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? 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.
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 )
It's true that I use the successor S, which I hear implicitly uses addition, but without S I don't see how to do much of anything at all.
I wonder how Nam defines * without using either + or S.
Come to think of it, if I can use S, why can't I just go ahead and define + in the usual (so far as I know) fashion x + 0 = x
x + Sy = S(x + y)
It seems to me that, if one is going to do arithmetic without addition, unique prime factorization becomes central. It might be useful to represent numbers by their prime exponents, so that 3 * 4 = 12 becomes ( 0, 1, ...) * ( 2, 0, ...) = ( 2, 1, ...) with special rules for 0, of course. It looks like countably many copies of N, with only finite many copies non-zero. Each copy has its own successor function S2(x) = 2*x, S3(x) = 3*x, ...
However, I am daunted by the prospect of defining * in this system. We would need to give rules that explain why 3 + 4 = 7 or, rather, why ( 0, 1, 0, 0, ...) + ( 2, 0, 0, 0, ...) = ( 0, 0, 0, 1, ...)