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

160 johannes hafner and paolo mancosupolynomial of odd degree has a root. More formally, a real closed field isdefined by the following axioms (the complete list can be found in theAppendix to this chapter).(i) Axioms for fields(ii) Order axioms(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)The theory of real closed fields, RCF, is the deductive closure of the aboveaxioms. A (first-order) sentence formulated in the language of RCF is called an‘elementary sentence’. An important result about RCF is that there is a decisionprocedure due to Tarski and Seidenberg (Tarski, 1951; Seidenberg, 1954), i.e.given any elemantary sentence ϕ the algorithm outputs 1, ifRCF proves ϕand 0, ifRCF proves ¬ϕ. Also, this decision procedure works uniformly forevery sentence ϕ in the language of RCF.Examples of real closed fields include the set of real numbers R and theset of real algebraic numbers R alg , i.e. the set of all roots of the nonzeropolynomials with rational coefficients. The latter field is not complete, i.e.not every Cauchy sequence converges. The set C of complex numbersis not a real closed field as it does not admit of an ordering. There isan abundance of real closed fields even just within R. Artin and Schreiershowed that there are uncountably many pairwise non-isomorphic real closedsubfields of R whose algebraic closure is isomorphic to C (cf. Brumfiel 1979,p. 131).For a real closed field R we denote the ring of polynomials in n variableswith coefficients from R by R[X 1 , ... X n ].AsubsetofR n , the affine n-space over R, is called semi-algebraic if it belongsto the smallest family of subsets of R n containing all sets of the form{x ∈ R n | f (x) >0}, f (x) ∈ R[X 1 , ... X n ]and which is closed under taking finite intersections, finite unions, andcomplements.Semi-algebraic subsets of R (i.e. one-dimensional) are exactly the finiteunions of points and open intervals (bounded or unbounded).Let A ⊂ R m and B ⊂ R n be two semi-algebraic sets. A mapping f : A → Bis semi-algebraic if its graph is semi-algebraic in R m+n . It is clear that polynomials(on semi-algebraic domains) are semi-algebraic mappings.R n can be endowed with the Euclidean topology coming from the orderingon R. Letx = (x 1 , ... , x n ) ∈ R n , r ∈ R, r > 0. We set