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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 11, 2013 3:45 PM
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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
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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