
Re: Can addition be defined in terms of multiplication?
Posted:
Aug 18, 2013 6:09 AM


On Sun, 18 Aug 2013, 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. > > Then I think the onus is on you to produced definitions in one or both of > these forms: > x + y = ... > x + y = z <> ... > > where the only nonlogical symbols (baring punctuation) in the ... are from > this set: {*,S,0} or this set: {*,S,0,<}. I wouldn't be surprised if + can be > defined (in the way requested) from {*,S,0} or {*,S,0,<} but I would like > either to see it spelt out, or to be given a reference.
As Jim Burns said z = x + y iff 2^z = 2^x * 2^y
where 2^n is defined by induction 2^0 = 1, 2^1 = 1 and 2^(n+1) = 2*2^n all of which can be done with Peano's axioms.
> > If you want it for the reals, > > Which I don't. > > > then 2^x, x in is > > definable with <= and a whole lot of logical overhand.

