Infinity Definition and 42 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. D

    B Why 1 / ∞ = 0 but ∞ * 0 is not equal to 1?

    As we know those relations are true: if a/b = c, then a = b*c and b = a/c Therefore if 1/ ∞ = 0, ∞ * 0 should be equal to 1 and 1/0 = ∞
  2. T

    B Difficulty with understanding whether 1/n converges or diverges

    Hi, I have a quick question about whether or not the infinite series of 1/n converges or diverges. My textbook tells me that it diverges, but my textbook also says that by the nth term test if we take the limit from n to infinity of a series, if the limit value is not equal to zero the series...
  3. D

    B My argument why Hilbert's Hotel is not a veridical Paradox

    Hello there, I had another similar post, where asking for proof for Hilbert’s Hotel. After rethinking this topic, I want to show you a new example. It tries to show why that the sentence, every guest moves into the next room, hides the fact, that we don’t understand what will happen in this...
  4. D

    B Hilbert's Hotel: new Guest arrives (Infinite number of Guests)

    Hilberts Hotel has infinity numbers of rooms and in every room is exactly one guest. On Wikipedia Hilberts Hotel gets described as well: Suppose a new guest arrives and wishes to be accommodated in the hotel. We can (simultaneously) move the guest currently in room 1 to room 2, the guest...
  5. J

    Find the range of y in a DE

    From integration by parts, and using y(10) = 0, I get the equation ##2e^{3t-30} = \frac{|y-2|}{|y+1|}.## Let ##f(t) = 2e^{3t-30}##. Since it's for t>10, f(10) = 2, and we have ##2=\frac{|y-2|}{|y+1|}##. Depending on the sign I choose to use, I get either that y=-4 or y =0. Since ##t: 10...
  6. P

    I Completed infinity

    If actual infinity represents a completed set of infinite data points, wouldn't that be a contradiction of terms?
  7. 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...
  8. 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...
  9. 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...
  10. 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?
  11. 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.
  12. 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.
  13. Mr Indeterminate

    I Where has this proof gone wrong? ∞= 1/0

    Now I expect that most of you on this forum would be familiar with the equality between point nine reoccurring and one: 0.999...=1 If your not familiar please review Now this equality can be used to imply something else, which is rather heterodox...
  14. L

    I Infinite versus finite space

    I have read some of the other posts about this topic but am still left unsatisfied. Could just be me. :cool: Did the universe, one minute after the big bang, consist of a finite volume of spacetime? If so, then is it not logically inconsistent that the universe can possibly be infinite now...
  15. Snen

    Lim x->0+ (x^cos(1/x))

    Homework Statement Homework Equations The Attempt at a Solution let y = lim x->0+ x^cos(1/x) lny = cos(1/x)*lnx = (x*cos(1/x)) * (lnx/x) x*cos(1/x) = 0 (sandwich theorem) lnx/x = 0 (l'hopital) so lny = 0 and y = 1 Is this correct?
  16. franktherabbit

    Find the limit of the expression

    Homework Statement $$\lim_{x\to\infty} \left(\frac{n^2+2n+1}{n^2+2n+3}\right)^{\frac{2n^2}{n+1}}$$ Homework Equations 3. The Attempt at a Solution [/B] I tried ##\lim_{x\to\infty} \left(\frac{n^2+2n+3-2}{n^2+2n+3}\right)^{\frac{2n^2}{n+1}}=## ##\lim_{x\to\infty}...
  17. F

    I Where do we use infinity in physics, and why?

    Hello! I'm doing a school project, where I am writing about Infinity in Math and Physics. I've got the math part settled, but it's the physics part that has begun to bother me. One part of my task is to write about some of the mathematical expressions in physics, where we use infinity - but the...
  18. A

    B What are surreal numbers?

    Hey guys! I have heard of this concept in various places and sort of understands what it attempts to do. Can anybody please explain it to me in more detail like how it works, how to notate it, and how to expand it to infinities and infinitesimals. Thanks in advance! Aakash Lakshmanan
  19. Radu Mitroi

    B What percentage of the universe do we know

    Hello. First of all, I must say that I'm new to this forum, so I apologize if I'm posting in the wrong section. I'm a 17 year old with not that much knowledge about physics, so if what I'm talking about makes no sense or is completely stupid, just let me know. A couple of days ago I asked...
  20. A

    Inifinity limit with natural log

    Homework Statement Limx--> ∞ Ln(x^2-1) -Ln(2x^2+3) Homework Equations The Attempt at a Solution Ln(x^2-1)/(2x^2+3) Then I divided the top and bottom by x^2 so in the end I got (1/2). Is this right?
  21. A

    Limit at infinity with radicals

    Homework Statement lim as x tends to -∞ (x)^3/5 - (x)^1/5 Homework Equations The Attempt at a Solution The first thing I did was convert it into a radical so it becomes fifthroot√x^3 - fifthroot√x. Then I rationalized to get ( x^3-x)/(fifthrt√x^3+fifthroot√x) . I then divided the top by...
  22. D

    B Can the space (or else measurable) be actually infinite?

    The (most popular) flat model of Universe is space-infinite. How the infinity is measured? Can you give me references to the papers about the actual infinity of space?
  23. P

    I When observable Universe was the size of a baseball was its gravitational influence bigger?

    When the observable universe was the size of a baseball, did its gravity (field?) extend to (as opposed to towards) infinity?
  24. D

    B Ratio of circumference to diameter for infinitely large circ

    If you divided the circumference of an infinitely large circle by its diameter, would the result be pi?
  25. P

    B Why do we use infinity in Physics?

    When we talk about a particular problem in Physics. For instance, let's say that light is coming from somewhere to hit the earth. We often say that the light is coming from "infinity." Let's say that we're tackling a black hole and we have a person somewhere as an example and we say that let's...
  26. Rectifier

    Infinity question - limits

    Is ## "\frac{0}{\infty}"=0 ## ?
  27. newrd

    B Is the Universe finite?

    The universe- from our understanding, is expanding, thus the regions (for lack of a better word) particles have not yet reached do not exist. How far our universe can/ will expand is unknown, it may be infinite, but we can conclude at this time, as it is still expanding, that it is finite. True...
  28. Bran

    B Expanding from and eventually to a singularity?

    I know this thread, about why the Universe can't expand inward, is fairly old; but I stumbled across it today and there was something mentioned here that sparked a question I feel like people here would be qualified to answer. What was mentioned, was that a singularity is a point at which our...
  29. N

    A Infinity in the real world?

    To what extent is the term infinity used in the physical world. When talking in terms of mathematics we can have a set of all natural numbers called an infinity, then we can have a value that comes after this set of infinity (lets call it 'a'). After 'a' comes 'a+1' then after this set of...
  30. Dopplershift

    I Determining the Rate at Which Functions approach Infinity

    With basic fractions, the limits of 1/x as x approaches infinity or zero is easily determine: For example, \begin{equation} \lim_{x\to\infty} \frac{1}{x} = 0 \end{equation} \begin{equation} \lim_{x\to 0} \frac{1}{x} = \infty \end{equation} But, we with a operation like ##\frac{f(x)}{g(x)}##...
  31. L

    I can't seem to find this limit

    Homework Statement Homework Equations The Attempt at a Solution I tried using the rule of multiplying with the "conjugate", for example what's above multiplied by (√n^3+3n)+(√n^3+2n^2+3)/(√n^3+3n)+(√n^3+2n^2+3). But I'm left with a huge mess :( I also tried dividing the top and the bottom...
  32. F

    What is the electric field due to hollow sphere at R=z?

    So I derived the E-field of a hollow sphere with a surface charge σ at z and I got: E(r)=\hat{z}\frac{\sigma R^2}{2\varepsilon _{0}z^2}\left ( \frac{R+z}{\left | R+z \right |}-\frac{R-z}{\left | R-z \right |} \right ) at z>R, the equation becomes: E(r)=\hat{z}\frac{\sigma R^2}{\varepsilon...
  33. C

    How do you transition from finite to infinite?

    So, according to, "In the centre of a black hole is a gravitational singularity, a one-dimensional point which contains infinite mass in an infinitely small space, where gravity become (sic) infinite and space-time curves infinitely, and where the laws of physics as we...
  34. Hamza Abbasi

    How to define infinity?

    Infinity is not a real number right? Then where do infinity stand (complex no?) . Why infinity is not a real number , I thought of it as a very very big real number! Ignore my poor communication skills.
  35. J

    1+2+3+4+5+6+7+...... =-1/12?

    I was some youtube videos and i got sucked into this channel called "numberphile". They were talking about infinite sets. In particular the set that is the sum of all natural numbers. Through some creative algebra they demonstrate the proof. Somehow the set that is equal to the sum of all...
  36. zachx

    1+2+3+4+....=-1/12 weirdness

    I've recently accepted (reluctantly) that 1+2+3+4+...=-1/12, but wouldn't that mean the limit as x→∞ of f(x)=x not be ∞ but some maybe negative number because if it were ∞ then 1+2+3+4+... should also be ∞. Also, do numbers get so positive they become negative lol.
  37. Stephanus

    Infinity compared to Infinity

    [Mentor's Note: Thread moved to Astrophysics since it concerns black hole singularities] Dear PF Forum, I am just wondering about this. See if anyone can help me. Is \frac{2}{0} is twice as much as \frac{1}{0}? Is the above question wrong? Is the number of points in 2 cm lines twice as much...
  38. kostoglotov

    Three point charges with a central fourth charge

    Homework Statement It is the last part, part (c) that I'm having trouble with, but I'll post the entire question for clarity. Three charges (q) form the vertices of an equilateral triangle. A fourth charge Q ( Q = -q ) is placed at the center of the triangle. (a) will the charges at the...
  39. Tungamirai

    What happens if you flip an infinite heads and tails coin?

    What happens if you flip a coin with infinite heads and infinite tails? I am not sure if this is the right place to post this question, or if my question even makes sense! just thought about it after reading about the simulation argument
  40. zerospin

    Could black holes have no singularity?

    I have a problem with the concept of a singularity, defined as something that has a property which is infinite. Infinities do not belong in our reality, and in my opinion are just hints that our understanding of the phenomenon is incomplete or wrong. From my understanding, during the collapse...
  41. camilus

    Anything actually physically infinite within the universe?

    Hi guys, I'm a mathematician from Miami Florida working in paraquaternionic and symplectic differential geometry, but I come from a very extensive physics background, pretty much well-versed in all modern physics. But my favorite of all is probably the philosophy of mathematics and science as...
  42. G

    Limit problem

    Homework Statement [Broken] In this image, I have no idea what the math behind this problem is. What exactly is happening here? Homework Equations The Attempt at a Solution I multiplied n/10 in order to get 10000n^9 as a denominator but the ln(2)...