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Topic: I Bet \$25 to your \$1 (PayPal) That You Can’t P
rove Naive Set Theory Inconsistent

Replies: 20   Last Post: Mar 19, 2013 1:32 PM

 Messages: [ Previous | Next ]
 Graham Cooper Posts: 4,495 Registered: 5/20/10
Re: I Bet \$25 to your \$1 (PayPal) That You Can’t P
rove Naive Set Theory Inconsistent

Posted: Mar 19, 2013 12:27 AM

On Mar 15, 11:58 am, Charlie-Boo <shymath...@gmail.com> wrote:
> On Mar 14, 6:06 pm, Graham Cooper <grahamcoop...@gmail.com> wrote:
>

> > Though not complete in any sense, this is the
>
>  > SMALLEST FORMAL SYSTEM possible - 12 lines of PROLOG.
>
> How do you know that is the smallest possible?
>

Well I did choose an arbitrary rubric of defining:

{ X | X div 2 = 0 }

since it uses a standard 2 line recursive definition, one of which is
in the inference rule form.

NOT & AND form a complete BASE of logic gates.

You could use a single logic gate NAND or NOT

but then I would lose NOT() which I need as it stands for FALSE-WFF.

>
> I can tell you plenty of smaller ones.
>

but somehow you cannot just post them.

(<12 lines remember!)

>
>

> > tru(t).
> > not(f).
> > and(X,Y)            :- tru(X),tru(Y).
> > and(X,not(Y))       :- tru(X),not(Y).
> > and(not(X),Y)       :- not(X),tru(Y).
> > and(not(X),not(Y))  :- not(X),not(Y).
> > even(0).
> > not(and( even(X) , not(even(s(s(X)))) )).
> > e(A, evens) :- tru(even(A)).
> > tru(even(X)) :- even(X).
> > tru(e(A,S)) :- e(A,S).
> > tru(R) :- not(and(L,not(R))) , tru(L).
> > **************************

>
> > by using a small subset of boolean input predicates (and, not)
>
> > You can enter this command into any PROLOG software
>
> > ?-  tru(  e(   s(s(s(s(0)))) ,  evens )).
>
> > YES
>
> > [4 e EVENS]  is a Theorem.
>
> > ***************************

This is a variation of modus ponens

tru(R) :- not(and(L,not(R))) , tru(L).

tru(R) :- if(L,R) , tru(L).

L
L->R
_____
R

Herc

Date Subject Author
3/13/13 Graham Cooper
3/13/13 Graham Cooper
3/14/13 Charlie-Boo
3/14/13 Charlie-Boo
3/14/13 Graham Cooper
3/14/13 Charlie-Boo
3/14/13 Graham Cooper
3/14/13 Charlie-Boo
3/14/13 Graham Cooper
3/15/13 Charlie-Boo
3/19/13 Graham Cooper
3/19/13 Charlie-Boo
3/19/13 Charlie-Boo
3/15/13 Graham Cooper
3/15/13 Charlie-Boo
3/15/13 Graham Cooper
3/19/13 Charlie-Boo