Math and Science for the Mathematically Uninclined

The Sizes of Infinities

In the last article, “How Big is Infinity?”, we discussed what it means when something is infinite — that it has no bound on how large it gets. But when it comes to infinitely large things, is there any way to measure them against each other? Is there a way to see which infinite collections are larger?

The answer, surprisingly, is yes, and delves into a topic of math called **set theory** — the mathematics of sets, or collections of objects, usually numbers. In order to understand this discussion, we’ll first tone it back a bit and work with finite things.

Probably some grad-level things or higher. I have plans to discuss relativity (physics), cardinality (set theory/mathematics), logical independence (logic), and various levels of cryptography, but I’m not against talking about lower-level materials, too, if that interests the readers.

If you have a particular topic you’d like me to discuss, please tell me! If I’m not familiar with it, I will do my best to research it and explain it as best as I can.

How Big is Infinity?

Most people understand that the symbol ∞ means “infinity”, and understand that it represents a really big number. But just *how* big is it? What does it actually mean? Can you do things to infinity?

In mathematics, infinity doesn’t actually represent a number in the typical way we think about numbers. With typical numbers, we can add, subtract, multiply, and do a whole range of calculations to them, and typically get another answer. But what happens when we do things forever? What does, say, 1 + 1 + 1 + … end up adding to?