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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 13, 2013 3:34 PM
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On Feb 11, 3:45 pm, Graham Cooper <grahamcoop...@gmail.com> wrote:
> On Feb 12, 4:53 am, Charlie-Boo <shymath...@gmail.com> wrote:
>
> > > I view set theory as being about the existence
> > > of mathematical objects. Naive set theory failed,
>
> > Failed meaning? There is nothing wrong with naïve set theory.
>
> > A. A wff maps SETS to SETS. E.g. if P(x,y) is a set then (exists
> > M)P(M,x) is a set.
> > B. x ~e x is not a set.
> > C. x = y is a set.
> > D. For any set M, x e M is a set.
>
> In NAIVE SET THEORY {x | x ~e x} *parses* as a Set.
>
> NST
>
> ALL(SET) EXIST(p):[T|F]
> ALL(x) x e SET <-> p(x)
>
> ----------------------
>
> Any DEFINABLE (p) COLLECTION is a SET.
>
> Since
> p <-> x ~e x
x ~e x is not a collection. When you call something a "collection"
you are just trying to hide the fact that it is a set and are using a
synonym for "set". Changing the syntax won't gain you anything.
Collection = Set. The sets that are not elements of themselves are
the collections that are not collected in themselves.
Do the collections that are not collected in themselves form a
collection? If you say so, then you are inconsistent. If you say no,
then your proof is not valid and Naïve Set Theory is still intact.
Set Theory is about what is intuitive and Naïve Set Theory is
intuitive.
Think of a WFF as a mapping from relations to relations. Then it
cannot be used on x ~e x.
If VARIABLE is a relation then (exists x)VARIABLE(x) is a relation.
C-B
> is DEFINABLE ... Russell's Set is a Definable Set
>
> ------------------------
>
> x e SET <-> x ~e x
>
> SET e SET <-> SET ~e SET
>
> CONTRADICTION
>
> NST |- thm, ~thm
>
> EX-CONTRADICTIONE SEQUITUR QUODLIBET
>
> http://blockprolog.com/EX-CONTRADICTIONE-SEQUITUR-QUODLIBET.png
>
> ------------------------
>
> Here is how a CONTRADICTORY SYSTEM (inconsistent) Proves *anything*.
>
> from MODUS PONENS formula you can derive
> EX-CONTRADICTIONE-SEQUITUR-QUODLIBET
>
> Some people like C. Boo think if you're using Natural Deduction
> anyway
> then there need not be this Huge Platonic Web of RULES of Set Theory
> to abide by... just use Naive Set Theory anyway.
>
> So it is really true that from a contradiction you can prove
> anything?
>
> Only if you keep MODUS PONENS!
>
> LHS->RHS ^ LHS -> RHS
> ------------------------------------
>
> Nve. Set THEORY |- RSeRS, ~RSeRS
>
> Now with
>
> THEORY |- FALSE
> INDUCTION RULE : LHS->RHS
> INDUCTION CHECK IF IT APPLI:ES : LHS? (MP)
> LHS -> RHS
> NOT(LHS) or RHS
>
> This version of IMPLIES means: if the LHS applies (is true)
> then the RHS must apply
>
> i.e. if the LHS is false, the induction rule doesn't MATCH any fact
> (with the bindings in use)
>
> so it has no effect on the RHS.
>
> So.... back to my previous derivation from MP.
> LHS->RHS ^ LHS -> RHS
> (!LHS or RHS) ^ LHS -> RHS
>
> (!LHS ^ LHS) v (RHS^LHS) -> RHS
>
> *** ~L ^ L -> RHS ***
>
> where L is any theorem
> as we are backward chaining to derive RHS
>
> So if the theory is inconsistent... there is 'likely' a inference
> rule LHS->RHS
>
> where LHS MATCHES the predicate pattern of RSeRS.
>
> So
>
> *MATCH* *MATCH*
> (~RSeRS) & (RSeRS) -> RHS
> i.e. a contradictory system proves anything!
>
> -------------------
>
> Do not confuse
>
> NATURAL LOGIC
> with
> DEDUCTIVE LOGIC
>
> Everyone here uses NATURAL LOGIC for their own calculations
> in NAIVE SET THEORY
> but you call it FIRST ORDER LOGIC
> as if it gives you some license to make any deductions without axioms.
>
> The "Standard Model", "In First Order LOGIC"
> this is just Natural Logic in Naive Set Theory
>
> *a Kangaroo just hopped past at my Weekender!*
>
> NATURAL LOGIC:
>
> LEGEND:
> thm(..X..) X is a Theorem
> L->R is a Inference Rule
>
> (LHS->RHS)
> ^ (LHS is true in some model)
> ^ (LHS is not false in any model)
> -> RHS
>
> It's very slow to check for errors with every deduction, which is how
> humans work with Natural Deductive logic!
>
> SHORT ANSWER: MODUS PONENS
>
> (LHS->RHS)
> ^ LHS
> ->RHS
>
> an *AUTOMATIC* Logic is incompatible with Naive Set Theory.
>
> Herc
> --www.BLoCKPROLOG.com
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