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lugita15@xxxxxxxxx wrote:
> Rupert wrote:
> > lugita15@xxxxxxxxx wrote:
> > > The simple theory of types, or TST, is a set theory equivalent to
> > > Zermelo set theory
> >
> > That's actually not quite true. In simple type theory, the types are
> > indexed by the natural numbers. In Zermelo set theory, there are a
> > hierarchy of levels which are indexed by the ordinal numbers up to
> > omega times two, and it may go beyond, it's just that you can't prove
> > that it does. Zermelo set theory can prove the consistency or the
> > arithmetical soundness of the deductive closure of the Peano axioms in
> > simple type theory. So it is a stronger theory.
> >
> Yes, Zermelo set theory is a stronger theory than TST, and I never said
> anything contrary to that. I said that TST is equivalent to Zermelo
> set theory with the seperation scheme restricted to formulas that use
> only bounded quantifiers. I believe you cut off my sentence by
> accident.
I parsed your sentence wrongly. I thought the "with" applied to type
theory, not Zermelo set theory.
> > > with seperation restricted to formulas with bounded
> > > quantification which assigns sets natural numbers called types. A set
> > > x can only be an element of a set y if y is one type higher than x.
> > > The first axiom is the same axiom of extensionality that the usual set
> > > theory has. The second axiom is a comprehension schema which defines a
> > > set y of type n+1 for every formula phi(x), where x is of type n.
> > >
> > > Quine's New Foundations, or NF, is a set theory in which the types are
> > > dropped from variables. In other words, in NF there is extensionality
> > > and a comprehension schema which assigns a set to every formula phi,
> > > where all the variables in phi can be consistently assigned types.
> > > What this means in effect is that NF is equivalent to TST plus a
> > > typical ambiguity scheme which states that every formula phi is
> > > equivalent to the corresponding formula phi+ which results from raising
> > > the the types of all the variables in phi by 1.
> > >
> >
> > I haven't studied NF in detail, but I think you'll find it's quite
> > different both to Zermelo set theory and the simple theory of types.
> > Its consistency is an open problem, for example.
> >
> I know that NF is different from both Zermelo set theory and TST.
> Rather, NF is equivalent to TST plus an axiom scheme which asserts that
> every formula phi is equivalent to phi+, which raises the types of all
> the variables in phi by 1.
Are you sure about this? Is this result anywhere in the literature?
> > > In the standard set theory of ZFC, something similar to the types in
> > > TST is done. Instead of types, sets in the cumulative hierarchy are
> > > assigned ranks, which may be transfinite ordinals. My question is, is
> > > there a set theory which is to ZFC as NF is to TST? That is, can you
> > > add a typical ambiguity scheme to ZFC which states that every wff phi
> > > is equivalent to the corresponding formula phi+ which results from
> > > increasing the ranks of all the variables in phi by 1? Is the
> > > resultant theory consistent? Is it perhaps equiconsistent with some
> > > large cardinal axiom?
> > >
> > > Any help would be greatly appreciated.
> > > Thank You in Advance.
> >
> > I'm not sure I understand the relation between NF and simple type
> > theory which you're asserting.
> Yes, I do. I apologize if I did not express myself very clearly.
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