C’mon, Verizon, You Can Do Better

I generally dislike Verizon’s current commercials, but the commercial about infinity has to be my least favorite, largely because it implies things that are mathematically wrong.

In the commercial, the man asks the kids, “what’s the largest number you can think of?” The commercial is of course a bit tongue-in-cheek, with one child answering “10”, but the part that I can’t get over is when one kid says “infinity”, then another says “infinity and one”, and then finally one says “infinity times infinity”, after which the man implies that his mind is blown.

Ignoring the fact that infinity isn’t strictly a number (but it’s often treated like a number, and I’ll likely confound this in my post; I’m also going to assume that infinity, unless otherwise specified, refers to aleph naught, or the cardinality of the set of natural numbers), this commercial teaches people bad math: infinity, infinity + 1, and infinity * infinity are all actually the same size. None is “larger” than any other, so you can’t one-up someone else by saying that infinity + 1 is larger than infinity or that infinity * infinity is larger than infinity + 1.

The part that should really blow your mind is not infinity * infinity by itself, but rather that infinity * infinity is the same size as infinity. An example: you can find a bijection between the rational numbers and the natural numbers, which means that the sets of the natural numbers and the rational numbers are the same size. And in general, assuming that you believe in the axiom of choice, any infinite cardinal number is equal to its square, so using “times” will not get you larger cardinalities.

But of course, there is something larger than infinity, and that’s the power set of infinity. Thanks to Cantor’s theorem, we know that for any set, the set of all subsets of that set has a strictly greater cardinality, which means that you can always find a greater infinity. Crazy, right? In fact, not only are there infinitely many infinities, but the “cardinality” of the collection of infinities is larger than any of the infinities that it contains.

I don’t expect a 30-second TV commercial to get into all of the intricacies of infinity, but I would hope that the makers of the commercial would at least not propagate misinformation about math.