Math CentralQuandaries & Queries


Question from Justin, a student:

Hello again, I was just wondering even in the context of set theory, how can other infinites can be larger than each other, I thought infinity itself is the largest possible quantity?

Thanks again for answering my math questions!

All the Best,



It all comes down to what we consider to be the "size" of a set. We say a finite set has size n if it can be put into 1-1 correspondence with the set {1, 2, ..., n}. This idea can be extended to say that two sets have the same size if they can be put into 1-1 correspondence with each other. For instance, the set of all squares of integers has the same size as the set of positive integers because they can be put into 1-1 correspondence. This leads to the idea of sets being the same size as the integers, or the reals, or whatever.

It turns out that no set, finite or infinite, can be put into 1-1 correspondence with its power set (the set of all subsets of the set). Therefore, if X is an infinite set, the power set of X is a "larger" infinite set. Since this process can be continued, there is no largest size of a set.

Hope this helps.


Many people had trouble with that idea when Cantor introduced it.
He defined equality of "cardinal numbers" to mean that there is a 1-1 pairing between them. (There are also "ordinal numbers" that count well-ordered sets; they are equal if there is a 1-1 order-preserving map. For infinite sets these definitions of equality are not equivalent, for finite sets they are.) If you look back at my first response I explain the proof of Cantor's theorem, which shows in particular that the set of subsets of any set - infinite or finite - is always strictly larger than the one you started with.


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