We will give the axioms defining the objects that exist in our
system. We will follow each axiom with the C `++` code to
implement the axiom or construct the objects that it defines from
other objects. These axioms will use recursive definitions that are
not in general valid in ZF. At each stage at which we can, we will
give a definition in ZF of the objects intended at that stage. The
ZF definition will exhaustively define the objects that meet a
given recursive definition. In particular the ZF definition will
disallow any infinite descending functional evaluations. In the
recursive definitions these are not explicitly prohibited (to do
requires quantification over the reals) but only objects that
satisfy this can be built up form the definitions. This is similar
to ZF where well founded sets are not explicitly excluded (the
axiom of regularity is generally not considered a part of ZF) but
only well founded sets can be constructed.

In this formulation objects from C `++` class
`Functional` take the place of sets in ZF. Everything, well
almost everything, is an instance of this class. Of course we have
the structure of C `++` as a preexisting framework for the
structures we define. It is also convenient to treat integers as a
special case. A `Functional` that represents an integer has
a member function that returns the C `++` notion of an
integer as a `very_long` object.
This can be defined as a standard C `++` integer object or
as a class object that implements arbitrarily long integers.

The axioms loosely follow the axioms for ZF defined in Appendix on page . The power set axiom is replaced by the well foundedness axiom. The replacement axiom is replaced by the combination of the well foundedness axiom and the induction axiom.

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