On the Variable Hierarchy of First-Order Spectra

Abstract

The spectrum of a first-order logic sentence is the set of natural numbers that are cardinalities of its finite models. In this article, we study the hierarchy of first-order spectra based on the number of variables. It has been conjectured that it collapses to three variables. We show the opposite: it forms an infinite hierarchy. However, despite the fact that more variables can express more spectra, we show that to establish whether the class of first-order spectra is closed under complement, it is sufficient to consider sentences using only three variables and binary relations.