On 16/08/2013 09:54, Peter Percival wrote: > Can addition be defined in terms of multiplication? I.e., is there a > formula in the language of arithmetic > > x + y = z <-> ... > > such that in '...' any of the symbols of arithmetic except + may occur? > Or, alternatively, is there a formula in the language of arithmetic > > x + y = ... > > with the same requirement? > > The symbols of arithmetic (for the purpose of this question) are either > > individual variables, (classical) logical constants including =, > S, +, *, and punctuation marks; > > or the above with < as an additional binary predicate symbol.
IIRC "arithmetic without addition" is a decidable theory (of course Presburger's famous theorem is that "arithmetic without multiplication" is decidable). I think this means that the theory of the language (0,1,*,=) in N is decidable. I'm not sure if we can allow S and/or <.
Of course if we have a language like this with a decidable theory, then we can't define addition, lest the full theory of Peano arithmetic be decidable.