Georg Cantor defined an infinite set as a set that could be mapped one-to-one onto a proper part of itself. For example the set of all postive integers can be mapped onto the set of all even numbers like this:

1 => 2
2 => 4
3 => 6
etc.

There are many "levels" of infinity. For example, the set of all real numbers cannot be mapped one-to-one onto the set of all integers. There are just too many of them, as can be shown by a diagonal argument.

There are not an infinite number of planets or stars in the universe. Scientists even debate on how large the universe is. There are theories about the Universe folding back in on itself meaning that it cannot go on forever. Forever and infinity are only conceptual as only conceptual things like numbers and what not may go on "forever" (pi) and even that is not absolute. Another interesting application of man trying to classify what it does not understand. Because the universe is larger than the scope of our knowledge does not make it infinite.

A brand of speakers, they make some midrange consumer parts as well as very highend speakers (such as the Overtures), expensive, but well worth the money. Owned by Harman International. They're not worth buying unless your other gear can back them up with quality input.

Well, I'd have figured my definition would be in here already, but it's not, so I'll take a stab at this.

The Value

The best definition for infinity that I know of is

             n
inf = lim   ___ 
      x->0   
            |x|

where n is any positive real number, and |x| approaches downward to zero (as noted by Professor Pi). I prefer to actually set n to 1 when I define infinity, because certain math problems arise where the limits of two functions f(x) and g(x) both approach infinity, but for any x g(x) is not the same value as f(x). For example, g(x) might be defined as twice f(x). In these cases, the limit of f(x) would be infinity, and the limit of g(x) would be 2 · infinity. To me, defining infinity as the limit of 1 divided by x as x approaches zero is easiest on my brain as it clears any ambiguity. I just use the above definition with n=1 and I know how large any infinite value is relative to any other. When you factor in the fact that you can have negative infinite values, well, I think it just makes sense.

Personally, I do think this definition makes the most sense because it can rearranged as a definition of zero:

                   n
zero =  lim       ___
       x->∞    
                   x

In my mind, n in this case is not so important since any number times zero is zero, but still, I prefer n to be one (at least, when dealing w/ limits). So there you have it. Well, my take on it at least.

The Symbol

The proper name for the symbol is actually infinity and it looks like an 8 rotated 90°. Supposedly, some browsers (supporting HTML 4.0) should be able to display infinity as ∞ or ∞ (but mine doesn't). In case yours does, the HTML for it is (or should be) ∞ or ∞ To use the former, your browser must support HTML 4. Ability to use the latter of course depends upon your system being able to display the ISO 10646 character set. My browser/OS combination doesn't allow either, but I've included it just in case yours does.

As for the name of the symbol, it is simply "infinity." It has no other fancy name, and in fact has no "official" name. Some do refer to it as lazy eight, and some call it lemniscate. As far as lemniscate goes, it is actually the name of a mathematical system that just looks like the symbol for infinity. The symbol itself was introduced by mathmetician John Wallis in 1655 when he wrote "De sectionibus conicis." The symbol was supposedly borrowed from a Roman symbol for 1 000 and was declared to represent infinity.

Now Txikwa tells me (in addition to the correct spelling of Wallis' treatise on conic sections) "Probably the thousand symbol mentioned is not ordinary M but an old alternative way of writing it: CID where the D is a backwards C. (And half of that gave normal D for 500.)" Sounds reasonable to me.


References
Ken "Dr." Math. Ask Dr. Math - Infinity Symbol. The Math Forum. <http://forum.swarthmore.edu/dr.math/problems/lazy8.html>
W3C. Character entity references in HTML 4. <http://www.w3.org/TR/REC-html40/sgml/entities.html>

Infinity isn't a number, and its not a thing. It is more appropriately, an idea. Infinity stands for the "never endingness" aspect of some things. We know that numbers never end, but there is no number labeled as "infinity". Rather, we use the word infinity to describe the nature of numbers; they never end. The same is also true for anything that is "infinite" by nature.

The name of an excellent book on mathematical philosophy, copyright 1953 by Lillian R. Lieber, PhD. The book is 359 pages long, and was published by Rinehart & Company, Inc., of New York. Its Library of Congress catalog card number is 53-5355.

Infinity is written in prose, but with line breaks that make it look like free verse. The author explains at the beginning that it is not meant to be read as verse, but some sections are so eloquent that it is difficult not to. The book also features funny, abstract, modernistic illustrations by Dr. Lieber's husband, Hugh G. Lieber, in addition to the requisite mathematical illustrations.

The book has the easiest to read explanations and readable proofs of complex mathematical concepts I've yet found. What Infinity lacks in formulae and strict proof, it makes up in clear, intuitive explanations, history, and references. It also serves as a treatise on the nature of science (observation), and its relationship with art (intuition), and pure mathematics (reason). Some of the topics it covers are: Potential Infinity; Hyperbolic, Elliptic, and Parabolic (Euclidean) Geometries; Cantor's proof of enumerability and non-enumerability of sets; Dedekind Cuts; the Differential and Integral Calculus; etc. You get the idea, it's mostly all to do with Cantor's set theory.

Here is the first part of chapter 14 (with some abridgments), to give you an idea of what the book is like:

So far then you realize that
man's yearning for
the infinite
has not been fulfilled in the
physical world.
Even the
entire physical universe
is not infinite,
so far as we know.
Even the total number of electrons
in the entire physical universe
is not infinite
The old-fashioned idea that
the earth was flat
and extends to infinity
turned out to be false,
for as we now know
the earth is a sphere
and thus is not infinite,
or, as we say,
it is unbounded but finite.
And similarly
our three-dimensional universe
has also turned out to be
unbounded but finite.
Thus, wherever we look in the
physical universe
we have not found infinity.

Infinite-Monkey Theorem = I = inflate

infinity n.

1. The largest value that can be represented in a particular type of variable (register, memory location, data type, whatever). 2. `minus infinity': The smallest such value, not necessarily or even usually the simple negation of plus infinity. In N-bit twos-complement arithmetic, infinity is 2^(N-1) - 1 but minus infinity is - (2^(N-1)), not -(2^(N-1) - 1). Note also that this is different from time T equals minus infinity, which is closer to a mathematician's usage of infinity.

--The Jargon File version 4.3.1, ed. ESR, autonoded by rescdsk.

Two men, identical twins, died at the same time and went to the stereotypical Christian Heaven. There, St. Peter explained that due to a clerical error, they could not be told apart. What was worse, only one of them was to be allowed in! The other would have to burn in Hell. Peter said, "however, we've come up with a solution that we think is equitable. You," he said, pointing to one of the twins, "will go to Hell for eternity. But every February 29th, you will change places with your brother for the day." The man was indignant. He cried "that's terrible! You mean that I might deserve to go to heaven, but my brother will go there and I'll only get out one day every four years?" St. Peter calmly responded, "that's correct, but in the long run you'll spend the same amount of time in both places.
This story serves to illustrate the difficulty arising from the equivalence of different sets that have the same number of elements as aleph-null, or the number of integers. Also, generally, the absurdity of instantiating an infinity of any empirical quantity such as time.

note: I did not invent this joke, but I don't know its origin
The universe cant be infinite, if it was we would have an infinite amount of stars and an infinite amount of galaxies and basically infinite everything including infinite size for the universe.

With a simple calculation we would have an infinite amount of star light coming down to us on earth and this will imply infinite amount of light. Tell me, is there a fixed amount of star light at night time or an infinite candela, I guess candela is the correct term to be used for light power if I remember my physics courses correctly. Or was it Lumina? In any case I can't imagine the universe is infinite. All I can imagine is that its big, Really big.

Consider this: something with a beginning I.E: Big Bang will definitely have a size and this size will continue to grow, now will this growth shrink back at one point or will it continue to grow? I vote for a Big Crunch.

Something with out a beginning I.E: God is infinite. How many objects can you name that has no beginning will we be able to call infinite? Even all the matter in the universe can be quantified even all the photons.

So, the huge gaseous material that started the Big Bang is not infinite, I don't even think that black holes have infinite gravity or infinite matter. People back in the days used to think that the number of sand on earth was infinite!!! Lets not make the same mistake they did!.

Interestingly, the Qur'an talks about the big bang and the big crunch in these two statements. The vapor word used in Arabic is "dukhan" which means hot smoke, a perfect analogy for the hot gaseous material that started the big bang.

41.11: Then He directed Himself to the heaven and it is a vapor, so He said to it and to the earth: Come both, willingly or unwillingly. They both said: We come willingly.

21.104: The day when We shall roll back the skies, like a scribe rolls up a written document. As We began the first creation, We shall surely repeat it. It is a promise (binding) upon us. Indeed it will happen.

The most naive approach to infinity is treating it like a number and using it in calculations like any other number. This lies at the heart of many a false mathematical proof and is, generally speaking, dead wrong.

However, it is also not quite right to say that infinity is not a number, because it is, it's just a transfinite one, and those behave differently than your normal everyday numbers. But there are certain operations that can be performed on them, and there is more than one.

To get to the point, there are two commonly encountered infinities:

  • countable infinity, i.e. the number of all natural numbers. This is, strangely enough, the same as the number of all integers and even the same as the number of all fractions, because it is possible to find bijective functions that map between these sets. So even though it intuitively seems like there are twice as many integers as natural numbers, and infinitely as many fractions, the numbers are really the same, and you see why infinities have to be treated specially.
  • uncountable infinity is, however, different: there are actually more real numbers than there are fractions or integers, because it is impossible to create a bijective mapping between them - the diagonal argument proves this
There are other, even larger cardinalities, but you are unlikely to ever encounter them unless you choose to do so. However, the two above (and the difference between them), come up quite frequently in many fields of mathematics.
There are two misconceptions that I have been noticing on infinity in astronomy. The first is that the universe cannot be infinite and the second is that it will definitely collapse back upon itself.

The universe can be infinite without there being an infinite amount of light and thus energy. The problem of the universe being infinite with a limited amount of energy visible is known as Olbers' Paradox.

The problem is solved when you consider two things together: a) even though the universe may be infinite its age is taken to be finite because the universe is expanding and b) the speed of light is finite at 299,792,458 m/s. This means that light can only be received from a distance in light-years equal to the age of the universe (around 10 to 15 billion years depending on the value used for the Hubbell constant which at this time is uncertain). Therefore the universe may very well be infinite, but since we cannot see beyond somewhere between 10 and 15 light-years away we cannot know.


The universe will not necessarily collapse back onto itself in a big crunch. The fate of the universe fits into three categories (assuming a cosmological constant of zero): expansion approaches zero, the universe collapses, or it expands forever. This depends on the amount of mass in the universe which is very uncertain, but the best guesses at this point put the universe's mass is under the critical value needed for a big crunch. As a result the best estimate is that it will expand forever becoming nothing more than infrared radiation.

Developing a mathematical sense of infinity can be fraught with peril and contradiction - many false mathematical proofs depends on an obscured assertion about infinity (or its close relation, zero) that allows nonsense to follow. Attempts to juggle notions such as division by zero, or to make sense of '0/0' tend to crash and burn horribly too. Renditions such as 'a number larger than any other' or 'the number of times zero has to be added to itself to get something non-zero' turn out to make as much sense as 'a square circle'. A passing familiarity with calculus, limits and infinitesimals often confuses things further, and of course people feel that, intuitively, there is something that should be called infinity. So rather than just asserting that things like '1/0=infinity' simply don't work, I'd like to give an overview of why they're misguided approaches. To do so, it seems necessary to give some insight into what it is mathematicians do with numbers (when we bother with them at all), and to prevent anyone feeling cheated out of something valuable, give some examples of where a notion of infinity can be usefully bolted to a mathematical framework.

Mathematical Playgrounds

My first observation is that the problem doesn't really arise from a problem with infinity but rather with the casual usage of mathematics, outside of mathematical disciplines. Whilst numeracy is a vital day-to-day skill, high level mathematics isn't, and people can usually get by with various rules of thumb for arithmetic. Unfortunately, rules like 'anything divided by itself is 1' or 'if a/b = c, then a = bc' should really be qualified by various regulations and exceptions that, when omitted, cause problems whenever a zero crops up.

In fact, even claiming the right to use division can be a step too far, mathematically. Often, mathematicians find it helpful to tackle problems not as they stand, but in terms of a more general framework. They choose a playground (somewhere to do the maths) and some toys (something to do maths with) then see what they can create as a result. Sometimes it can be even more general than that- a set of toys to use in any playground. For instance, the theory of metric spaces equips a space X (and we don't care what - it could be anything from the set containing 0 and 1 to the entire complex plane, or not numerical at all- nodes on E2, or Elephants in Africa) with a rule known as the metric, which, given two things from X, returns a number meant to indicate the 'distance' between them. All that we ask is that this metric be well-behaved in certain ways- an object is no distance from itself, x is as far from y as y is from x, and so on.

Why bother? Well, if we can prove something in this seemingly sparse environment, then it's true for all examples of metric spaces we care to think of. For example, both the real numbers and the complex numbers can be thought of as metric spaces in a fairly natural way. But there are many things that are true of the complex plane that do not hold for the reals, and vice versa. So a theorem you prove as a real analyst is worthless to complex analysts unless they can prove it too. If you leaned on some of the special properties of the reals, this might not even be possible. If, however, your argument was couched entirely in terms of metric space properties, then it readily transfers over to the complex numbers too. This abstraction also helps to throw into relief the fundamental differences and similarities between different mathematical environments. Some of the most powerful mathematics arises when links are found between seemingly disconnected topics that allow difficult problems in one to be tackled as easy problems in the other.

A playground for arithmetic

Having hopefully convinced you of the power of abstraction, it's time to look at an environment where we do basic arithmetic on numbers - the kind of place where it's tempting to throw infinity into the mix. Whilst my previous example (metric spaces) was from analysis (it's actually a special case of the even more abstract study of topology, which goes even further by ditching the notion of a metric and works in set-theoretic terms), the natural place for arithmetic is Algebra. So here's an algebraic structure.

Groups (roughly)

We can turn a set G into a group by pairing it with a binary operation * - that is, one which takes pairs (hence binary) of objects from G and returns another object from G. We also demand that:

  • * is associative, meaning (a*b)*c=a*(b*c)
  • * has an identity e in G, meaning applying * to any object and e just gives you the object back
  • any object in G has an inverse, such that applying * to a and the inverse of a gives us, magically, that identity again.

Which probably sounds like so much gibberish, so an example is in order.

The Integers, Z, are the 'whole numbers', including 0 and all the negative numbers. If we use addition as our operation *, and write a+b to denote the action of adding a to b (that is, more formally, applying the operation * to the pair a,b), then we can see that a group has emerged- 0 is the identity, since 0+a=a; and -a is the inverse of a, since a+-a is 0, which we just confirmed was the identity.

The notation * suggests we could try multiplication instead of addition and get a group, but actually we'll come unstuck. An identity is fine, since 1*a=a for any a we might chose from Z (this stipulation is important!); but you can't find a value in Z that serves as an inverse to (for instance) 2- we all know that 2*1/2 is 1 as desired, but 1/2 isn't in our playground Z.

In fact, we don't even have a sense of division yet. The sensible approach is to take a/b to mean a*b-1, where b-1 is the multiplicative inverse of b, if such a thing exists. In Z, you can only divide in this way by 1 or -1 if you intend to stay in Z. We can expand our attention to Q, the set of rational numbers, or even R, the real numbers, and by having more playthings we find more multiplicative inverses. But there will always be one element that refuses to play ball - you can't find a real number which multiplies by 0 to give 1. Here's why.

Arithmetic axioms of the Reals

In addition to these 9 rules, there are three order axioms and a completeness axiom which are necessary to precisely determine the reals. The following list is also satisfied by, for instance, the rationals, but I don't want anyone to feel I'm pulling a fast one by working with a restricted set.

  1. a+(b+c)=(a+b)+c for all a,b,c in R (+ is associative)
  2. a+b=b+a for all a,b in R (+ commutes)
  3. 0+a=a+0=a for all a in R (0 is the additive identity for the group consisting of R with +)
  4. Given any a in R, there is (-a) in R such that a + (-a) = 0. (Existence of additive inverses; this really is a group under +)
  5. a(bc)=ab(c) for all a,b,c in R (multiplication is associative; we suppress the * or . for ease of notation so ab means a times b)
  6. ab=ba for all a,b in R
  7. 1a=a1=a for all a in R
  8. Given any a in R other than zero, there is a-1 in R such that aa-1=1
  9. a(b+c)=ab+ac for all a,b,c in R (multiplication distributes over addition)

Note that I'm not establishing a self-fulfilling prophecy by asserting axiom 8 above. By saying that every non-zero real number has an inverse, I'm not ruling out the possibility of a 0-1, just the necessity of one - that is, wonderful if you can find one, but things won't break without it. Sadly, it turns out that the combined weight of these axioms removes the possibility of an inverse for 0, as I claimed:

Theorem: a0 = 0 for all a in R
0+0 = 0 by axiom 3.
So a(0+0) = a0, multiplying by a.
Hence a0+a0 =a0 by axiom 9.
If you believe in subtraction, you can now conclude a0=0. But we never asserted anything about subtraction, so for the sake of rigor: (a0+a0) + (-a0) = (a0)+(-a0), adding (-a0) to each side.
Then (a0+a0)+(-a0) = 0 by axiom 4.
So (a0)+ (a0+(-a0))= 0 by axiom 1.
So a0 + 0 =0 by axiom 4
Hence, after copious axiom chasing, a0 = 0 by axiom 3.
Corollary: 0 has no multiplicative inverse.
Suppose a0 = 1.
By Theorem, a0 = 0.
So, 0=1.
But 0 is not 1 (this is axiomatic for a field (which the reals turn out to be) or just bleedingly obvious if you're not a mathematician.

So we find ourselves banned from writing 1/0 at all. Nor can we write 1/∞, because we don't even have an ∞ in our playground in the first place, to even contemplate finding an inverse of. So that's why these notions are meaningless in every day 'real' mathematics.

Building a better playground?

But still the question arises - why can't we just define 1/0 as ∞? (or equivalently 1/∞ as 0) Well, if you really wanted to, you could - posit a set with a nifty name (such as the extended reals, or hyper reals) that consists of the real numbers and a new symbol, ∞, and give yourself an inverse for 0. Great stuff. But if you've got a bigger playground, it better be at least as fun as the old one, right?

Theorem: if ∞ is a real number, all real numbers are the same
Let a and b be real numbers. Then a0=0=b0 by the previous theorem.
So a0∞=0∞=b0∞
But 0∞=1. So a1=1=b1.
But a1=a and b1=b. So a=b=1.
Corollary: this new creation is a rubbish playground.

This brings us to the central objection. You can try anything you like with mathematics. Sometimes it will be useful and sometimes it won't be. The axioms of the reals are useful, but so is the ability to tell them apart. By tacking on ∞ as just another real number, one either has to abandon the axioms (lest the above result hold) or lose useful properties (since the above will then hold); whereas adding it as something other than a number obviously complicates things (working out how to weave it into the existing rules for calculating things), and still feels like dodging the question. Ultimately, it's rarely helpful to adopt either approach.

Some useful uses

Most mathematical uses of infinity don't involve treating it as just another number (be it one 'bigger than all others' or 'the inverse of 0'). Rather, infinity is often encountered as infinite - an adjective for describing mathematical objects, rather than being a mathematical object in its own right. In particular, an entire hierachy of infinities known as cardinality is required to describe the size of sets such as the natural numbers or the real numbers (neither is finite, but the reals are in a sense so much more infinite than the naturals). The following environments, however, roll up their sleeves and make an attempt at using infinity as an object.

The degree of a polynomial

A polynomial in a variable x is a sum of powers of x (with non-negative integers as exponents), such as x2-1 or x3 + 3x -7. Broadly speaking, the degree of the polynomial is the highest power of x which appears- in the examples, 2 and 3 respectively. But numbers themselves are polynomials - '5' is the polynomial 5x0, a degree 0 polynomial. Given two polynomials and their degrees, it'd be nice to know something about the degree of their sum, or product, or difference... and but for a small wrinkle, this is possible.

At first glance, it seems that deg(f+g) or deg(f-g) ≤ max{deg(f), deg(g)} and deg(fg)=deg(f)+deg(g). But this breaks when you consider the zero polynomial (the constant 0) since 0*f=0 for any polynomial f. We get around this by defining deg(0) to be -∞, where -∞ is a symbol with the properties that

  • -∞ < z for any integer z
  • -∞ + -∞ = -∞
  • -∞ + x = x + -∞ = -∞ for any integer x.

Using this trick, one can make assertions about the degree of a polynomial with confidence that the zero polynomial, an often useful tool, can't break things.

Tending to Infinity

In analysis (especially, calculus), the notion of a limit is vital - sequences of numbers can tend to a limit, or a function may have limiting behaviour as it's arguments grow (or shrink). But a sequence needn't get arbitrarily close to a particular value, nor need a function be all that well behaved.

We say that a sequence tn tends to infinity if, for any real number P, there is an index N such that for any term tn after tN, tn is greater than P. That is to say, any attempt to put an upper bound on the sequence fails. Note that it is the limit which is described as being ∞ - any given term, no matter how large, is still an actual number.

Projective Geometry

Algebraic geometry is (in my experience) simultaneously fascinating and very difficult, probably for the same reason- you routinely bump into the idea of 'stuff at infinity'. For instance, in describing the straight lines through the origin in 3-dimensional space, you can almost always characterise the line with just two parameters, rather than 3. This is accomplished by fixing a plane (for instance, Z=1) then noting the x,y intercept of the line with that plane. However, lines parallel to the plane will never meet it, so there is no intercept - however, by knowing this parallel condition you can describe it as a line in the 2-dimensional plane Z=0, again with only two parameters. This illustrates the result that N-dimensional projective space can be thought of as, mostly, N-dimensional affine space, in disjoint union with the 'stuff at infinity' - a projective space of dimension N-1. This region can also be thought of as 'where parallel lines meet', and makes possible the study of, for instance, elliptic curves, in a consistent way.


So, I hope that clarifies a few things, and you feel the gains are worth abandoning the 'obvious but wrong' approach. I've tried to make this as accessible to a non-mathematician as possible (hopefully without being too patronising), but after a few years at University it's hard to gauge just what level that should be. Any feedback much appreciated.

In*fin"i*ty (?), n.; pl. Infinities (#). [L. infinitas; pref. in- not + finis boundary, limit, end: cf. F. infinit'e. See Finite.]

1.

Unlimited extent of time, space, or quantity; eternity; boundlessness; immensity.

Sir T. More.

There can not be more infinities than one; for one of them would limit the other. Sir W. Raleigh.
<-- now known to be false! -- See aleph null, etc.-->

2.

Unlimited capacity, energy, excellence, or knowledge; as, the infinity of God and his perfections.

Hooker.

3.

Endless or indefinite number; great multitude; as an infinity of beauties.

Broome.

4. Math.

A quantity greater than any assignable quantity of the same kind.

⇒ Mathematically considered, infinity is always a limit of a variable quantity, resulting from a particular supposition made upon the varying element which enters it.

Davies & Peck (Math. Dict. ).

5. Geom.

That part of a line, or of a plane, or of space, which is infinitely distant. In modern geometry, parallel lines or planes are sometimes treated as lines or planes meeting at infinity.

Circle at infinity, an imaginary circle at infinity, through which, in geometry of three dimensions, every sphere is imagined to pass. -- Circular points at infinity. See under Circular.

 

© Webster 1913.

Log in or registerto write something here or to contact authors.