Infinite = ?

[ronthepon]

I'm asking here because have not found anything on google.

 

Is there a way to use infinite values in maths? Say, like in equations?

 

Is there any branch of maths working on infinity?

 

Is there a number system that defines infinity?

[Qfwfq]

There are two basic notions of infinity: one is how many elements a set has and the other is infinity in limits. The first is in set theory and the second is in calculus. There is also a developing study of so-called hyper-real numbers, this is a quite new thing though and still hasn't been fully settled.

[ronthepon]

Any source of good info that I can get on this?

[Qfwfq]

Look up countable and uncountable cardinality, look up limits in calculus...

[Pyrotex]

Any source of good info that I can get on this?

For a first step, try

http://scidiv.bcc.ctc.edu/Math/infinity.html

 

It contains links to specific information.

 

One of the most important and elusive things to remember is that inifinity is NOT a number. It is meaningless to say there are an infinite number of things (whatever) unless you are dealing with well-defined mathematical entities, such as points. You cannot "count" to infinity; infinity is not a counting number. No measurement can be 'infinite'; no length or duration can be said to 'be infinite' as this would be treating infinity as a number. It is more like a concept, meaning "uncountable", or an abstract description, like the "limit" in calculus.

 

Let me demonstrate the problems with using 'Infinity' in an argument. This example is from Dennett's book, "Darwin's Dangerous Idea", and was just too good to pass up, so here is a condensed version, which I give in the form of a conversation between a Priest and a Physicist. **DISCLAIMER** This is not a "proof" and is not intended to reflect badly on religion. It is merely to demonstrate the use of "infinity" in ANY argument is begging for trouble

 

========

 

Priest: The probability of Life coming into existence out of non-living matter is just astronomical. It has been calculated that the odds of a protein forming out of the so-called Primordial Soup is only one chance out of ten to the hundredth power! That's a one followed by a hundred zeros! It makes much more sense that God created the Earth -- an Infinite God that had no beginning and has no end.

 

Physicist: I will grant you that the probabilities of Life forming out of non-living matter by mere chance is indeed astronomical. But you insist that your God must be Infinite?

 

Priest: Yes. Absolutely. There can be no doubt about that.

 

Physicist: Then grant me a random universe and an Infinite amount of time. Let's take the spontaneous creation of a protein by 'accident'. If the odds against it are ten to the 100th power to one, then over a period of ten to the 100th power years, that accident would be sure to happen. If Human Life with Intelligence has odds of ten to the millionth power to one against it of ever having formed 'accidently', then over a period of ten to the millionth power years, that accident becomes a certainty. Indeed, if you merely grant me the same attribute for a random universe that you insist for your God, namely that it is of Infinite duration, then every event, no matter how rare, no matter how improbable, not only MUST happen, it must happen an Infinite number of times.

 

Priest: Oh my, but I cannot let you get away with that. Certainly the universe cannot be of infinite duration!

 

Physicist: It can be just as easily as God can be of infinite duration. And in my universe, infinite time combined with chaos and random movement produces this Earth with you and me having this same conversation just as easily and with just as much certainty. If God is indeed infinite, then in an infinite universe, He becomes indistinguishable from random chance. They both produce exactly the same results.

[ronthepon]

That makes sense.

 

So infinity is still a concept and cannot be worked out with definiteness in maths.

 

So if infinity did exist, some rules like this would be kinda valid...

 

∞ + 1 = ∞

 

Now that would make holes in maths.

[Pyrotex]

That makes sense.

So infinity is still a concept and cannot be worked out with definiteness in maths.....

Actually, infiniteness IS worked out in math. It's just NOT a number you can use in most equations. If any.

inf + 1 = inf ... is true, but useless

 

infinity doesn't come into its own until you get to Calculus, where you calculate the "limit" of the value of a repeating Sum of algebraic terms, such as 1/2 + 2/5 + 3/14 + 4/41 + 5/122 + ...

expressed as: SUM(n/3n-1) where n goes from 1 to infinity

 

Note, infinity is not used to measure or count, but to express an abstract "limit".

 

In Integral Calculus, you want to know the total area under a curve, let's say

y = e^f(x)

That's "e" (2.781828...) to the power of a function of x

The area under that curve is

Integrate (x goes from 0 to infinity){ e^f(x)*dx }

 

Here again, infinity is a "limit" not a measure or count.

 

Analytical Calculus studies the behavior of certain equations containing "singularities", like where the denominator goes to zero, and the value of the equation "goes to infinity". You never speak of the value "being infinite" -- only that it "goes to infinity (in the limit)"

 

Check out the links I gave you, and look Infinity up in the Wikipedia

[CraigD]

Any source of good info that I can get on this?

As usual, wikipedia is a good starting point – try http://en.wikipedia.org/wiki/Transfinite. As with any hypertext, try following all the links, and see what you can absorb.

 

To get a decent understanding of mainstream mathematical ideas of infinity, which, amounts mostly to the set theory work or Cantor, most folk find they need a decent foundation in set theory. Though its possible to gain this by browsing the internet, many people find a conventional textbook to be more helpful. An actual AP or college class to go along with such a text is, IMHO, best of all, if the opportunity to take one presents itself.

[ronthepon]

I get all that. So infinity is not ever used as a value or a number. It is used as a limit.

[ughaibu]

I think mathematicians also use infinity as a non-value in uncountability, but I'll defer to Qfwfq, et al, on that question.

[Farsight]

Infinity is just a concept. Some things are very very large. Some things are very very numerous, but nothing is infinite. And it's not something you can use in mathematics, because like you said

 

∞ + 1 = ∞

 

and subtracting ∞ from both sides means your mathematics falls apart with

 

1 = 0

[Qfwfq]

Here comes...

 

An infinite cardinality may be countable or uncountable. Two sets have the same cardinaltiy be definition if there is any way of mapping their elements one-to-one and onto, which means a distinct element of one set for each one of the other, and vice versa. Briefly, one may call such a map a bijection between the two sets. To state that a set has an infinite cardinality is equivalent to stating that there exists a bijection between it and some proper subset.

 

Any set A which can be placed in bijection with that of the natural numbers is said to be countable, i. e. you can exhaustively associate each element of A with a counting number. As Pyro implies, no element of A will be "the infinite-th one", this makes no sense. The cardinality of the naturals is also called aleph_0. It is quite trivial to show that the set of integers has the same cardinality, just associate negative integers with odd naturals according to n <--> -2z + 1 and the others to even naturals according to n <--> 2z.

 

Cantor's diagonal argument proves that there are more real numbers than natural ones, even for just the real numbers in a finite interval. The cardinality of the reals is also called the "cardinality of the continuum" and is equal to n raised to aleph_0, for some n > 1. Try answering these questions Ron, before asking or looking up the answers:

 

Are there more rational numbers than naturals (or integers)?

 

Are there as many rationals as reals?

 

Is the cardinality of the rationals closer to aleph_0 or to that of the continuum?

 

subtracting ∞ from both sides means your mathematics falls apart with

 

1 = 0

Actually, it only means that ∞ - ∞ is indeterminate, by virtue of the fact that

∞ + a = ∞ for any a.