Re: Multilayer Set Theory:

On May 27, 10:59 am, Aatu Koskensilta <aatu.koskensi...@xxxxxxxxx>
On 2007-05-27, in sci.math, zuhair wrote:

I don't know, regarding this point.
But as I see neither Vi nor Vj are constants
they are variables that range over constants V1,V2,V3,...,Vn ( the
finite version) or
over V1,V2,V3,...... in the infinite version.

There is no apparatus in first order logic for quantifying over syntactic
elements of the first order language in question. AViAvj(...) is simply
not well-formed.

Accordingly axiom 3 should be schematized to:

3) Layers schema: For every i,j such that j>i the sentence
Ax(xeVi -> xeVj) & ~ Vi=Vj
is an axiom.

What about the other axioms should they be schams also
Like for example pairing

AaeViAbeViExeViAy(yex<->(y=a v y=b)).

Should that be changed to the following schema:

For every i the sentence

AaeViAbeViExeViAy(yex<->(y=a v y=b))

is an axiom.

I always like to speak in terms of MK rather than ZF.
so essentially what I did is Omega-order MK type theory.

What you're doing is omega-order MK, not MK in omega-order type theory. As
said, in type theory comprehension is restricted so that we can only
quantify over elements of Vi when defining sets in Vi. In your theory, and
in omega-order MK, there is no such restriction.

Aatu Koskensilta (aatu.koskensi...@xxxxxxxxx)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus