Math CentralQuandaries & Queries


Question from Justin, a student:
I was just wondering is the limit x->0, y=1/x=infinity,
the biggest uncountable infinity according to Cantor's cardinal
numbers in set theory?

No. The "infinity" introduced in the one-point compactification of the positive reals is not a cardinal number at all. Neither are any of the real numbers cardinals; the cardinal 1, the ordinal 1, the integer 1, the rational number 1, and the real number 1 are all subtly different entities [differing mainly in what operations and relations are defined upon them.]

By embedding the finite cardinals (0,1,2,3,...) in the reals, we can obtain a sequence of reals that corresponds to the sequence of finite cardinals in a fairly natural way and converges to "infinity". Making this identification (which I stress is not vacuous), the "positive real infinity" would correspond to aleph-null, the smallest transfinite cardinal (and the only one which is not uncountable.)

Note also that in Cantor's set theory there is no biggest transfinite cardinal. For every cardinal, its powerset (collection of subsets) is (by Cantor's theorem) a strictly larger set [not necessarily its successor].

If you know set theory a bit beyond this point you might ask whether we can get a largest cardinal by taking the union of all cardinals (to make this persuasive you need to know about ordinals too.) The answer in many versions of set theory is that you can, but that it is not a set, but something called a "proper class" (that is, a class that is not a set.) Classes form a sort of backstop for set theory.

We can easily avoid all paradoxes in set theory by allowing very few operations that construct new sets. For instance, if we don't allow any set to be part of the definition of a set element, then Russell's paradox about "the set of all sets that are not members of themselves" is ruled out. Unfortunately, so is most interesting math. If on the other hand we allow all "obvious" operations then we certainly do get paradoxes. One way to get out of the problems is to allow quite a lot of construction methods but to say that some of them yield not sets but "classes", and that the problematic operations are not defined on classes (or yield a still higher-level structure.) Thus there is no set of all sets; there is a class of all sets, with a subclass of all sets which are not members of themselves. This class is not a set, so obviously not a member of itself (even though it's not a member of itself.)

Do you have a good text on set theory? I would suggest Halmos' "Naive Set Theory" (reprinted by Springer) if you're working alone.

Good Hunting!

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