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Re: Peano-like Axioms for the Integers in DC Proof
Posted:
Feb 24, 2013 5:39 PM
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On Feb 25, 7:35 am, Dan Christensen <Dan_Christen...@sympatico.ca>
wrote:
> > 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
AND 2nd ORDER LOGIC!
[DAN] - I make no claim that it is first-order.
...
Sure, given ALL(X) ...formula..
it's so easy!
----------------------------
This just uses DOUBLE VARIABLE NAME COMPATIBLE BINDING
and THE COMPUTER SEARCHES for the Proofs using only 10 lines!
EG. Are all Nats Even?
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).
**** QUERY NOT(ALL(NATS are EVEN)) ***
?- !(subset( nats , evens ))
>>!(!insect((nats, !(evens)).
>> insect(nats,odds).
>> TRUE
?- e(A,nats) , e(A,odds)
A = s(0)
The 10 lines of PROLOG can find the counterexample NUMBER 1!
This *pattern matching*
e(A,nats) :- nat(A).
is an INTERFACE from 4GL "WHAT IF" requests
to 3GL recursive programs in a set theoretic framework.
The SCOPE of syntactic proof checkers tends to be limited to meta-
facts about the theory itself.
Herc
--
www.BLoCKPROLOG.com
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