RCF4: Inconsistent Quantification

TL;DR

This paper explores the consistency of a minimal quantified theory extended with well-ordering and self-evaluation, ultimately demonstrating its inconsistency through a diagonal argument.

Contribution

It introduces a new approach to the consistency problem using choice maps, mu-recursion, and well-ordering, leading to a proof of inconsistency.

Findings

The minimal quantified theory Q's mu-operator aligns with that of partial PR maps.

Forcing a well-order on omega^omega enables a non-infinitary PR descent schema.

A self-evaluation subsystem is shown to be inconsistent via diagonal argument.

Abstract

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.