Universal Algebra and Mathematical Logic

TL;DR

This paper interprets first-order logic within universal algebra using clone theory, defining structures and theories through algebraic constructs, and states key theorems like completeness and incompleteness in this framework.

Contribution

It introduces a novel algebraic framework for first-order logic based on clone theory, connecting logical concepts with universal algebra.

Findings

Defines free clone of terms in first-order logic

Represents structures as valuations of algebraic formulas

States Godel's completeness and incompleteness theorems within this framework

Abstract

In this paper, first-order logic is interpreted in the framework of universal algebra, using the clone theory developed in three previous papers. We first define the free clone T(L, C) of terms of a first order language L over a set C of parameters in a standard way. The free right algebra F(L, C) of formulas over T(L, C) is then generated by atomic formulas. Structures for L over C are represented as perfect valuations of F(L, C), and theories of L are represented as filters of F(L). Finally Godel's completeness theorem and first incompleteness theorem are stated as expected.