On Sun, 18 Aug 2013 02:18:14 -0700, William Elliot <firstname.lastname@example.org> wrote:
>> 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.
You simply don't know what "a definition in the language of arithmetic" means. But don't let that stop you.
> >If you want it for the reals, then 2^x, x in is >definable with <= and a whole lot of logical overhand.