“Nearly Infinite”
Eugene Volokh had a post today on whether something can be “nearly infinite”. I think it can— but only if it actually is infinite. First, here’s what he says:
“Nearly Infinite”: An otherwise very good item that I read a while back refers to a certain set of things as “nearly infinite.” It then pointed out, as evidence, that there were over 100 elements in that set.
Now that’s defining infinity down. Of course, even if there were thirty-seven googol elements, that would still be infinitely far from infinity.
There’s no such thing as “nearly infinite.” The problem isn’t like the asserted (but in my view overstated) problem with “more perfect” (as in “more perfect Union”) or “more round.” You can get materially more perfect or round than you were before. But so long as something is finite, it’s not nearly infinite, no matter how much it grows.
I think something *can* be “nearly infinite”. I understand this as being the same as “almost infinite”, which I understand as being within some small amount Epsilon of “infinite”, where the listener gets to choose Epsilon to be as small as he chooses, so long as it is greater than zero.
So, first ask the listener: “What is Epsilon?” Suppose he says, “Epsilon equals .01″. Then you can say that any number greater than “Infinity minus .01″ is “almost infinite”. That includes Infinity, of course. Whether a number such as “infinity minus .009″ is different from “infinity” is a question too hard for me. I think the conventional answer is that it is not.
(Eugene’s statement can be made correct if we clarify my definition of “almost infinite” by adding that if number X is “within distance Epsilon” of Y, we do not allow X to be equal to Y.)
All that sounds a bit silly. But where that approach is actually important when we switch to a number being “almost infinitely small”. By my logic above, this would mean that the number is “less than Epsilon”, where the listener first chooses Epsilon. There definitely *do* exist numbers in the interval (0, Epsilon), and all of them are greater than zero.
These very small numbers are the stuff of calculus and measure theory, and they allow, for example, there to be an industry composed of infinitesimally small firms producing, in aggregate, 100 tons of output when each produces only an infinitesimal amount.
Next day: Prof. Volokh got two comments that are relevant here. First, a way to look at this with measure theory:
I think what you’re looking for is an “outer measure.” To get technical about it, let the underlying set be Z, the positive integers, consider the ring of all possible subsets of Z (set-theoretic definition of ring, not the algebra definition of ring….), and define the outer measure, M, of any set to be n/(n+1) for any finite set, where “n” is the number of elements in the set, and 1 for any infinite set. (Outer measures need only be finitely SUB-additive, not additive, i.e. M(evens U odds) = 1 < = M(evens) + M(odds) = 2, since both sets are infinite.)
Then you could talk about finite sets being within some particular bound of infinity, if 1 - bound < M(set) < 1.
This makes sense. The measure of the set of all odd integers is 1, then, the same as the measure of all even integers or of all real numbers. The measure of a set of 800 elements is 800/801, which is “almost 1″, so it is “almost infinite”. The user may define “almost” rigorously by saying how large [1-M(set)] can be.
Second, more simply,
I mean if we are comfortable saying (for real numbers) that y is almost 0 if |y|< epsilon then why not just say that z is almost infinity if |1/z|< epsilon (and if you want + infinity z is positive).