| > There are only two types of infinities: countable and uncountable. There are cardinal transfinite numbers (starting at aleph-null) and ordinal transfinite numbers (starting at omega). There are infinitely many transfinite cardinals, and likewise for the ordinals. There are also other infinite numbers in mathematics, distinct from the transfinite numbers of set theory, such as infinities in the projectively and affinely extended reals, the surreals, etc Usually, discussions of transfinite numbers assume ZFC set theory (Zermelo–Fraenkel with the axiom of choice). You also have large cardinals which only exist if you add additional axioms to ZFC (large cardinal axioms). There is a hierarchy of larger and larger large cardinals, which corresponds to the ordering of the consistency strength of the large cardinal axioms. How many cardinals exist, in ZFC? Well, "the set of all cardinals" isn't a set in ZFC – for the same reason that ZFC lacks a universal set (that's how it avoids Russell's paradox) – so we cannot speak of which cardinal is its size. Of course, if we adopt a suitable extension of ZFC (such as proper classes), then maybe we can, but then "the cardinal measuring the number of cardinals in ZFC" wouldn't be the same type of mathematical object (category?) as the usual cardinals are. And then I assume if you replace ZFC with some alternative set theory, such as NBG or NFU, at some point you'll get different cardinals and ordinals arising. But that's a question that has always intrigued me but I've never known the answer to. I'm sure the answer is in some graduate maths text somewhere which is going to go completely over my head. |