Math CentralQuandaries & Queries


Question from Justin, a student:

Hello there, I was wondering if one infinity is larger than another in the extended real number system (just like in the transfinite ordinals and cardinals with respect to infinite sets) or are all infinities the same size in the extended real number system? Thanks sooo much for answering my question! I greatly appreciate it!

All the Best,



If you mean the usual extended real number system of first year calculus, there are only two infinities (positive and negative). One would say (if one needed to, which one doesn't usually) that positive infinity is larger than negative infinity, but has the same absolute value, in the same sense that 1 is larger than -1 but has the same absolute value.

Now, there are other ways to extend the real number system. In particular, there is the "nonstandard real number system" which can be used as the basis for a completely different foundation for calculus; it contains infinitely many different sizes of infinite element. See


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