Undefinability of addition from one unary operator

Abstract

It is the object of this paper to prove that the binary operator of addition of the natural numbers is not arithmetically definable in terms of a single unary operator. An arithmetical (or elementary) definition is one in which no variables ranging over sets of natural numbers are permitted; all variables range over just the natural numbers themselves. It is actually easier to prove something more than this: that a single unary operator will not suffice even when any number of one-place predicates of natural numbers are added. The method of proof is by elimination of quantifiers, originally due to Presburger. A by-product of the method used is the subsidiary result that addition is not definable without quantifiers in terms of any set of unary operators, one-place predicates and two-place predicates. The interpreted well-formed formulas herein considered have the following as symbols: =, identity, interpreted in the usual way; f, a unary functor, interpreted as a unary operator over the natural numbers; truth functions and quantifiers; and predicate letters, each interpreted as a definite property of natural numbers. a, b, c, d, x, y, z are variables. A term will be either a variable x or f'(x), i.e.,f(f (... (x) ... )), in which f occurs i times; thusfo(x) is simply the variable x. t and s, with and without subscripts and superscripts, will be arbitrary terms. The symbols T and I are propositional constants standing, respectively, for truth and falsity. m, n, h, i, j, k are natural numbers. The class of formulas that are allowed can be made precise by the following recursive characterization: (1) if tl, t2, .. are terms and F is an n-ary predicate letter (n > 1) then t1 = t2, Ft, ... tn, T and I are allowed formulas; (2) if 0, and 02 are formulas and x is a variable then -((D), (DI & 02, ( 3x)DI and (x)D I are all formulas; and (3) nothing else is a formula.