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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-placedoi:10.1090/s0002-9947-1965-0176923-1 fatcat:3vcbagrt2jgehop4dsu3bv653u