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Topic: Peano-like Axioms for the Integers in DC Proof
Replies: 11   Last Post: Mar 3, 2013 6:13 PM

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 Graham Cooper Posts: 4,495 Registered: 5/20/10
Re: Peano-like Axioms for the Integers in DC Proof
Posted: Feb 24, 2013 5:39 PM

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

Date Subject Author
2/23/13 Graham Cooper
2/24/13 Graham Cooper
2/24/13 Graham Cooper
2/28/13 Graham Cooper
3/3/13 Graham Cooper
2/24/13 Graham Cooper