set theory - Denumerable sets

Here is a summary of standard parlance in set theory.

• A set is finite if it is equinumerous with a natural number. Otherwise, the set is infinite.




• A set is countable if it is equinumerous with a subset of the natural numbers. (This includes the finite sets.)




• A set is countably infinite if it is countable and infinite. This is also called countably enumerable.




• A set is Dedekind finite if it is not bijective with any proper subset of itself. This is equivalent to the set not containing any countably infinite subset. If the Axiom of Choice holds, it is equivalent to being finite.




• The word enumerable is often used with countable sets, but does not by itself imply countability. For example, set theorists often enumerate uncountable sets in a well-ordered sequence. The word enumeration carries a connotation of being well-ordered, but this is not strict, since one might enumerate a set by reals, meaning simply to have a surjection from the reals onto the set.




• Some set theorists use denumerable synonymously with countable (e.g. Moschovakis, Notes on Set Theory, undergraduate textbook). Many or most set theorists seldom use the word denumerable. In a quick perusal, I couldn't find use of it in Jech's book Set Theory or Kanamori's book on large cardinals.

Of course, all these concepts originate with Cantor, and I looked up his Contributions to the founding of the theory of Transfinite Numbers (Dover, in translation by Jourdain). I couldn't find the word denumerable at all, but Cantor does use enumerable at first just to mean countable, but then later extended to include uncountable well-ordered enumerations, as I mentioned above.