First order logic without equality on relativized semantics

TL;DR

This paper investigates relativized diagonal free set algebras of dimension , showing that most free algebras are atomless and lack zero-dimensional elements, implying certain logical limitations in the associated first-order logic without equality.

Contribution

It proves that almost all free algebras in  are atomless and contain no zero-dimensional elements other than zero and top, revealing limitations of the corresponding logic.

Findings

Almost all free algebras are atomless.

Free algebras lack zero-dimensional elements besides zero and top.

No finitely axiomatizable, complete, and consistent theory exists in this logic.

Abstract

Let $\alpha\geq 2$ be any ordinal. We consider the class $\mathsf{Drs}_{\alpha}$ of relativized diagonal free set algebras of dimension $\alpha$. With same technique, we prove several important results concerning this class. Among these results, we prove that almost all free algebras of $\mathsf{Drs}_{\alpha}$ are atomless, and none of these free algebras contains zero-dimensional elements other than zero and top element. The class $\mathsf{Drs}_{\alpha}$ corresponds to first order logic, without equality symbol, with $\alpha$-many variables and on relativized semantics. Hence, in this variation of first order logic, there is no finitely axiomatizable, complete and consistent theory.