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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

 Messages: [ Previous | Next ]
 Charlie-Boo Posts: 1,635 Registered: 2/27/06
Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle
to Resolve Several Paradoxes

Posted: Feb 4, 2013 7:26 AM

On Feb 3, 11:53 pm, camgi...@hush.com wrote:
> On Feb 4, 2:19 pm, Charlie-Boo <shymath...@gmail.com> wrote:
>
>
>

> > > 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.

>
> if  e(x,y) is a predicate
> then  not(e(x,y)) is a predicate

And more importantly not(e(x,x)) is a predicate (diagonalization.)

Yes, that is Naïve Set Theory, which is correct. But the IF fails.

"e(x,y) is a predicate" is not correct due to diagonalization. There
is no Russell Paradox, only Russell's Diagonalization.

If e(x,y) were a predicate then not(e(x,x)) would be a predicate but
because of diagonalization it is not.

What I believe about Prolog consists only of its being a database and
query language similar to predicate calculus (aka FOL.)

Then I take it that:

if [ not [ e rs rs ]] [ e rs rs ] Line 1 X => rs
if [ e rs rs ] [ not [ e rs rs ]] Line 2 X => rs
[ not [ e rs rs ]] = = [ e rs rs ] Def = = , if
~P = = P [ e rs rs ]
=> P

Then some axiom about ~P = = P.

In Combinatory Logic:

M x => N x x Def M
M M => N M M x => M

There is no M M = = N M M because it is executed (no Turing Machine
can halt both yes and no because it stops as soon as it halts either
cf Rosser 1936.) M M and N M M occur at different points in time!
The system simply changes its mind as it goes along learning using AI
techniques. The circle is amicable.

In CBL:

M # P(x) / Q(a,b) means program/wff M enumerates/represents set P and
is written in programming language/logic Q meaning Q(x,y) iff program/
wff M outputs at some point/is provable on input y. Then Q(M,x) = =
P(x) (all x). Abbreviated M#P/Q means P=Q(M).

P(x) / Q(a,b) means (exists M) M # P(x) / Q(a,b) There is a program/
wff that enumerates/represents set P.

Let SE(x,y) iff set x contains element y.

Then the Russell Paradox is:

~SE(x,x) / SE(a,b) There is a set of all sets that do not contain
themselves.
M # ~SE(x,x) / SE(a,b) Let M be the set of all sets that do not
contain themselves.
SE(M,x) = = ~SE(x,x) Def # /
SE(M,M) = = ~SE(M,M) x => M
P = = ~P SE(M,M) => P

This is encapsulated in axiom (lower level theorem) - ~P/P where - E
means expression E is not true and P/Q abbreviates P(x)/Q(a,b) or
P(x,x)/Q(a,b) if P is 2-place. Then if P/Q then P differs from Q and
we prove truth, provability and unrefutability distinct (Godel,
Rosser, Smullyan) - see 18 Word Proof in FOM July 2010.

C-B

> http://en.wikipedia.org/wiki/First_order_logic#Formation_rules
> Negation. If f is a formula, then f' is a formula.
>
> ---------------------------------------------
>
> In BLOCK PROLOG Russell's Set would be:
>
> if    [ not [ e X X ]]    [ e X rs ]
> if    [ e X rs ]            [ not [ e X X ]]
>
> I'm trying to forward MODUS PONENS to a contradiction!
>
> Herc
> --www.BLoCKPROLOG.com

Date Subject Author
2/1/13 Graham Cooper
2/3/13 Charlie-Boo
2/3/13 Graham Cooper
2/3/13 Charlie-Boo
2/3/13 Graham Cooper
2/3/13 Graham Cooper
2/3/13 Charlie-Boo
2/3/13 Graham Cooper
2/3/13 Charlie-Boo
2/3/13 camgirls@hush.com
2/4/13 Charlie-Boo
2/4/13 billh04
2/4/13 Charlie-Boo
2/4/13 William Hale
2/4/13 Lord Androcles, Zeroth Earl of Medway
2/9/13 Graham Cooper
2/5/13 Charlie-Boo
2/4/13 Graham Cooper
2/5/13 Charlie-Boo
2/5/13 Graham Cooper
2/5/13 Brian Q. Hutchings
2/6/13 Graham Cooper
2/6/13 Charlie-Boo
2/4/13 fom
2/4/13 Charlie-Boo
2/4/13 fom
2/5/13 Charlie-Boo
2/7/13 fom
2/9/13 Charlie-Boo
2/9/13 Graham Cooper
2/11/13 Charlie-Boo
2/10/13 fom
2/10/13 Graham Cooper
2/10/13 fom
2/10/13 Graham Cooper
2/11/13 Charlie-Boo
2/11/13 Charlie-Boo
2/11/13 Charlie-Boo
2/11/13 Graham Cooper
2/13/13 Charlie-Boo
2/11/13 Charlie-Boo
2/11/13 fom
2/5/13 Charlie-Boo
2/5/13 fom
2/6/13 fom
2/11/13 Charlie-Boo
2/11/13 fom
2/13/13 Charlie-Boo
2/13/13 fom
2/4/13 Graham Cooper
2/4/13 Charlie-Boo
2/5/13 Charlie-Boo