Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-13T07:00:17.429Z Has data issue: false hasContentIssue false

Algebraic theories with definable Skolem functions

Published online by Cambridge University Press:  12 March 2014

Lou van den Dries*
Affiliation:
The Institute for Advanced Study, Princeton, New Jersey 08540

Extract

(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 bR ∪ {∞} maximal such that (a, b) ⊂ D. Four cases have to be distinguished:

(i) a = − ∞ and b = + ∞; then we choose 0;

(ii) a = − ∞ and bR; then we choose b − 1;

(iii) aR and b ∈ = + ∞; then we choose a + 1;

(iv) aR and bR; 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.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1984

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[B]Bacsich, P., Defining algebraic elements, this Journal, vol. 38 (1973), pp. 93101.Google Scholar
[C-D]Cherlin, G. and Dickmann, M., Real closed rings, Annals of Mathematical Logic (to appear).Google Scholar
[C-K]Chang, C. and Keisler, J., Model theory, North-Holland, Amsterdam, 1973.Google Scholar
[D]Delzell, D., A constructive, continuous solution of Hilbert's 17th problem, and other results in semialgebraic geometry, Ph.D. Thesis, Stanford University, Stanford, California, 1980.Google Scholar
[vdD]van den Dries, L., Model theory of fields, Thesis, Utecht, 1978.Google Scholar
[H-J-L]Herrmann, C., Jensen, C. and Lenzing, H., Applications of model theory to representations of finite-dimensional algebras, Mathematische Zeitschrift, vol. 178 (1981), pp. 8398.CrossRefGoogle Scholar
[K.]Kochen, S., Integer valued rational functions over the p-adic numbers: A p-adic analogue of the theory of real fields, Number theory, Proceedings of Symposia in Pure Mathematics, vol. 12, American Mathematical Society, Providence, Rhode Island, 1969, pp. 5773.CrossRefGoogle Scholar
[M]Macintyre, M., On definable subsets of p-adic fields, this Journal, vol. 41 (1976), pp. 605610.Google Scholar
[P]Poizat, B., Une théorie de Galois imaginaire, this Journal, vol. 48 (1983), pp. 11511170.Google Scholar