Mathematics, Infinity,

Looking up some mathematical issue regarding infinity I've come across the following page and I'll assume rightly or wrongly that it represents some kind of broader opinion, and to be honest I'm a bit shocked at the intellectual level at play. People should have a bit more humility when dealing supposedly with infinity, as it seems likely they'll end up otherwise blandly substituting it with something very finite, and postulating confidently and ever more wrongly from there.

http://gwydir.demon.co.uk/jo/numbers/interest/infinity.htm

What is infinity? It is bigger than the biggest number, but it is not a number itself.

What is wrong here is blatantly enough the description of infinity being bigger than the biggest number, as of course within the realm of infinity there is no such thing as a biggest number. It is only within a finite realm that there is a biggest number.

If you could do arithmetic using infinity, then you would end up proving that 1 = 2, which is not a good idea! So you cannot do arithmetic using infinity. It's where the number system breaks down.

The only means by which you would prove 1=2 is if you have decided for the sake of some childish convenience to pretend infinity is something falsely other than itself, which intellectual gibberish in turn can justify and produce such resulting gibberish as 1=2.

Surely there are more rationals (fractions) than natural numbers. 

Again the same obvious criticism. How can there be more numbers than an infinite number? There can only be more than a finite number. So again something finite, humanly conceived and definite has replaced the endless world of infinity.

Countable sets

So how can we say anything about infinity at all? In fact, we can say more than you'd think. First, we can say that there are infinitely many natural numbers. Now we have a way of counting infinite sets of numbers. Wait a minute - how can we count something that's infinite? Surely it would take a infinite amount of time, even for a computer? 

What has a computer got to do with the, surely by any kind of meaningful definition, impossibility of counting a never-ending stream of numbers to a conclusion?

Are all infinite sets of numbers the same size? No. The set of irrationals and the set of reals are not countable. There is no way that you can lay them out so there is a one-to-one correspondence with the natural numbers. This means that there are different types of infinity. The countable sets of natural numbers and rationals are smaller than the sets of irrationals and reals.

Again no infinite set can be countable because to be able to count all of the numbers within a set again necessitates a definite, limited and finite number. And so naturally in truth there is no such thing as a set containing infinity, as infinity cannot be contained within form. So a closer look at the notion of "all the numbers within infinity." Take one of these numbers within the infinity of numbers; this being Pi. Pi is itself infinite, and so one cannot speak of 'all the numbers of Pi'; 'all' being an inclusive term, whereas infinity spills endlessly beyond 'all'. You cannot write down all the numbers as all of something requires a completed totality, whereas infinity again reaches no conclusion . A totality is finite, while infinity cannot be circumscribed within a system or set, else it is certainly not infinity despite whatever claims.

Follow-up to this here.