Investigations of isotropy and homogeneity of spacetime in first-order logic

We investigate the logical connection between (spatial) isotropy, homogeneity of space, and homogeneity of time within a general axiomatic framework. We show that isotropy not only entails homogeneity of space, but also, in certain cases, homogeneity of time. In turn, homogeneity of time implies hom...

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Bibliographic Details
Published in:Annals of pure and applied logic 2022-10, Vol.173 (9), p.103153, Article 103153
Main Authors: Madarász, Judit X., Stannett, Mike, Székely, Gergely
Format: Article
Language:English
Subjects:
Axiomatisation
Classical spacetime
First-order logic
Homogeneity
Isotropy
Relativity theory
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Summary:We investigate the logical connection between (spatial) isotropy, homogeneity of space, and homogeneity of time within a general axiomatic framework. We show that isotropy not only entails homogeneity of space, but also, in certain cases, homogeneity of time. In turn, homogeneity of time implies homogeneity of space in general, and the converse also holds true in certain cases. An important innovation in our approach is that formulations of physical properties are simultaneously empirical and axiomatic (in the sense of first-order mathematical logic). In this case, for example, rather than presuppose the existence of spacetime metrics – together with all the continuity and smoothness apparatus that would entail – the basic logical formulas underpinning our work refer instead to the sets of (idealised) experiments that support the properties in question, e.g., isotropy is axiomatised by considering a set of experiments whose outcomes remain unchanged under spatial rotation. Higher-order constructs are not needed.
ISSN:0168-0072
DOI:10.1016/j.apal.2022.103153