TY - JOUR

T1 - Definability in rationals with real order in the background

AU - Gurevich, Yuri

AU - Rabinovich, Alexander

PY - 2002/2

Y1 - 2002/2

N2 - 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 φ(X, Y) and ψ(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 φ(Q, J) holds over the real line. The question arises whether, for every φ, F can be defined by means of an appropriate ψ(Y) interpreted over the rational order. We answer the question negatively. The answer remains negative if the first-order logic is strengthened 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.

AB - 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 φ(X, Y) and ψ(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 φ(Q, J) holds over the real line. The question arises whether, for every φ, F can be defined by means of an appropriate ψ(Y) interpreted over the rational order. We answer the question negatively. The answer remains negative if the first-order logic is strengthened 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.

KW - Definability

KW - Expressibility

KW - Monadic logic of order

UR - http://www.scopus.com/inward/record.url?scp=0036465841&partnerID=8YFLogxK

U2 - 10.1093/logcom/12.1.1

DO - 10.1093/logcom/12.1.1

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AN - SCOPUS:0036465841

SN - 0955-792X

VL - 12

SP - 1

EP - 11

JO - Journal of Logic and Computation

JF - Journal of Logic and Computation

IS - 1

ER -