Some of this stuff has already been mentioned, but hopefully this is an explanation which will be readable for everyone? Not everyone has done math courses and knows what cardinality and ordinality are.
That was sort of what I was trying to say in my post. Meta-concepts like infinity and time are actually impossible to perceive outside of themselves. Infinity, for example, can only be described by infinity. For another example, think about how many numbers are between 0 and 1 divided by infinity. The answer is infinity. Even though 1 over infinity is an infinitely small number, there's still an infinity of numbers that can be crammed between each infinite incrementation. In other words, any problem involving infinity actually contains an infinite amount of infinity, which means that it's impossible to directly perceive or describe infinity.
As has been mentioned, you can't divide a number by infinity. Infinity is not a number. You can 'append' infinitely large and small numbers to the real numbers to get the hyperreals (among other fields), which are a proper extension of the reals and you can derive calculus with them in a way analogous to real numbers, but they do not include 'infinity.' One over infinity (the multiplicative inverse of infinity?) only gives you zero when you write it on a math exam. I'm not going to talk about those, though, because typically we have more experience with real numbers, and those are more familiar for non-math people, so I'm going to talk about them instead. For those who are scared of math, this is basically rigour free: there is only one proof and you can actually skip it without too much worry (although I recommend looking at it, at least).
Part I -- Infinite Sets:It is true that there are the same 'number' of elements in the real numbers in the interval between zero and one as in all of the real numbers. In fact, there is a set called the 'Cantor set', which you get by taking 'almost' everything (by a very specific definition of almost)
out of the interval in between zero and one, and it turns out it contains the same 'number' of elements as the entire real line.
The reason I'm putting 'number' in quotes is because it's sort of a naïve term for what we're talking about, and it is what makes a lot of people confused about the topic. It is very important to be specific here on
what we are counting and how we are counting it. If I say I have an infinite number of apples and an infinite number of oranges, what do I mean? I certainly can't mean that I have 'infinite' in the same way that I could have, say, 'five'. What I mean, when I say I have infinite, is that there is no end to the number I have. So, no matter how many I give you, I can always give you another one.
If we take that to be the definition of infinite, then how do we compare infinite values? With our infinite apples and infinite oranges, we could start by developing a process of comparison which assigns every apple to an orange. If we can construct a process such that every apple goes to an orange with no two apples going to the same orange, regardless of whether we pick apple #1 or apple #1,000,000,000, we can say that we have the same 'number' or, in math terms, that there exists a one to one correspondence between apples and oranges.
Now it turns out by this logic that we can develop a one to one correspondence between the natural numbers, the kind you use for counting, and the integers, which are the counting numbers including zero and negative numbers. The same goes for the rational numbers, which are ratios of integers. We cannot do the same for any of these and the real numbers (I hope you have a general, intuitive idea of what real numbers are, because defining them rigorously from scratch actually requires more than I am inclined to write here).
So there are 'more' real numbers than there are natural numbers, but as I mentioned before we can establish that there are the same number of real numbers between zero and one as there are real numbers. It sounds weird, but by the above definition it actually makes sense.
It does not mean that you can count them all up and get the same number. It just means that you can construct a relation between them such that our apple/orange condition is met, such that, no matter which number in the interval between zero and one you pick, it is assigned to exactly one real number and there is no real number such that a number on the interval between zero and one is not assigned to it.
Part II -- Limits and Convergence for Dummies:That takes care of infinite collections of objects (takes care of = barely touches on), but what do we do with them? If we have infinite numbers of things, can we do operations on all of them and get some kind of meaningful result?
The basic foundation of doing math with infinite numbers of things is a sequence. A sequence is just a function (meaning you put one thing in and you get one thing out the other end) between natural numbers (counting numbers, remember) and some other set, in this case real numbers. An infinite sequence has a way of taking any counting number and giving you a real number value back. For example, we can take the sequence {a_n}, such that a_n=1/n. That is to say, for every element of our sequence, we take the number of that element and divide one by it (or take its multiplicative inverse or however you please).
To bring this back in touch with something meaningful, the
idea behind this exercise is that, if we start looking at what happens to this sequence when n gets larger and larger and larger, we get some kind of meaningful way to define a concept
like '1/infinity' in some way that we get an actual answer besides 'come see me during office hours.'
To do this, we must define the notion of a limit. If we can, for some number L, take any interval of any size we would like around L and show that there is a tail end of the sequence which lies completely within this interval, we say that the sequence is 'convergent' and that its limit is L. In math terms, we say that for any e (pretend that's a Greek epsilon) where e>0, there exists some $N$ such that |a_n-L|<e whenever n>=N. In other words, given any positive number e, we can pick a number N such that every element in our sequence associated with N or a number larger than N is within a distance of e from the number L, the limit. In other other words, the sequence gets 'closer and closer' to the limit (although this naïve definition is not as descriptive as we would like).
The simplest way to do this, to quote a professor I once had, is to 'cough up N.' Just show that you can find one for your L, given any e>0. For the case of our series (which we would hope has a limit of zero), it's easy: take N to be the
smallest number such that N is greater an 1/e (I'm skipping a lot of the rigour in this explanation, but this guaranteed to exist since we have an infinite number of numbers and they are all ordered). Since N>1/e, e>1/N and |1/N-0|<e. Of course, 1/N is also greater than 1/(N+1), or any other, larger n, so the statement above is true and we can say that 'the limit of 1/n as n goes to infinity is 0'. Ta da!
So we haven't just said that 1/infinity is zero, we've actually defined what it means to have infinite numbers of things and defined a meaningful limiting process which gives us a result which we can prove (incidentally, you can easily prove that the limit of a convergent sequence is unique).
You can take the above definitions and start deriving theorems from them basically right away. You can say 'what if we define a sequence by adding up the terms of another sequence' and you can get infinite
sums, you can look at limits of functions in an analogous way (this is how calculus is built), etc. As a simpler exercise, you can also describe what happens when there is no L such that the above is true and come up with the idea of divergence.
Part III -- What Does It All Mean?As has been mentioned before, there are many ways to describe infinitely large or small values, but the above is the most intuitive, I think, because it doesn't require you to picture a number larger than any number or anything funny like that. You just have to picture the idea of a process that keeps going as long as you would like.
This is a mathematical truth, and it takes for granted certain axioms. The idea that you can have an infinite number of numbers is kind of an axiom, in this case, as there are meaningful algebras (groups, for you math-types) on, for example, sets of four numbers (actually, you can prove there are only two of these, although that's another story). In fact, there is a meaningful algebraic structure you can define on one number (but it's pretty boring because everything gives you the same number).
Asking whether this has any deeper truth which is integral to the universe itself runs the same risks as asking the same question for any other concepts, but if you're going to talk about infinity and expect to be taken seriously, it helps to have a consistent definition, like the above, so you don't appear to be 'subtracting a tree and five oxen from a cat times happiness'.
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