The Definitional Equivalence of Gal(1, 3) and Gal F (1, 3) Jeffrey Ketland * April 25, 2023 Abstract There has been some recent interest and debate about the notion of equiva- lence for physical theories. In particular, one notion of equivalence of theories (as axiomatic systems) concerns the notion of definitional equivalence. Some au- thors, though, have raised doubts about whether such equivalences are sufficient for genuine, or perhaps metaphysical equivalence (North (2021)). Here we focus on Galilean spacetime. In a separate article Ketland (2023), a second-order axioma- tization Gal(1, 3) of Galilean spacetime was given. In Field (1980), Hartry Field gave slightly different axioms for Galilean spacetime, which we call Gal F (1, 3). The primitive concepts of these theories are {Bet, ∼, ≡ ∼ } and {Bet, ∼, ≡ S } respectively, where the congruence primitives, ≡ ∼ and ≡ S , are distinct. In this article, we show that the primitives ≡ ∼ and ≡ S are inter-definable, and that the axiom systems Gal(1, 3) and Gal F (1, 3) are definitionally equivalent. It is then argued that this is a case where the definitional equivalence is genuine physical equivalence. Keywords: Galilean spacetime, Theoretical equivalence. Contents 1 Synthetic Geometry and Spacetime 2 2 The Standard Coordinate Structures 5 3 Gal(1, 3) and Gal F (1, 3) 6 4 Proving the Axioms of Gal(1, 3) in Gal F (1, 3) 10 5 Proving the Axioms of Gal F (1, 3) in Gal(1, 3) 16 6 The Equivalence: ≡ ∼ versus ≡ S 18 * Faculty of Philosophy, University of Warsaw, 3 Krakowskie Przedmieście, 00-927 Warszawa, POLAND. Email: jeffreyketland@gmail.com. 1