Date: Feb 1, 2013 3:35 PM
Author: Graham Cooper
Subject: Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle<br> to Resolve Several Paradoxes

On Feb 2, 4:09 am, Charlie-Boo <> wrote:
> There is a peculiar parallel between Semantic Paradoxes, Set Theory
> Paradoxes and ordinary formal Arithmetic.
> Consider the following 3 pairs of expressions in English, Set Theory
> and Mathematics:
> A
> This is false.
> This is true.
> B
> 1/0
> 0/0
> 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}
> A is the Liar Paradox, B is simple Arithmetic, and C is Russell?s
> Paradox.

This is Russells Paradox

{x | x ~e x} e {x | x ~e x}
{x | x ~e x} ~e {x | x ~e x}

To make a consistent set theory the formula { x | x ~e x }
must be flagged somehow. ZFC

{ x | x e y } IFF E(z) y C z

Axiom of Specification.

there is no z such that RS C z
so RS is impossible to define in ZFC.

I'm working on an automated Proof By Contradiction of RS.

You need to forward chain MODUS PONENS
which is new to PROLOG LOGIC since it requires
depth limiting. (which is simple enough and we've
got DL working on backward chaining MP already)