We accept numbers as infinitely precise - for example, pi cannot be fully expanded in digits. Yet we balk at numbers that are infinitely wide - for example, the number of integers.

The number of integers is less than the number of complex numbers. That we can compare them this way, and discuss them this way, suggests they are numbers, even though they are not finite.

Of course, we can define numbers to be anything we want. My right pinky's nail is a number because I say so, and according to the little game of conversation we're playing, in which we get to choose the terms, that's actually true.

But I'm not saying that. I'm saying that infinity is remarkably like other math objects we call numbers.