On the Syntax of Logic and Set Theory

TL;DR

This paper extends propositional calculus with predicate abstracts and quantifiers, introduces a new consistent set theory framework, and explores its relation to classical systems and foundational paradoxes.

Contribution

It presents a novel logical system with a unified rule and comprehension schema, avoiding traditional set theory postulates, and analyzes its consistency and relation to existing theories.

Findings

Proves the system's consistency in finite Boolean lattices

Addresses classical set-theoretic paradoxes within the new framework

Discusses methods for higher order quantification and abstraction

Abstract

We introduce an extension of the propositional calculus to include abstracts of predicates and quantifiers, employing a single rule along with a novel comprehension schema and a principle of extensionality, which are substituted for the Bernays postulates for quantifiers and the comprehension schemata of ZF and other set theories. We prove that it is consistent in any finite Boolean subset lattice. We investigate the antinomies of Russell, Cantor, Burali-Forti, and others, and discuss the relationship of the system to other set theoretic systems ZF, NBG, and NF. We discuss two methods of axiomatizing higher order quantification and abstraction, and then very briefly discuss the application of one of these methods to areas of mathematics outside of logic.