```Date: Feb 23, 2013 12:53 AM
Author: Graham Cooper
Subject: Re: Peano-like Axioms for the Integers in DC Proof

>> I don't know if they are anything new, but I have presented a set of axioms for the integers based on a clearly defined successor function. It differs considerably from the usual presentations that you see online. As such, I thought readers might be interested.>> Also, I'm not sure how these other sets of axioms were arrived at other than simply being as a wishlist of requirements for integer arithmetic that just seems to work.And they DO work, as is!nat(0).nat(s(X)) :- nat(X).even(0).even(s(s(X))) :- even(X).odd(s(0)).odd(s(s(X))) :- odd(X).e(A,nats) :- nat(A).e(A,evens) :- even(A).e(A,odds) :- odd(A).insect(S1,S2) :- e(A,S1),e(A,S2).?- insect(nats,evens).?- insect(nats,odds).This will return  TRUE TRUE on any PROLOG machine.******************************?- e(A, nats) , e(A, evens)will returnA = 0A = s(s(0))A = s(s(s(s(0))))A = s(s(s(s(s(s(0))))))*******************************?- e(A,nats) , e(A,odds)will returnA = s(0)A = s(s(s(0)))A = s(s(s(s(s(0)))))A = s(s(s(s(s(s(s(0)))))))NOTE:  ?-insect( odds, evens )  will CRASH!I need to get a TRACE working to see how Prolog solves simultaneousequations..[DAN] My axioms, however, were justified by the  application of formalaxioms and rules of logic and set theory beginning with relativelysimple structures as I have described here. I show that that aninteger-like structure with its own a principle of mathematicalinduction is actually embedded in such simple structures. Admit it,Charlie, that's pretty cool!The principle of mathematical induction for integers14  ALL(p):[Set(p)    & ALL(a):[a @ p => a @ z]    & 0 @ p    & ALL(a):[a @ p => next(a) @ p]    & ALL(a):[a @ p => next'(a) @ p]    => ALL(a):[a @ z => a @ p]][HERC]So succ(X) is next(X)and right is  1,2,3,4...and left is -1,-2,-3,-4....I like how you define the Domain z for the inductive predicate    & ALL(a):[a @ p => a @ z]    => ALL(a):[a @ z => a @ p]]Herc--www.BLoCKPROLOG.com
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