
Re: Peanolike Axioms for the Integers in DC Proof
Posted:
Feb 23, 2013 12:53 AM


> > 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 return
A = 0 A = s(s(0)) A = s(s(s(s(0)))) A = s(s(s(s(s(s(0))))))
*******************************
? e(A,nats) , e(A,odds)
will return
A = 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 simultaneous equations..
[DAN]
My axioms, however, were justified by the application of formal axioms and rules of logic and set theory beginning with relatively simple structures as I have described here. I show that that an integerlike structure with its own a principle of mathematical induction is actually embedded in such simple structures. Admit it, Charlie, that's pretty cool!
The principle of mathematical induction for integers 14 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

