Definability of Initial Segments

Let T be a first order theory formulated in the language L and P a new relation symbol not in L. Let φ(P ) be an L ∪ {P}-sentence. Let us say that φ(P ) defines P implicitly in T if T proves φ(P ) ∧ φ(P ) → ∀x(P (x) ↔ P (x)). Beth’s definability theorem states that if φ(P ) defines P implicitlly in T then P (x) is equivalent to an L-formula. However, if we consider implicit definability in a given model alone, the situation changes. For a more precise explanation, let us say that a subset A of a given modelM of T is implicitly definable if there exists a sentence φ(P ) such that A is the unique set with (M,A) |= φ(P ). It is easy to find a structure in which two kinds of definability (implicit definability and first order definability) are different. For example, let us consider the structureM = (N∪Z, <), where < is a total order such that any element in the Z-part is greater than any element in the N-part. The N-part is not first order definable in M , because the theory of M admits quantifier elimination after adding the constant 0 (the least element) and the successor function to the language. But the N-part is implicitly definable in M , because it is the unique non trivial initial segment without the last element. On the other hand, for a given structure, we can easily find an elementary extension in which two notions coincide. In this paper, we shall consider implicit definability of the standard part {0, 1, ...} in nonstandard models of Peano arithmetic (PA). It is needless to say that the standard part of a nonstandard model of PA is not first order definable. As is stated