# Re: Multilayer Set Theory:

*From*: zuhair <zaljohar@xxxxxxxxx>*Date*: 27 May 2007 12:05:37 -0700

On May 27, 10:59 am, Aatu Koskensilta <aatu.koskensi...@xxxxxxxxx>

wrote:

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

.

**Follow-Ups**:**Re: Multilayer Set Theory:***From:*Aatu Koskensilta

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