Definability in Rationals with Real Order in the Background

The paper deals with logically definable families of sets (or point-sets) of rational numbers. In particular we are interested whether the families definable over the real line with a unary predicate for the rationals are definable over the rational order alone. Let (X, Y ) and (Y ) range over formulas in the first-order monadic language of order. Let Q be the set of rationals and F be the family of subsets J of Q such that (Q, J) holds over the real line. The question arises whether, for every , F can be defined by means of an appropriate (Y ) interpreted over the rational order. We answer the question negatively. The answer remains negative if the firstorder logic is strengthen to weak monadic second-order logic. The answer is positive for the restricted version of monadic second-order logic where set quantifiers range over open sets. The case of full monadic second-order logic remains open.

• Type: Journal
• Year: 2002
• Where: LOGCOM
• Authors: Yuri Gurevich, Alexander Moshe Rabinovich
• Keywords: First-order Monadic Language, LOGCOM 2002, Rational Order, Real Line