Question

Is infinity represented only by Cantor’s full hierarchy of transfinite numbers (Alephs), since no Aleph is large enough to number all the Alephs in the hierarchy? Has Cantor has not captured, mathematically, “genuine” infinity?

Answer 1

The notion of infinity gets refracted into a spectrum of different transfinite cardinalities when it is passed through the prism of Cantor’sSet Theory. Using the criterion that countably infinite sets are the ones that can be put into an infinite sequence or placed in one-to-one correspondence with N+, we were able to prove several results about such sets. The familiar sets of numbers, Z, Q, and were all shown to be countably infinite, and the set-theoretic operations of union and Cartesian product applied to two (and thus finitely many)countably infinite sets were shown to yield countably infinite sets. Given the evidence amassed on countably infinite sets, one might wonder whether all the hoopla about countably infinite sets isn’t just so much sophisticated hot air, whether infinite sets can’t all be shown to be countably infinite, given enough ingenuity. Sets like didn't look like they could be listed at the outset, but given an inventive rearrangement of the elements’natural order, it was found to be possible. Maybe this can always be done. Can’t you just choose some element as the first one, then pick another as the second, and so on, until eventually everything in the entire set is listed? The answer to this is mostly “no.” It is a little bit “yes,” in the highly qualified sense that every set can be well-ordered or sequentialized, provided the axiom of Choice accepted. This is the result that originally got Zermelo started on axiomatizingSet Theory 1908. But even given this very surprising result, which is too involved in getting into here, it is still “no” concerning numerosity. A listing procedure can be used to demonstrate that every infinite set contains a countably infinite subset , but unless you can show that your countably infinite sequence eventually catches every element of the set, you do not know whether the whole set can be enumerated. And, in fact, not all infinite sets are countably infinite, as we will show in a powerful way below. Some infinite sets are so big that they are uncountable, in the strong, human-independent sense that they cannot be listed by any countably infinite sequence, no matter how ingeniously devised. Such sets are more numerous than the set of natural number. No , Cantor has not captured, mathematically, “genuine” infinity