
Re: Can addition be defined in terms of multiplication?
Posted:
Aug 16, 2013 12:14 PM


On Fri, 16 Aug 2013, Robin Chapman wrote:
> 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?
A similar problem I have asked some years ago is the following:
Given a multiplication on a set (e.g. defined as a commutative and associative operation allowing cancellation (ab = ac implies b = c)), is there an addition so that the set becomes a ring with both operations? I have no clue how to tackle such questions.
An example: Let M = {x elem Z : x == 1 mod 3} with ordinary multiplication. Could this be the multiplication in a ring, if addition is suitably defined? I guess, no, but it is but a guess.
> 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.
I'll look into this argument a bit closer.
 Helmut Richter

