> Jim Burns wrote: > > On 8/16/2013 4:54 AM, 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? > > > > > > 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. > > > > How about > > x + y = z <-> 2^x * 2^y = 2^z > > > > where 2^x is just an abbreviation for the function 2pwr: N -> N, > > defined by > > 2pwr(0) = 1 > > 2pwr( Sx ) = 2 * 2pwr( x ) > That goes beyond what I defined as the language of arithmetic.
It does not. It quite definable with Peano's axioms which may be presumed to be what you intend because of the inclusion of S in the symbols of arithematic.
If you want it for the reals, then 2^x, x in is definable with <= and a whole lot of logical overhand.