Variations on a Visserian Theme

TL;DR

This paper explores the concept of tightness in first order theories, demonstrating that several foundational theories like Z_2, ZF, and KM are also tight, extending Visser's original theorem about PA.

Contribution

The paper proves that Z_2, ZF, and KM are tight theories, broadening the class of theories known to have this property.

Findings

Z_2, ZF, and KM are tight theories

Extends Visser's theorem from PA to other foundational theories

Shows bi-interpretability implies equality for these theories

Abstract

A first order theory T is said to be "tight" if for any two deductively closed extensions U and V of T (both of which are formulated in the language of T), U and V are bi-interpretable iff U = V. By a theorem of Visser, PA (Peano Arithmetic) is tight. Here we show that Z_2 (second order arithmetic), ZF (Zermelo-Fraenkel set theory), and KM (Kelley-Morse theory of classes) are also tight theories.