
Re: Can addition be defined in terms of multiplication?
Posted:
Aug 18, 2013 6:12 PM


On Friday, August 16, 2013 1:54:40 AM UTC7, 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 <> ... > >
Yes but you just get an equivalent formula for standard peano addition.
X + 0 = X X + s(Y) = s(Z)

This works in www.phpPROLOG.com
n^(X) is just a Prefix Function symbol like f(x)
PROGRAM LIST
1 + [ n^ X ] [ n^ 0 ] [ n^ X ]
2 + [ n^ X ] [ n^ [ s Y ] ] [ n^ [ s Z ] ] : + [ n^ X ] [ n^ Y ] [ n^ Z ]
3 + X Y Z : + [ n^ X ] [ n^ Y ] [ n^ Z ]
TRACE + [s 0] [s[s 0]] ANS ?
HEAD 1 + X Y Z TAIL 1 + n^ X n^ Y n^ Z + n^ [ s 0 ] n^ [ s s 0 ] n^ Z HEAD 1 + [ n^ X ] [ n^ [ s Y ] ] [ n^ [ s Z ] ] TAIL 1 + n^ X n^ Y n^ Z + n^ [ s 0 ] n^ [ s 0 ] n^ Z HEAD 1 + [ n^ X ] [ n^ [ s Y ] ] [ n^ [ s Z ] ] TAIL 1 + n^ X n^ Y n^ Z + n^ [ s 0 ] n^ [ 0 ] n^ Z HEAD 1 + [ n^ X ] [ n^ 0 ] [ n^ X ] MATCH TRUE 1 MATCH TRUE 1 MATCH TRUE 1 MATCH
+ [s 0] [s[s 0]] ANS ?
ANS = s s s 0
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