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Definability and undefinability with real order at the background

Published online by Cambridge University Press:  12 March 2014

Yuri Gurevich  and
Alexander Rabinovich
Yuri Gurevich
Affiliation:
Microsoft Research, One Microsoft Way, Redmond, WA 98052-6399, USA
Alexander Rabinovich
Affiliation:
Department of Computer Science, Beverly Sackler School of Exact Sciences, Tel Aviv University, Israel69978
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Extract

We consider the monadic second-order theory of linear order. For the sake of brevity, linearly ordered sets will be called chains.

Let = ⟨A <⟩ be a chain. A formula ø(t) with one free individual variable t defines a point-set on A which contains the points of A that satisfy ø(t). As usually we identify a subset of A with its characteristic predicate and we will say that such a formula defines a predicate on A.

A formula (X) one free monadic predicate variable defines the set of predicates (or family of point-sets) on A that satisfy (X). This family is said to be definable by (X) in A. Suppose that is a subchain of = ⟨B, <⟩. With a formula (X, A) we associate the following family of point-sets (or set of predicates) {P : P ⊆ A and (P, A) holds in } on A. This family is said to be definable by in with at the background.

Note that in such a definition bound individual (respectively predicate) variables of range over B (respectively over subsets of B). Hence, it is reasonable to expect that the presence of a background chain allows one to define point sets (or families of point-sets) on A which are not definable inside .

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 65 , Issue 2 , June 2000 , pp. 946 - 958
Copyright
Copyright © Association for Symbolic Logic 2000

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References

REFERENCES

[1]Büchi, J. R., On a decision method in restricted second order arithmetic, Proceedings of the international congress on logic, methodology and philosophy of science (Nagel, E.et al., editors), Stanford University Press, 1960, pp. 1–11.Google Scholar
[2]Gurevich, Y., Modest theory of short chains, I, This Journal, vol. 44 (1979), pp. 481–490.Google Scholar
[3]Gurevich, Y., Monadic second order theories, Model theoretical logics (Barwise, J. and Feferman, S., editors), Springer Verlag, 1986, pp. 479–506.Google Scholar
[4]Gurevich, Y. and Shelah, S., Monadic theory of order and topology in ZFC, Annals of Mathematical Logic, vol. 23 (1982), pp. 179–198.CrossRefGoogle Scholar
[5]Rabin, M. O., Decidable theories, Handbook of mathematical logic (Barwise, J., editor), North-Holland, 1977.Google Scholar
[6]Shelah, S., The monadic theory of order, Annals of Mathematics, vol. 102 (1975), pp. 349–419.CrossRefGoogle Scholar
[7]Thomas, W., Ehrenfeucht games, the composition method, and the monadic theory of ordinal words, Structures in logic and computer science: A selection of essays in honor of A. Ehrenfeucht, Lecture Notes in Computer Science, no. 1261, Springer-Verlag, 1997, pp. 118–143.CrossRefGoogle Scholar
[8]Trakhtenbrot, B. A. and Barzdin, Y. M., Finite automata, North Holland, Amsterdam, 1973.Google Scholar