```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 likethm( X )  <-   if( not(X),  P )  ^  if( not(X), not(P) )but this is forward chaining so it needs to be depth limited or PROLOGwill 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 usingNOT( PROVABLE( T ) )  <->  DERIVE( NOT(T) )EXISTS( SET( S ) )  <->  PROVABLE ( EXIST (SET (S) ) )AXIOM OF SET SPECIFICATIONHerc--www.BLoCKPROLOG.com
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