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arXiv · 1807.00690

First order logic without equality on relativized semantics

Amitayu Banerjee·Mohamed Khaled
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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.

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Amitayu Banerjee, Mohamed Khaled. 2018-06-29. First order logic without equality on relativized semantics. https://arxiv.org/abs/1807.00690

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