A note on the expressive power of linear orders

TL;DR

This paper demonstrates that certain pairs of linear orders can replicate the expressive power of first-order logic with the Bit predicate, revealing insights into the expressive capabilities of linear orders in logic.

Contribution

It shows that specific linear orders can match the expressive power of FO(Bit), introducing new perspectives on the relationship between linear orders and logical expressiveness.

Findings

Existence of two linear orders equivalent to FO(Bit)

A built-in permutation can replicate FO(Bit)

Linear orders can encode Bit-predicate logic

Abstract

This article shows that there exist two particular linear orders such that first-order logic with these two linear orders has the same expressive power as first-order logic with the Bit-predicate FO(Bit). As a corollary we obtain that there also exists a built-in permutation such that first-order logic with a linear order and this permutation is as expressive as FO(Bit).