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lugit...@xxxxxxxxx wrote:
> Aatu Koskensilta wrote:
> > lugita15@xxxxxxxxx wrote:
> > > Aatu Koskensilta wrote:
> > >> lugita15@xxxxxxxxx wrote:
> > >>> 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?
> > >> No. Setting aside the fact that it's not clear what is meant by the
> > >> "rank of a variable" in general,
> > > It is very clear what the rank of a set is. This is defined by
> > > transfinite recursion.
> >
> > Certainly. What isn't clear is what the "rank" of a *variable* occurring
> > in a formula in the language of set theory is.
> >
> > >> the formula "all sets of rank 0 are empty" is certainly not equivalent
> > >> to "all sets of rank 1 are empty".
> > >
> > > This is not what is meant by a typical ambiguity scheme. We do not
> > > replace all uses of the term "rank n" in a wff with "rank n+1."
> > > Rather, we raise the ranks of all the set variables in a wff by 1.
> >
> > How is the rank of a set variable in a formula in the language of set
> > theory determined?
> >
> That's the point. Instead of considering TST which is an obvious typed
> theory, let us instead consider Zermelo set theory with the seperation
> scheme restricted to formulas wih bounded quantification. The same
> difficulty as in ZFC is how to assign types (ranks) to the variables.
> That is the motivation for a typical ambiguity scheme; it merely relies
> on the obvious intuition we have that as long as types (ranks) can be
> assigned consistently for the variables in a formula, it does not
> matter what particular type (rank) each variable in each formula is
> assigned.
> > > A correct example of typical ambiguity is that for each ordinal n, the
> > > following is a theorem: there exists a set which contains all sets of
> > > rank n.
> >
> > Given an arbitrary ordinal alpha there is in general no sentence
> > expressing "there exists a set which contains all sets of rank alpha".
> >
> Yes there is. That set is called V_alpha.
> > In any case, the scheme introduced by Specker,
> >
> > Phi <--> Phi+
> >
> > says that every Phi is equivalent to Phi+, not only that if Phi is
> > provable so is Phi+ (and hence any formula obtained from Phi raising the
> > types by a fixed amount). Indeed, while Specker's scheme is not provable
> > in TST, the inference rule Phi |- Phi+ is conservative over TST.
> >
> > > I'm not precisely sure how to answer your objection concerning "all sets
> > > of
> > > rank 0 are empty," but if your objections are correct then won't there be
> > > the
> > > same objections to adding a typical ambiguity scheme to TST?
> >
> > My objections do not apply to the simple theory of types in any obvious
> > sense. For any type n the formula "all sets of type n are empty" is false.
> >
> Then by similar reasoning, shouldn't "all sets of type 0 are empty"
> also be false?
I meant rank here, sorry.
> > --
> > Aatu Koskensilta (aatu.koskensilta@xxxxxxxxx)
> >
> > "Wovon man nicht sprechen kann, daruber muss man schweigen"
> > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
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