infinity) (1 + 1/n)^n = e

    Would you please proove that (lim n -> infinity) (1 + 1/n)^n = e ?
  46. T

    Reaching Infinity: Exploring Realization and Creation

    We are all a part of reality. Reality is the state which results from events which we percieve as being real. To be real implies that events can be defined. Events that have the characteristic of being real can be replicated. If we are able to define something, then we can recreate it...
  47. C

    Limit of 1/x as x Approaches 0: Infinity?

    What is the limit of 1/x as x approaches 0. Is it infinity?
  48. D

    Confused About Infinity: Help Understanding a Math Limit

    My book tolds me that: \lim_{n\rightarrow\infty} \frac{n(n+1)}{2n^2}= \frac{1}{2}. I don't get it. Maybe this with infinity, I dunno... Please, help!
  49. T

    Need help with lim sqrt(n)/n as n-> infinity

    Hi, could you help me a bit with this limit? \lim_{n \rightarrow \infty} \sqrt[n]{n!} Sure it should be more than \lim_{n \rightarrow \infty} \sqrt[n]{n} But, when I write it as \lim_{n \rightarrow \infty} \sqrt[n]{n!} = \lim_{n \rightarrow \infty} \sqrt[n]{n} ...
  50. T

    A philosophy problem about infinity

    does it make sense if i define that the shape of a circle is infinity, which i write its equation as: x^2+y^2 = ∞ i know that infinity means very very big until we cannot define how big is it. in this case, is the equation above still valid? can i still say that the shape of this...
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