On the variable hierarchy of first-order spectra

Eryk Kopczynski; Tony Tan

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arXiv · 1403.2225

On the variable hierarchy of first-order spectra

Eryk Kopczynski·Tony Tan

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Abstract

The spectrum of a first-order logic sentence is the set of natural numbers that are cardinalities of its finite models. In this paper we study the hierarchy of first-order spectra based on the number of variables. It has been conjectured that it collapses to three variable. 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.

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Eryk Kopczynski, Tony Tan. 2014-03-10. On the variable hierarchy of first-order spectra. https://arxiv.org/abs/1403.2225

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