Undefinability of addition from one unary operator

Robert McNaughton
1965 Transactions of the American Mathematical Society  
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
more » ... f 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;/, a unary functor, interpreted as a unary operator over the natural numbers; truth functions and quantifiers; and predicate letters,
doi:10.1090/s0002-9947-1965-0176923-1 fatcat:3vcbagrt2jgehop4dsu3bv653u