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Topic: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle
to Resolve Several Paradoxes

Replies: 53   Last Post: Feb 13, 2013 3:53 PM

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Graham Cooper

Posts: 4,321
Registered: 5/20/10
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
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

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





Date Subject Author
2/1/13
Read Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle
to Resolve Several Paradoxes
Graham Cooper
2/3/13
Read Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle
to Resolve Several Paradoxes
Charlie-Boo
2/3/13
Read Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle
to Resolve Several Paradoxes
Graham Cooper
2/3/13
Read Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle
to Resolve Several Paradoxes
Charlie-Boo
2/3/13
Read Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle
to Resolve Several Paradoxes
Graham Cooper
2/3/13
Read Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle
to Resolve Several Paradoxes
Graham Cooper
2/3/13
Read Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle
to Resolve Several Paradoxes
Charlie-Boo
2/3/13
Read Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle
to Resolve Several Paradoxes
Graham Cooper
2/3/13
Read Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle
to Resolve Several Paradoxes
Charlie-Boo
2/3/13
Read Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle
to Resolve Several Paradoxes
camgirls@hush.com
2/4/13
Read Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle
to Resolve Several Paradoxes
Charlie-Boo
2/4/13
Read Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle
to Resolve Several Paradoxes
billh04
2/4/13
Read Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle
to Resolve Several Paradoxes
Charlie-Boo
2/4/13
Read Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle to Resolve Several Paradoxes
William Hale
2/4/13
Read Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle to Resolve Several Paradoxes
Lord Androcles, Zeroth Earl of Medway
2/9/13
Read Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle
to Resolve Several Paradoxes
Graham Cooper
2/5/13
Read Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle
to Resolve Several Paradoxes
Charlie-Boo
2/4/13
Read Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle
to Resolve Several Paradoxes
Graham Cooper
2/5/13
Read Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle
to Resolve Several Paradoxes
Charlie-Boo
2/5/13
Read Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle
to Resolve Several Paradoxes
Graham Cooper
2/5/13
Read Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle
to Resolve Several Paradoxes
Brian Q. Hutchings
2/6/13
Read Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle
to Resolve Several Paradoxes
Graham Cooper
2/6/13
Read Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle
to Resolve Several Paradoxes
Charlie-Boo
2/4/13
Read Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle
to Resolve Several Paradoxes
fom
2/4/13
Read Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle
to Resolve Several Paradoxes
Charlie-Boo
2/4/13
Read Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle
to Resolve Several Paradoxes
fom
2/5/13
Read Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle
to Resolve Several Paradoxes
Charlie-Boo
2/7/13
Read Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle
to Resolve Several Paradoxes
fom
2/9/13
Read Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle
to Resolve Several Paradoxes
Charlie-Boo
2/9/13
Read Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle
to Resolve Several Paradoxes
Graham Cooper
2/11/13
Read Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle
to Resolve Several Paradoxes
Charlie-Boo
2/10/13
Read Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle
to Resolve Several Paradoxes
fom
2/10/13
Read Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle
to Resolve Several Paradoxes
Graham Cooper
2/10/13
Read Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle
to Resolve Several Paradoxes
fom
2/10/13
Read Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle
to Resolve Several Paradoxes
Graham Cooper
2/11/13
Read Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle
to Resolve Several Paradoxes
Charlie-Boo
2/11/13
Read Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle
to Resolve Several Paradoxes
Charlie-Boo
2/11/13
Read Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle
to Resolve Several Paradoxes
Charlie-Boo
2/11/13
Read Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle
to Resolve Several Paradoxes
Graham Cooper
2/13/13
Read Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle
to Resolve Several Paradoxes
Charlie-Boo
2/11/13
Read Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle
to Resolve Several Paradoxes
Charlie-Boo
2/11/13
Read Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle
to Resolve Several Paradoxes
fom
2/5/13
Read Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle
to Resolve Several Paradoxes
Charlie-Boo
2/5/13
Read Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle
to Resolve Several Paradoxes
fom
2/6/13
Read Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle
to Resolve Several Paradoxes
fom
2/11/13
Read Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle
to Resolve Several Paradoxes
Charlie-Boo
2/11/13
Read Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle
to Resolve Several Paradoxes
fom
2/13/13
Read Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle
to Resolve Several Paradoxes
Charlie-Boo
2/13/13
Read Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle
to Resolve Several Paradoxes
fom
2/4/13
Read Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle
to Resolve Several Paradoxes
Graham Cooper
2/4/13
Read Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle
to Resolve Several Paradoxes
Charlie-Boo
2/5/13
Read Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle
to Resolve Several Paradoxes
Charlie-Boo

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