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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?
> --
> Aatu Koskensilta (aatu.koskensilta@xxxxxxxxx)
>
> "Wovon man nicht sprechen kann, daruber muss man schweigen"
> - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
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