![]() ![]() This should imply that it is definable in Hyperbolic geometry, but one would need extra arguments to see it directly. Once we know that $\rho$ is semialgebraic we know that the equidistance relation on our Hyperbolic model of Euclidean geometry is semialgebraic. I think it should be pretty clear that $\rho$ is semialgebraic. (On Wikipedia there is a factor of $2$ in from the arcsinh, but that is just a scaling factor, so we can drop it). The Hyperbolic distance $r$ of $p$ from the origin is not a semialgebraic function of $p$, but it is the arcsinh of a semialgebraic functions, see Wikipedia. Tarski works in a system where the domain is $\mathbb, ,\times,<)$. In my opinion the right axioms for Euclidean/Hyperbolic geometry are the Tarski axioms. There are a lot of things to check here and I haven't. Here is an expanded version of my previous comments. Walsberg's comment is really an answer both to the question as titled and to my actual concern, though he correctly saw I had another issue about methods also in mind. Some of my text concerning synthetic methods was less relevant (though those questions too intrigue me). So my title was true to my actual concern. The work that led me to this question is about interpretation in first order logic, and not about compass (or horocompass) and straightedge or other such construction methods, and really not about avoiding coordinates. ![]() First order algebraic considerations on fields (notably pythagorean fields) are already implicit in Euclid and central to successful first-order elementary geometry. It is between using or not using higher order notions like point-set continuity and limits. Hilbert, Tarski, and Greenberg all show that the important logical distinction characterizing elementary methods is not between using or not using coordinates in some field. My text was ambiguous about "synthetic" methods, in just the way that Tarski What is elementary geometry? means when he says "In colloquial language the term elementary geometry is used loosely no well determined meaning." Judging from the ones I do have (including Hartshorne) it seems likely that his discussion of equiconsistency for the two geometries (pp.213-214) refers to analytic presentations of geometry.Ĭan I find mutually interpretable axioms for synthetic Euclidean and hyperbolic geometry?Įdit: Erik Walsberg's comment about Tarski's axioms answers my title question, even though not in the way I had in mind when I wrote the text. But I do not have all his references at hand. I know Greenberg's discussion of axiomatic issues in Greenberg. Maybe the analytic equiconsistency proofs really do not transfer well to synthetic geometry. Notably $n$-section of lines, see trisection-of-a-hyperbolic-line-segment. The only difference, in these axiomatizations, between Euclidean and hyperbolic geometry is in the axiom of parallels. But this synthetic hyperbolic geometry is incapable of some constructions that we take pretty much for granted both in Euclidean geometry and also (more to my point) in analytic hyperbolic geometry. ![]() They omit the non-elementary axiom of continuity, and they add something to assure that circles will actually have points of intersections with lines and other circles that they cross. The most common rigorous axiomatizations for synthetic Euclidean and hyperbolic geometry, so far as I can tell, are Hilbert's from his Foundations of Geometry, with two changes. Looking into the literature, though, I wonder if I am too optimistic about this. Those proofs are so robustly geometric that it seems like they must have synthetic analogues. Operator.There are familiar analytic equiconsistency proofs for Euclidean and hyperbolic geometry. Structures warped into nanocones by a disclination the nonzero size of theĭisclination is taken into account, and a boundary condition at the edge of theĭisclination is chosen to ensure self-adjointness of the Dirac-Weyl Hamiltonian In the crystal lattice, we consider quantum ground state effects in monolayer The tight-binding approximation for the nearest neighbor interaction of atoms The continuum model for long-wavelength electronic excitations, originating in ![]() Space, then a variety of quantum effects is induced in the vacuum. Sitenko and 1 other authors Download PDF Abstract: Space out of a topological defect of the Abrikosov-Nielsen-Olesen vortex type Download a PDF of the paper titled Non-Euclidean geometry, nontrivial topology and quantum vacuum effects, by Yurii A. ![]()
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