How Big is Infinity?

Most everyone is 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”?

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, ask questions like this and don’t usually get very good answers. For adults, these questions don’t seem to have very much meaning in daily life, so their “not so good” answers don’t matter too much.

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

In: Math