RCF3: Map-Code Interpretation via Closure

TL;DR

This paper develops a method to interpret map codes within higher-order Cartesian closed arithmetical theories, leading to a formalization of Richard’s paradox and demonstrating inconsistency in related set theories and categorical frameworks.

Contribution

It introduces a novel interpretation technique for map codes in higher-order theories, enabling code-self-evaluation and formalizing Richard’s paradox within these frameworks.

Findings

Formalizes Richard’s paradox in higher-order theories

Shows inconsistency in set theories like ZF and categorical theories

Provides a new interpretation method for map codes in Cartesian closed theories

Abstract

For a (minimal) Arithmetical theory with higher Order Objects, i.e. a (minimal) Cartesian closed arithmetical theory -- coming as such with the corresponding closed evaluation -- we interprete here map codes, out of [A,B] say,into these maps "themselves", coming as elements ("names") within hom-Objects B^A. The interpretation (family) uses a Chain of Universal Objects U_n, one for each Order stratum with respect to "higher" Order of the Objects. Combined with closed, axiomatic evaluation, this interpretation family gives code-self-evaluation. Via the usual diagonal argument, Antinomie RICHARD then can be formalised within minimal higher Order (Cartesian closed) arithmetical theory, and yields this way inconsistency for all of its extensions, in particular for set theories as ZF, of the Elementary Theory of (higher Order) Topoi with Natural Numbers Object as considered by FREYD, as well as already for the Theory of Cartesian Closed Categories with NNO considered by LAMBEK and SCOTT.