Semantic characterizations of number theories

Abstract: "We show that a number-theoretic formula is a theorem of First-Order Arithmetic if and only it is true, as a statement about numbers, in all Henkin-structures that are closed under abstract jump (i.e. strict II1/1 definitions), and that a number-theoretic formula is a theorem of Arithmetic with existential induction if and only if it is true in all Henkin-structures that contain their abstract RE (i.e. strict-II1/1 definable) sets."