posted on Apr, 8 2009 @ 02:36 PM
I don't understand how you go into that with infinity.
Infinity is fake, the way it's used in mathematics. You can't really add, divide, or do anything to infinity (though this is sort of a lie, you're
just not supposed to most of the time) because it's not a number and doesn't exist. And one infinity isn't the same as another...
When you use it in mathematics, you only use it as a limit. For those of you who need a refresher of precalculus or early calculus, a limit is
something you approach, get infinitely close to, but don't touch. A mathematical concept used to solve many problems such as the derivative or in
physics to make infinite approximations (ex. energy used to bring a charged particle to a distance x from another charged particle from infinity is
its electric potential energy). So you can't just say you multiply infinity by infinity, you have to multiply the limit of x as it goes to infinity
by the limit of y as it goes to infinity. Which would be bizarre, and infinity.
There's also different sorts of infinity, which I think I'll best understand after I take a couple upper level math courses (I'm fighting the good
fight in the 200 levels now in college), but there is a bizarre behavior of infinities that you can find through a test of surjection.
Surjection is a property of a function. A function is more than you've been taught in high school, or at least, I have. It has three distinct
parts. You need, as you know, some rule, say, f(x) = x + 3. You often see it as y = x + 3 if you're graphing it. But you also need a domain and
target space (or codomain, sometimes also called the range, but range can refer to either image or target space). Domain is the set of numbers that
you plug into the function. You can make it integers (whole numbers), numbers from 1-6, all complex numbers (integers and fractions), all real
numbers (all complex numbers plus irrational numbers such as the square root of two and pi), or anything else you want. Your target space is the set
of numbers for which you are defining the output of your function, which is just as arbitrary. You can make it whatever you want.
I will also define the image, which is every number the function "hits" for the domain it is defined. For example, you have a function, f(x) = x +
3, where the domain is integers from 0-4 and the target space is integers from 0-10. You plug in 0, 1, 2, 3, and 4 into the function and get 3, 4, 5,
6, and 7. Your image here is integers from 3-7. Your target space was 0-10, and your image obviously lies inside of this target space, but does not
span it entirely.
Whether or not a function is surjective is simply saying that the image spans the target space entirely. At first I wondered, why would you test for
such a thing, when the target space is something you arbitrarily define? Then I realized potential applications such as a test for surjection as I'm
about to describe.
Since we got through the muck (interesting muck if you're me) I can now simply ask, does the number of positive integers including zero (infinite)
equal the number of all integers (also infinite)?
To test this, you see if you can create a function which is surjective, where the domain is all positive integers and zero and the target space is all
integers. The answer is, yes, you can, if you make some function (we don't need to figure out how to write it mathematically) where 0 is 0, 1 is 1,
2 is -1, 3 is 2, 4 is -2. and so on, until you go to infinity.
Remember, however, that surjection doesn't mean only that you go from one end of the target space to the other. You need to hit EVERYTHING
inbetween. If it's integers from 1-10, you have to hit 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10. But if it's real numbers from 1-10, you have to hit
EVERY SINGLE DECIMAL (there are infinite) between 1 and 10. Your image (output of your function you get when you input all numbers in your domain)
must go from one end of the target space to the other without leaving gaps for a function to be surjective.
You do this again asking, does the number of all integers then equal the number of complex numbers (integers and fractions)? It seems like it should
be no! But you can, in fact, define every fraction as nothing more than a ratio of integers (as what a fraction is, say, 1/2 is the ratio of 1 to 2),
and you can make a function where, as you proceed to infinity, you hit not only every integer but also every complex number. That is, using only the
set of integers (whole numbers), you can get out every ratio of integers. So the number of integers and the number of complex numbers are the same,
as you can define a surjective function from integers to complex numbers.
But it gets weird here. Can you define a function from complex numbers, or integers, to all real numbers? Remember that real numbers are all complex
numbers AND numbers that cannot be described as a fraction (irrational numbers)...which there are, in fact, more of than of rational numbers. This is
a famous proof my Euler (who did many, many great mathematical things), which shows that no, you CANNOT define a function from integers to real
numbers which is surjective. That is, no matter how you define some function, you will ALWAYS have some numbers inbetween that you did not hit. This
is another way of saying that no matter how precise you are, you can always get more precise -- if you hit .0001 and .0002, you just missed .00011,
.00012, and so forth. Therefore, and to my surprise, this means that you have an infinite number of integers, an infinite number of fractions
(complex numbers), and an infinite number of decimals (real numbers)...but the former two infinities, the infinities of integers and fractions, are
LESS than the infinity of real numbers! So infinity, through this test of surjection, cannot equal infinity; therefore, there must be different sorts
of infinity. It gets even weirder when you try and add and subtract infinities from one another...but you're not really supposed to do that, since
infinity is, in fact, some conceptual construct created to solve problems, and not really a number at all.
[edit on 8-4-2009 by Johnmike]