The concept of theories of arithmetic whose integers are the true mathematical integers is captured by ω-logic. Let "T" be a theory in a countable language which includes a unary predicate symbol "N" intended to hold just of the natural numbers, as well as specified names 0, 1, 2, ..., one for each (standard) natural number (which may be separate constants, or constant terms such as 0, 1, 1+1, 1+1+1, ..., etc.). Note that "T" itself could be referring to more general objects, such as real numbers or sets; thus in a model of "T" the objects satisfying "N"("x") are those that "T" interprets as natural numbers, not all of which need be named by one of the specified names.