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Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle
to Resolve Several Paradoxes
Posted:
Feb 5, 2013 9:10 AM
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On Feb 4, 4:14 pm, Graham Cooper <grahamcoop...@gmail.com> wrote:
> 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) )
What does that say in English?
> 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)"
And don't forget strings that imply it.
> 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
What does this say in English?
C-B
> Herc
>
> --www.BLoCKPROLOG.com
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