Re: Han's startling new set theory.

Han.deBruijn@xxxxxxxxxxxxxx writes:

> Jesse F. Hughes wrote:
>
>> There is no Russell's paradox in this setting. You can form the
>> class of all sets that don't contain themselves. But that class is
>> not a set, and so Russell's paradox doesn't go through. The
>> set/class distinction is analogous to Russell's type theory (but it
>> is extraordinarily rare that anyone needs to discuss extensions of
>> classes, so the hierarchy is usually cut off at level 2).
>
> Of course "there is no Russell's paradox in this setting". Because
> every time you need a set of all sets, you replace it by a "class of
> all sets" and you say: "Ah, but a class is not a set". "Consistency"
> by switching a label, attaching another name. It's the cheapest trick
> I've ever seen.

In fact, the formal theory ZF has no classes at all. When I speak of
classes, it is a handy tool with no formal status. But we can
understand classes very easily. Consider the formulas of ZF (with one
free variable) and quotient out by the equivalence relation:

Phi R Psi <=> |- (A x)(Phi(x) <-> Psi(x)).

Call the elements of that quotient the *classes* of ZF.

Now prove Russell's paradox with this introduced terminology.

Trick? Doesn't seem much like a trick to me. It is a handy
conceptual tool, on the other hand.
--
Jesse F. Hughes

"Would you please stop talking and start talking?"
-- Vincent Price as the Saint
.

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