Theory of Infinity?

Question

Ok, so I have this idea about infinity and its relationship with 0:

From hereon in, infinity will be referred to as INF.
From hereon in, any possible number will be referred to as x

x / INF = 0
x / 0 = INF
Therefore 0 * INF = x

0/INF = 0 * 1 / INF = 0 * 0 = 0
INF/0 = INF * 1 / 0 = INF * INF = INF

Also:
INF * x = INF
INF / x = INF
INF + x = INF
INF - x = INF

Just as:
0 * x = 0
0 / x = 0
0 + x = x
0 - x = -x

Also note that
x / INF = 0 * x
x / 0 = INF * x

Now, because x can be both positive and negative, let's say x is negative.

-x / INF = 0 (just as x / INF = 0)
-x / INF = x / -INF
Therefore, INF = -INF

It's a little cryptic, but I think you guys can get the gist...
What do you think?

Oh, I made a little mistake: x is any number BESIDES 0 and infinity.

And I've defined Infinity as the following:

INF = x + x + x + x +... (goes on forever)
INF = x * x * x * x *...

Answer 1

There actually is a formal way of dealing with and at times computing with infinity, if you are interested. There are in fact infinitely many different infinities. The one you are talking about is called "countably infinite" and is the number of natural numbers 1, 2, 3, ..., and is often represented as "Aleph null" (a fancy N with a 0 subscript), since I do not know how to make an aleph in Yahoo!, I will represent it as N_0.

Marvin Gardner has a delightful treatment of N_0 is a book, which I believe is called Aha! Insight!, which uses a hotel with N_0 rooms to explain why N_0 + n = N_0 and n*N_0 = N_0. Basically he says, suppose you have a hotel with N_0 rooms, but it is full. That is all rooms are occupied, so there are N_0 people. Suppose n people arrive, then to accomadate them he ask each current resident to more from their current room i to room i+n. Then he can put the new arrivals in rooms 1...n. Now suppose N_0 people arrive. The manager then asks the current occupants to move from room i to room 2i, then he can put the N_0 people in the odd numbered rooms (person i goes in room 2i-1).

Note that N_0 is also the number of rational numbers, and the number of algebraic numbers (numbers that are roots of polynomials with integer coefficients and finite degree). However, it can be shown that for any number n (and this includes infinities), that n < 2^n. So we can say that N_0 < 2^N_0 < 2^(2^N_0) < ... Thus there are infinitely many infinities. Note that 2^N_0 is the number of real numbers.

Hope that helps your exploration of infinity!

Answer 2

No, it doesn't. If Big Bang theory is correct, then the universe began at a certain point in time and is continually expanding. Therefore it can never be infinite - since there are always going to be moments when it is bigger than it was before. Also, if Big Bang is followed at some point by Big Crunch, then again, the universe can't be regarded as infinite, no matter how massive. I suppose it could continue to expand and contact an infinite number of times - but that wouldn't mean that the universe itself was infinite. For example, I could sit here till judgement day continually blowing up and then deflating a balloon - I could do that an infinite number of times, but it wouldn't mean that the balloon itself was infinite in size.

Answer 3

You're assuming that your symbol INF will behave as any other symbol in an algebra.
It will not, because other symbols in algebra have well defined meanings. Yours does not. So you're just playing around with pseudo math here.
If you want a theory of infinity, read the works of G. Cantor. They're a bit technical, but they have been interpreted by various popular science writers such as Isaac Asimov and Marvin Gardner.

Answer 4

There is a problem on the first couple of lines. x/0 is not necessarily infinity. It depends on the sign of x, as well as how the limit, as the denominator approaches zero, is taken. It could be infinity, but it could also be -infinity.

That's the reason why x/0 is undefined, instead of just being infinity.

Answer 5

to expand your knowledge and awareness of infinity and how it "behaves", try exploring:

transfinite numbers
cardinality
modern algebra (infinite sets)
number theory

some of the operations and algebra that you mention are undefined and/or indeterminate

Answer 6

You would benefit from reading this:

http://www.mathpath.org/concepts/division.by.zero.htm