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- W2010971114 abstract "(1.1) A well-known example of a theory with built-in Skolem functions is (first-order) Peano arithmetic (or rather a certain definitional extension of it). See [C-K, pp. 143, 162] for the notion of a theory with built-in Skolem functions, and for a treatment of the example just mentioned. This property of Peano arithmetic obviously comes from the fact that in each nonempty definable subset of a model we can definably choose an element, namely, its least member. (1.2) Consider now a real closed field R and a nonempty subset D of R which is definable (with parameters) in R. Again we can definably choose an element of D , as follows: D is a union of finitely many singletons and intervals ( a, b ) where – ∞ ≤ a < b ≤ + ∞; if D has a least element we choose that element; if not, D contains an interval ( a, b ) for which a ∈ R ∪ { − ∞}is minimal; for this a we choose b ∈ R ∪ {∞} maximal such that ( a, b ) ⊂ D . Four cases have to be distinguished: (i) a = − ∞ and b = + ∞; then we choose 0; (ii) a = − ∞ and b ∈ R ; then we choose b − 1; (iii) a ∈ R and b ∈ = + ∞; then we choose a + 1; (iv) a ∈ R and b ∈ R ; then we choose the midpoint ( a + b )/2. It follows as in the case of Peano arithmetic that the theory RCF of real closed fields has a definitional extension with built-in Skolem functions." @default.
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- W2010971114 title "Algebraic theories with definable Skolem functions" @default.
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