ABSTRACT
6.1 We now have sets and numbers at hand, so it is tempting to associate a number to a set in order to measure its size. That is the idea to define the cardinality of a set. But this is not a trivial matter, in fact why should any relation between sets and the axiomatically defined numbers be natural or even possible? The obvious question that prompts itself is: are there enough numbers to distinguish sets up to a bijection? A good definition of “finite set” will take care of those, see Definition 6.3, but a correct treatment of infinite sets presents deep problems. We mention some of the recent development in 6.26; the term recent is relative but compared to the fundamental nature of the problems it may be surprising to learn that these were only settled in the 1960's.
