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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 3, 2013 11:19 PM
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On Feb 3, 9:02 pm, Graham Cooper <grahamcoop...@gmail.com> wrote:
> On Feb 4, 9:12 am, Charlie-Boo <shymath...@gmail.com> wrote:
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> > On Feb 3, 4:56 pm, Graham Cooper <grahamcoop...@gmail.com> wrote:
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> > > On Feb 4, 7:18 am, Charlie-Boo <shymath...@gmail.com> wrote:
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> > > > On Feb 3, 4:03 pm, Graham Cooper <grahamcoop...@gmail.com> wrote:
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> > > > > On Feb 4, 3:01 am, Charlie-Boo <shymath...@gmail.com> wrote:
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> > > > > > On Feb 1, 3:35 pm, Graham Cooper <grahamcoop...@gmail.com> wrote:
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> > > > > > > On Feb 2, 4:09 am, Charlie-Boo <shymath...@gmail.com> wrote:
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> > > > > > > > There is a peculiar parallel between Semantic Paradoxes, Set Theory
> > > > > > > > Paradoxes and ordinary formal Arithmetic.
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> > > > > > > > Consider the following 3 pairs of expressions in English, Set Theory
> > > > > > > > and Mathematics:
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> > > > > > > > A
> > > > > > > > This is false.
> > > > > > > > This is true.
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> > > > > > > > B
> > > > > > > > 1/0
> > > > > > > > 0/0
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> > > > > > > > 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}
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> > > > > > > > A is the Liar Paradox, B is simple Arithmetic, and C is Russell?s
> > > > > > > > Paradox.
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> > > > > > > This is Russells Paradox
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> > > > > > > {x | x ~e x} e {x | x ~e x}
> > > > > > > <->
> > > > > > > {x | x ~e x} ~e {x | x ~e x}
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> > > > > > > To make a consistent set theory the formula { x | x ~e x }
> > > > > > > must be flagged somehow.
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> > > > > > 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.
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> > > > > > C-B
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> > > > > in the usual manner by Syntactic construction.
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> > > > > IF X is a WFF
> > > > > THEN ALL(Y) X is a WFF
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> > > > > and so on.
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> > > > The problem isn't with the connectives. What can X be for starters -
> > > > the most primitive wffs from which we build others?
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> > > > C-B
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> > >http://en.wikipedia.org/wiki/First_order_logic#Formation_rules
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> > > In PROLOG we use lowercase words for TERMS
> > > and uppercase words for VARIABLES
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> > > ATOMIC PREDICATE
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> > ATOMIC PREDICATE meaning relation?
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> > C-B
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> 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.
C-B
> ATOMIC PREDICATE
> p(a, b(c,d), e(f,g))
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> NON-ATOMIC PREDICATE
> a(b) -> d(c)
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> NON-ATOMIC PREDICATE
> All(a) p(a, b(c,d), e(f,g))
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> Relational Algebra is generally used to refer to ordinary tuples of
> terms. e.g SQL Tables.
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> QUANTIFIED LOGIC
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> ALL(n):N EXIST(m):N m>n
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> ------------------
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> AS A SUBSET
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> { n | n e N } C { n | m>n }
> every n here --- has a bigger m here
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> -----------------
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> AS A PROLOG PREDICATE
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> subset( N, nat(N), bigger(M,N) )
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> ------------------
>
> subset() is not easy to program though...
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> You can use the LISP addon to PROLOG
> or my record management addon.
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> PROLOG+ A CONCEPT DATABASE MANAGEMENT LANGUAGE (DBML)
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> We add simple breadth first extensions to PROLOG CLAUSES.
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> Y> next record
> Y< prev record
> Y>> last record
> Y<< 1st record
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> AXIOM OF FINITE SUBSETS
> -----------------------
>
> subs(A,X,Y) <- e(A>>,X) ^ e(A,Y).
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> subs(A,X,Y) <- e(A,Y) ^ e(A>,X) ^ subs(A,X,Y).
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> ss(X,Y) <- e(A<<,X) ^ subs(A,X,Y).
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> This is just using PROLOG RECURSION to do a FOR LOOP
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> A>> last record
> A> next record
> A<< first record
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> AXIOM OF FINITE SET EQUALITY
> ----------------------------
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> equals(X,Y) <- ss(X,Y) ^ ss(Y,X).
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> Now you can do Set Theory and Logic all in Atomic Predicates!
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> In BLOCK PROLOG the above would be a rule:
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> equals X Y
> ss X Y
> ss Y X.
>
> Herc
> --www.BLoCKPROLOG.com
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