2.3: Finite and infinite sets

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Sets can have an infinite number of members. That doesn’t make sense for the Davies family example, but for other things it does, of course, like: \[I = \{~k : \text{$k$ is a multiple of 3}~\}.\]

Obviously there are infinitely many multiples of 3, and so $I$ has an unlimited number of members. Not surprisingly, we call $I$ an infinite set. More surprisingly, it turns out that there are different sizes of infinite sets, and hence different kinds of infinity. For instance, even though there are infinitely many whole numbers, and also infinitely many real (decimal) numbers, there are nevertheless more real numbers than whole numbers. This is the thing that drove Cantor insane, so we won’t discuss it more here. For now, just realize that every set is either finite or infinite.

You might think, by the way, that there’s no way to define an infinite set extensionally, since that would require infinite paper. This isn’t true, though, if we creatively use an ellipsis: \[I = \{~3,6,9,12,15,\dots~\}\] This is an extensional definition of $I$, since we’re explicitly listing all the members. It could be argued, though, that it’s really intensional, since the interpretation of “…" requires the reader to figure out the rule and mentally apply it to all remaining numbers. Perhaps in reality we are giving an intensional definition, cloaked in an extensional-looking list of members. I’m on the fence here.