Date: Feb 5, 2013 5:34 PM
Author: Graham Cooper
Subject: Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle<br> to Resolve Several Paradoxes

On Feb 6, 12:10 am, Charlie-Boo <shymath...@gmail.com> wrote:
> On Feb 4, 4:14 pm, Graham Cooper <grahamcoop...@gmail.com> wrote:

>>  thm( X )  <-   if( not(X),  P )  ^  if( not(X), not(P) )
>What does that say in English?


Negate an Hypothesis and prove it by contradiction

(~x->p) & (~x->~p)  --> x

[1]


>  >  Sure you can syntactically eliminate strings like  "~e(X,X)"
>
> And don't forget strings that imply it.
>
> > in NFU Set Theory.
>
> > I'm working on using
>
>  >  NOT( PROVABLE( T ) )  <->  DERIVE( NOT(T) )

[2]

a WFF 'T' is not "provable" (a True or False predicate)
if there is a proof of NOT(T)

e.g.  T = EXIST(rs) rs={x|x~ex}

T is false
hence there cannot be a proof of T


>
>  >  EXISTS( SET( S ) )  <->  PROVABLE ( EXIST (SET (S) ) )
>  >  AXIOM OF SET SPECIFICATION

[3]


>
> What does this say in English?
>

Set existence is dependant upon that set's existence being provable.

-------------------------------------------------------

e.g.
   PROOF(  NOT(EXIST(RUSSELLS-SET))    by [1]
   NOT(PROVABLE(RUSSELLS-SET))         by [2]
   NOT(EXIST(RUSSELLS-SET))        by [3]


---------------------------------

So far my PROLOG FORMAL SYSTEM has 1 type - PREDICATES
with 2 variations

thm( .... predicate...)
and
not(.... predicate...)

I think by adding a SETorELEMENT  type:

[X1]
exist( rs )

I can finish my PROVABLE-SET-THEORY by using:

[X2]
e(MEM,SET)  :-  exist(MEM)  ,  exist(SET)


then PROLOG should be able to INSTANTIATE rs
into its e(X,X) definition just like you did here:

> Then I take it that:
> if    [ not [ e rs rs ]]      [ e rs rs ]              Line 1 X => rs
> if    [ e rs rs ]               [ not [ e rs rs ]]     Line 2 X => rs


by using rule [X2]  and the fact  exist(rs)

it should be able to derive the contradiction...
and keep the Theory consistent via that!


Herc