Saturday, July 17, 2010

How Big is Infinity?


Most of us are familiar with the infinity symbol – the one that looks like the number eight tipped over on its side. The infinite sometimes crops up in everyday speech as a superlative form of the word many. But how many is infinitely many? How far away is “from here to infinity”? How big is infinity?

You can’t count to infinity. Yet we are comfortable with the idea that there are infinitely many numbers to count with: no matter how big a number you might come up with, someone else can come up with a bigger one: that number plus one – or plus two, or times two. Or times itself. There simply is no biggest number. Is there?

Is infinity a number? Is there anything bigger than infinity? How about infinity plus one? What’s infinity plus infinity? What about infinity times infinity? Children, to whom the concept of infinity is brand new, pose questions like this and don’t seem to have very much bearing on daily life, so their unsatisfactory answers don’t seem to be a matter of concern.

At the turn of the century, in Germany, the Russian – born mathematician George Cantor applied the tools of mathematical rigor and logical deduction to questions about infinity in search of satisfactory answers. His conclusions are paradoxical to our everyday experience, yet they are mathematically sound. The world of our everyday experience is finite. We can’t exactly say where the boundary line is, but beyond the finite, in the realm of the transfinite, things are different.

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