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Definability and undefinability with real order at the background

Published online by Cambridge University Press:  12 March 2014


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

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 : PA 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
Copyright
Copyright © Association for Symbolic Logic 2000

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References

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