Mancosu - Philosophy of Mathematical Practice (Oxford, 2008).pdf

eyond unification 177AppendixAxioms for real closed fields(i) Axioms for fields∀x∀y∀z(x + (y + z) = (x + y) + z)∀x∀y(x + y = y + x)∀x(x + 0 = x)∀x∃y(x + y = 0)∀x∀y∀z(x · (y · z) = (x · y) · z)∀x∀y(x · y = y · x)∀x(x · 1 = x)∀x∀y∀z(x · (y + z) = (x · y) + (x · z)∀x(x ̸= 0 →∃y(x · y = 1)0 ̸= 1(ii) Order axioms∀x∀y∀z(x ≤ y&y ≤ z → x ≤ z)∀x∀y(x ≤ y&y ≤ x → x = y)∀x(x ≤ x)∀x∀y(x ≤ y ∨ y ≤ x)∀x∀y∀z(x ≤ y → x + z ≤ y + z)∀x∀y∀z(x ≤ y&0 ≤ z → x · z ≤ y · z)(iii) ∀x∃y(x = y 2 ∨−x = y 2 )(iv) For each natural number n, the axiom∀x 0 ∀x 1 ... ∀x 2n ∃y(x 0 + x 1 · y + x 2 · y 2 + ... + x 2n · y 2n + y 2n+1 = 0)BibliographyAndradas, C., Bröcker,L.,andRuiz,J.M.(1996), Constructible Sets in Real Geometry(Berlin: Springer).Bochnak, J., Coste, M.,andRoy, M.-F. (1998), Real Algebraic Geometry (Berlin:Springer).Brumfiel, Gregory W. (1979), Partially Ordered Rings and Semi-Algebraic Geometry(Cambridge: CUP).Courant, Richard (1971), Vorlesungen über Differential- und Integralrechnung, 2 vols. 4thedn (Berlin: Springer).