
Re: Can addition be defined in terms of multiplication?
Posted:
Aug 18, 2013 7:13 AM


On Sunday, August 18, 2013 3:17:34 AM UTC7, Peter Percival wrote: > William Elliot wrote: > > > 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. > > > > And the magic formulae > > > > 2^x = ... > > > > 2^x = y <> ... > > >
First you need to define a power sequence..
instead of s(s(s(0)))
t(t(t(1))))
is 2^3

pot [t[t[t 1]]] ?
****** QUERY ****** 1 pot 21 t 221 t 2221 t 2222 1 *******************
1 X (22:1) = t 2 X (22:21) = t 3 X (22:22) = 1 HEAD 1 pot [ t X ] TAIL 1 pot X pot [ t t 1 ]
****** QUERY ****** 1 pot 21 t 221 t 222 1 *******************
4 X (22:1) = t 5 X (22:2) = 1 HEAD 1 pot [ t X ] TAIL 1 pot X pot [ t 1 ]
****** QUERY ****** 1 pot 21 t 22 1 *******************
6 X (:) = 1 HEAD 1 pot [ t X ] TAIL 1 pot X pot [ 1 ]
****** QUERY ****** 1 pot 2 1 *******************
HEAD 1 pot 1 MATCH TRUE 1 MATCH TRUE 1 MATCH TRUE 1 MATCH
as opposed to 'addition based' arithmetic.
http://phpprolog.com/demo/pp7.png http://phpprolog.com/demo/pp9.png http://phpprolog.com/demo/pp1.png
Herc 
www.phpPROLOG.com

