On Sep 28, 3:29 pm, Graham Cooper <
grahamcoop...@gmail.com> wrote:
> I see no useful distinction between:
>
> E(x) !(x=x)
>
> and
>
> E(r) A(x) xer<->!(xex)
>
> when both can be formulated into something meaningful.
There's a huge distinction. = doesn't have to be part of logic.
First-order logic comes in both "with equality" AND a "without
equality" variant.
Without equality is better.
In that case "=" becomes re-definable. It's not usually defined in a
way that loses reflexivity, but it is at least possible that you could
have a reason for
wanting to do it that way.
The Russell's Paradox sentence by contrast IS JUST FALSE.
> A(x) x=x
^^^^^^^^
This IS NOT "making Ex[ ~x=x ] into something meaningful!
This is THE DENIAL OF Ex[ ~x=x ] !! DENYING something IS THE
OPPOSITE of "making it meaningful"!
It's the purest TRASHING of it that you can have!
> E(r) A(x) xer<-> ( !(xex) ^ (x=/=r))
This is meaningless until you come up with all 3 of an alternative set
theory that allows cyclic membership relations,
a theory of equality within that set theory, and some proofs that this
r is good for something.
> THE SET OF ALL SETS THAT DON'T CONTAIN THEMSELVES.
>
> The container set excluded for consistency.
Excluding the container set by itself won't help you with consistency.
In ZFC, NO set contains itself.
You have to come up with some OTHER set theory in which that's
POSSIBLE, first.
This other set theory will still REQUIRE lots of other sets to exist.
It may FORCE the existence
of r INside this set after all, which would again be inconsistent.
Naive set theory doesn't become consistent just because you decide to
put ~x=r on the end.