First order logic without equality on relativized semantics

Abstract

Let α ≥ 2 be any ordinal. We consider the class Drsα of relativized diagonal free set algebras of dimension α. With same technique, we prove several important results concerning this class. Among these results, we prove that almost all free algebras of Drsα are atomless, and none of these free algebras contains zero-dimensional elements other than zero and top element. The class Drsα corresponds to first order logic, without equality symbol, with α-many variables and on relativized semantics. Hence, in this variation of first order logic, there is no finitely axiomatizable, complete and consistent theory.2010 Mathematics Subject Classification. Primary 03G15, 03B45, 03C95. Secondary 03G25, 03C05, 08B20.

Distances Between Formal Theories

“…In a future algebra-oriented article, we plan to discuss in detail the correspondence between our logical definition herein and the above algebraic idea. We note that it happens quite often that one obtains interesting results in mathematical logic by using algebraic tools, e.g., [17], [7], [1], [9], and [22]. We start with a quick review for the notions of logic that we are going to use.…”