On Feb 25, 7:35 am, Dan Christensen <Dan_Christen...@sympatico.ca> wrote: > On Sunday, February 24, 2013 12:13:17 PM UTC-5, Charlie-Boo wrote: > > [snip] > > > > 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. 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 integer-like structure with its own a principle of mathematical induction is actually embedded in such simple structures. Admit it, Charlie, that's pretty cool! > > > If you can show you have a logically different set of axioms than > > > Peano's and they prove Peano's axioms (i.e. they prove what Peano's > > > axioms prove) then that would be cool - even I would be interested. > > > How do you rate Peano's attempt? > > It was a real breakthrough in its day, but it was a essentially an exercise in pattern recognition: Here are some facts about the natural numbers from which it may be possible to derive all others. What I have done differently is to derive these facts/axioms (including induction) starting with nothing more than assuming that there exists an injective fuction defined on a set. I have also derived similar facts/axioms for the integers from only a pair of such functions. > > Dan
Your system is INCONSISTENT!
> > Your system should be able to prove: > > ~p & (~p -> p) -> ~p > Easy. > 1 ~P & [~P => P] > Premise > 2 ~P > Split, 1 > 3 ~P & [~P => P] => ~P > > > > > In fact by modus ponens it is easy to see that p > > derives from ~p & (~p -> p). > Also easy. > 1 ~P & [~P => P] > Premise > 2 ~P > Split, 1 > 3 ~P => P > Split, 1 > 4 P > Detach, 3, 2 > 5 ~P & [~P => P] => P
Really Dan there are dozens of Micro Data structures and fine grain computing models, you are knocking the very System you modelled in a convoluted syntax based on ZFC, a much larger framework than Peano Arithmetic.
Pattern Matching as you put it - UNIFY( f1(a,b,c) f2(a,b,c) )
is considerably more efficient and more utility than syntax checking.
Here is a MICRO TURING MACHINE - only 2 Fetch Cycle commands!