We exhibit canonical middle-inverse Choice maps within categorical
(Free-Variable) Theory of Primitive Recursion as well as in Theory of partial
PR maps over the Theory of Primitive Recursion with predicate abstraction.
Using these choice-maps, defined by mu-recursion, we address the Consistency
problem for a minimal Quantified extension Q of latter two theories: We prove,
that Q's exists-defined mu-operator coincides on PR predicates with that
inherited from theory of partial PR maps. We strengthen Theory Q by
axiomatically forcing the lexicographical order on its omega^omega to become a
well-order: "finite descent". Resulting theory admits non-infinit PR-iterative
descent schema (pi) which constitutes Cartesian PR Theory piR introduced in
RCF2: Evaluation and Consistency.

A suitable Cartesian subSystem of Q + wo(omega^omega) above, extension of piR
"inside" Theory Q + wo(omega^omega), is shown to admit code self-evaluation:
extension of formally partial code evaluation of piR into a "total"
self-evaluation for the subSystem. Appropriate diagonal argument then shows
inconsistency of this subsystem and (hence) of its extensions Q +
wo(omega^omega) and ZF.