Posts tagged infinity

I think Cantor was wrong when he defined infinity. I think all the mathematicians that built on his work are wrong. When and if you’ve finished laughing at me, read on ;)

Ok so I don’t think those incredibly intelligent guys are wrong about everything. It’s just some of the fundamentals of number theory that I have problems with.

Like the idea of infinity. And the idea that there are things bigger than infinity.

The way this was first explained to me was via Cantor’s diagonal argument. His argument goes as follows: if you write all the infinity of numbers in a list (adding zeros to the front of short numbers to make them all equal length), and then you form a number by cutting diagonally across this infinite square you have created, and then you change every digit in that number to some different digit, you form a new number that can’t be in your infinitely big list since it differs from every number there in at least one of the digits. So you’ve got infinity plus one - a set of numbers called ‘uncountable’.

My problem with this is a little bit technical but it’s not really that hard to understand. Basically, my argument is this: if your list of numbers is anything except infinitely long, then it isn’t square, and you can’t cut completely across it diagonally. Any diagonal number won’t include at least half the list (and that’s just for binary numbers). For any fixed list length, this is true. As the list length grows, this remains true. Why then do we allow this fact to suddenly change when the list reaches the (unreachable) length of infinity? Don’t you see an issue with that logic?

So then to generalise this a little bit further - what on earth makes us think that this idea of infinity makes any real sense? Where does it come from?

In number theory, the first mistaken supposition in my mind is the idea that one can keep counting forever. Sure, there is a pattern to how numbers change as you keep adding one, and adding one, and adding one. There is no reason to think that suddenly you’d be unable to add one - is there? So we can have an infinite number of integers. Oh but wait - we’ve assumed an infinite amount of time, and concluded that there is such a thing as infinity (at least, in a mathematical sense). What if, maybe, the universe comes to an end before we can count that high? What if we run out of energy, or perhaps die, before we get to infinity? Why would we assume that we could have an infinite amount of time?

So here is the crux of the problem: while we’re busy trying to explain the universe mathematically, we’re still holding on to ancient ideas about some pure land of mathematics that differs from this world we live in. It just doesn’t gel.