Categorical Pairs and the Indicative Shift

TL;DR

This paper introduces categorical pairs and the indicative shift, a framework that generalizes fixed point theorems in logic and categories, providing new insights into self-reference phenomena like Gödel's Incompleteness Theorem.

Contribution

It defines categorical pairs and the indicative shift, offering a unified approach to understanding fixed points and self-reference in logic and category theory.

Findings

Generalizes fixed point theorems using categorical pairs

Shows the indicative shift explains Gödel's self-reference

Connects categorical structures to logical fixed points

Abstract

This paper introduces the notion of a categorical pair, a pair of categories (C,C') such that every morphism in C is an object in C'. Categorical pairs are precursors to 2-categories. Arrows in C' can express relationships among the morphisms of C. In particular we show that by using a model of the linguistic process of naming, we can ensure that every morphism in C has an indirect self-reference of the form a -----> Fa where this arrow occurs in the category C'. This result is shown to generalize and clarify known fixed point theorems in logic and categories, and is applied to Goedel's Incompleteness Theorem, the Cantor Diagonal Process and the Lawvere Fixed Point Theorem. In particular we show that the indirect self-reference that is central to Goedel's Theorem is an instance of a general pattern here called the indicative shift.