
Re: This is False. 0/0 {x  x ~e x} e {x  x ~e x} A single Principle to Resolve Several Paradoxes
Posted:
Feb 3, 2013 11:19 PM


On Feb 3, 9:02 pm, Graham Cooper <grahamcoop...@gmail.com> wrote: > On Feb 4, 9:12 am, CharlieBoo <shymath...@gmail.com> wrote: > > > > > > > > > > > On Feb 3, 4:56 pm, Graham Cooper <grahamcoop...@gmail.com> wrote: > > > > On Feb 4, 7:18 am, CharlieBoo <shymath...@gmail.com> wrote: > > > > > On Feb 3, 4:03 pm, Graham Cooper <grahamcoop...@gmail.com> wrote: > > > > > > On Feb 4, 3:01 am, CharlieBoo <shymath...@gmail.com> wrote: > > > > > > > On Feb 1, 3:35 pm, Graham Cooper <grahamcoop...@gmail.com> wrote: > > > > > > > > On Feb 2, 4:09 am, CharlieBoo <shymath...@gmail.com> wrote: > > > > > > > > > There is a peculiar parallel between Semantic Paradoxes, Set Theory > > > > > > > > Paradoxes and ordinary formal Arithmetic. > > > > > > > > > Consider the following 3 pairs of expressions in English, Set Theory > > > > > > > > and Mathematics: > > > > > > > > > A > > > > > > > > This is false. > > > > > > > > This is true. > > > > > > > > > B > > > > > > > > 1/0 > > > > > > > > 0/0 > > > > > > > > > C > > > > > > > > {x  x ~e x} e {x  x ~e x} > > > > > > > > {x  x e x} e {x  x ~e x} > > > > > > > > {x  x ~e x} e {x  x e x} > > > > > > > > {x  x e x} e {x  x e x} > > > > > > > > > A is the Liar Paradox, B is simple Arithmetic, and C is Russell?s > > > > > > > > Paradox. > > > > > > > > This is Russells Paradox > > > > > > > > {x  x ~e x} e {x  x ~e x} > > > > > > > <> > > > > > > > {x  x ~e x} ~e {x  x ~e x} > > > > > > > > To make a consistent set theory the formula { x  x ~e x } > > > > > > > must be flagged somehow. > > > > > > > How do you define a wff  precisely? That is the problem. Frege was > > > > > > right, Russell was wrong, and all you need is an exact (formal) > > > > > > definition of wff. > > > > > > > CB > > > > > > in the usual manner by Syntactic construction. > > > > > > IF X is a WFF > > > > > THEN ALL(Y) X is a WFF > > > > > > and so on. > > > > > The problem isn't with the connectives. What can X be for starters  > > > > the most primitive wffs from which we build others? > > > > > CB > > > >http://en.wikipedia.org/wiki/First_order_logic#Formation_rules > > > > In PROLOG we use lowercase words for TERMS > > > and uppercase words for VARIABLES > > > > ATOMIC PREDICATE > > > ATOMIC PREDICATE meaning relation? > > > CB > > RELATION > p(a, b, e)
If wffs are built on relations then { x  x ~e x } is not a wff because ~e is not a relation.
We don?t need ZF  at all. All we need is Naïve Set Theory, a complete formal definition of wff and recognition that x ~e x is not a relation due to diagonalization on sets.
Logic = Set Theory
Logic = NOT AND OR EXISTS simple, easy
ZF Set Theory = a dozen messy axioms for which people can?t even agree on the specifics ??
There are a dozen set theories and a dozen interpretations of the most popular set theory, and 2 or 3 versions of it (with or without Choice, etc.) none of which decide any of the important questions of set theory due to exhaustive work (a waste!) by Godel and Cohen.
CB
> ATOMIC PREDICATE > p(a, b(c,d), e(f,g)) > > NONATOMIC PREDICATE > a(b) > d(c) > > NONATOMIC PREDICATE > All(a) p(a, b(c,d), e(f,g)) > > Relational Algebra is generally used to refer to ordinary tuples of > terms. e.g SQL Tables. > > QUANTIFIED LOGIC > > ALL(n):N EXIST(m):N m>n > >  > > AS A SUBSET > > { n  n e N } C { n  m>n } > every n here  has a bigger m here > >  > > AS A PROLOG PREDICATE > > subset( N, nat(N), bigger(M,N) ) > >  > > subset() is not easy to program though... > > You can use the LISP addon to PROLOG > or my record management addon. > > PROLOG+ A CONCEPT DATABASE MANAGEMENT LANGUAGE (DBML) > > We add simple breadth first extensions to PROLOG CLAUSES. > > Y> next record > Y< prev record > Y>> last record > Y<< 1st record > > AXIOM OF FINITE SUBSETS >  > > subs(A,X,Y) < e(A>>,X) ^ e(A,Y). > > subs(A,X,Y) < e(A,Y) ^ e(A>,X) ^ subs(A,X,Y). > > ss(X,Y) < e(A<<,X) ^ subs(A,X,Y). > > This is just using PROLOG RECURSION to do a FOR LOOP > > A>> last record > A> next record > A<< first record > > AXIOM OF FINITE SET EQUALITY >  > > equals(X,Y) < ss(X,Y) ^ ss(Y,X). > > Now you can do Set Theory and Logic all in Atomic Predicates! > > In BLOCK PROLOG the above would be a rule: > > equals X Y > ss X Y > ss Y X. > > Herc > www.BLoCKPROLOG.com

