  In previous papers on this project a general static logical framework for
formalizing and mechanizing set theories of different strength was suggested,
and the power of some predicatively acceptable theories in that framework was
explored. In this work we first improve that framework by enriching it with
means for coherently extending by definitions its theories, without destroying
its static nature or violating any of the principles on which it is based. Then
we turn to investigate within the enriched framework the power of the minimal
(predicatively acceptable) theory in it that proves the existence of infinite
sets. We show that that theory is a computational theory, in the sense that
every element of its minimal transitive model is denoted by some of its closed
terms. (That model happens to be the second universe in Jensen's hierarchy.)
Then we show that already this minimal theory suffices for developing very
large portions (if not all) of scientifically applicable mathematics. This
requires treating the collection of real numbers as a proper class, that is: a
unary predicate which can be introduced in the theory by the static extension
method described in the first part of the paper.
