Cantor's paradox

In set theory, Cantor's paradox is derivable from the theorem that there is no greatest cardinal number, so that the collection of "infinite sizes" is itself infinite. The difficulty is handled in axiomatic set theory by declaring that this collection is not a set but a proper class; in von Neumann–Bernays–Gödel set theory it follows from this and the axiom of limitation of size that this proper class must be in bijection with the class of all sets. Thus, not only are there infinitely many infinities, but this infinity is larger than any of the infinities it enumerates.

This paradox is named for Georg Cantor, who is often credited with first identifying it in 1899 (or between 1895 and 1897). Like a number of "paradoxes" it is not actually contradictory but merely indicative of a mistaken intuition, in this case about the nature of infinity and the notion of a set. Put another way, it is paradoxical within the confines of naïve set theory and therefore demonstrates that a careless axiomatization of this theory is inconsistent.

In order to state the paradox it is necessary to understand that the cardinal numbers admit an ordering, so that one can speak about one being greater or less than another. Then Cantor's paradox is:

This fact is a direct consequence of Cantor's theorem on the cardinality of the power set of a set.

Another consequence of Cantor's theorem is that the cardinal numbers constitute a proper class. That is, they cannot all be collected together as elements of a single set. Here is a somewhat more general result.

Since the cardinal numbers are well-ordered by indexing with the ordinal numbers (see Cardinal number, formal definition), this also establishes that there is no greatest ordinal number; conversely, the latter statement implies Cantor's paradox. By applying this indexing to the Burali-Forti paradox we obtain another proof that the cardinal numbers are a proper class rather than a set, and (at least in ZFC or in von Neumann–Bernays–Gödel set theory) it follows from this that there is a bijection between the class of cardinals and the class of all sets. Since every set is a subset of this latter class, and every cardinality is the cardinality of a set (by definition!) this intuitively means that the "cardinality" of the collection of cardinals is greater than the cardinality of any set: it is more infinite than any true infinity. This is the paradoxical nature of Cantor's "paradox".

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