In any nonstandard model of Peano arithmetic, the standard part is not first-order definable. But we show that in some model the standard part is definable as the unique solution of a formula $\varphi(P)$, where P is a unary predicate variable.
"Definability of Initial Segments." Notre Dame J. Formal Logic 43 (2) 65 - 73, 2002. https://doi.org/10.1305/ndjfl/1071509428