The Axiom of Infinity: Number Theory

Recall that we denote by S the category of (abstract, discrete, constant) sets and arbitrary maps between them that we have studied till now. In the various branches of mathematics (such as mechanics, geometry, analysis, number theory, logic) there arise many different categories χ of (not necessarily discrete, variable) sets and respectful maps between them. The relation between S and the χ's is (at least) threefold:

1. (0) S is “case zero” of an χ in that in general the sets in χ have some sort of structure such as glue, motion and so on, but in S this structure is reduced to nothing. However, the general X often has a functor determining the mere number (Cantor) |X| of each such emergent aggregate X.

2. (1) A great many of the mathematical properties of such a category χ of variable sets are the same or similar to properties of the category S of constant sets. Thus, a thorough knowledge of the properties of S, together with some categorical wisdom, can be indispensable in dealing with problems of analysis, combinatorics, and so forth. The main common properties include the concepts of function spaces XT and of power sets P(X).

3. […]