Date: Feb 4, 2013 4:14 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 4, 10:26 pm, Charlie-Boo <shymath...@gmail.com> wrote:
> If e(x,y) were a predicate then not(e(x,x)) would be a predicate but
> because of diagonalization it is not.
>
> What I believe about Prolog consists only of its being a database and
> query language similar to predicate calculus (aka FOL.)
>
> 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
> [ not [ e rs rs ]]  = =    [ e rs rs ]               Def = = , if
> ~P = = P                                                 [ e rs rs ]
> => P
>
> Then some axiom about ~P = = P.

Yes, something like

thm( X ) <- if( not(X), P ) ^ if( not(X), not(P) )

but this is forward chaining so it needs to be depth limited or PROLOG
will just recurse the first if(L,R) rule it finds!

>
> In Combinatory Logic:
>
> M x => N x x  Def M
> M M => N M M  x => M
>
> There is no M M = = N M M because it is executed (no Turing Machine
> can halt both yes and no because it stops as soon as it halts either
> cf Rosser 1936.)  M M and N M M occur at different points in time!
> The system simply changes its mind as it goes along learning using AI
> techniques.  The circle is amicable.
>
> In CBL:
>
> M # P(x) / Q(a,b) means program/wff M enumerates/represents set P and
> is written in programming language/logic Q meaning Q(x,y) iff program/
> wff M outputs at some point/is provable on input y.  Then Q(M,x) = =
> P(x) (all x).  Abbreviated M#P/Q means P=Q(M).
>
> P(x) / Q(a,b) means (exists M) M # P(x) / Q(a,b)  There is a program/
> wff that enumerates/represents set P.
>
> Let SE(x,y) iff set x contains element y.
>
> Then the Russell Paradox is:
>
> ~SE(x,x) / SE(a,b)  There is a set of all sets that do not contain
> themselves.
> M # ~SE(x,x) / SE(a,b)  Let M be the set of all sets that do not
> contain themselves.
> SE(M,x) = = ~SE(x,x)  Def # /
> SE(M,M) = = ~SE(M,M)  x => M
> P = = ~P    SE(M,M) => P
>
> This is encapsulated in axiom (lower level theorem) - ~P/P where - E
> means expression E is not true and P/Q abbreviates P(x)/Q(a,b) or
> P(x,x)/Q(a,b) if P is 2-place.  Then if P/Q then P differs from Q and
> we prove truth, provability and unrefutability distinct (Godel,
> Rosser, Smullyan) - see 18 Word Proof in FOM July 2010.
>
> C-B
>

Sure you can syntactically eliminate strings like "~e(X,X)"

in NFU Set Theory.

I'm working on using

NOT( PROVABLE( T ) ) <-> DERIVE( NOT(T) )

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

Herc

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