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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. Let V_0 be the empty set. Let V_(a+1) be the
powerset of V_a. For limit ordinals a, let V_a be the union of all the
sets V_b where b is less than a. The rank of a set x is the smallest
ordinal a such that x is a subset of V_a. This definition is perfectly
clear and precise.
>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. 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. If a typical ambiguity scheme is accepted, then a theory could
be created in which there was a universal set, similar to NF except
bearing the same relation to ZFC as NF bears to 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?
Any futher help would be greatly appreciated.
Thank You in Advance.
> --
> Aatu Koskensilta (aatu.koskensilta@xxxxxxxxx)
>
> "Wovon man nicht sprechen kann, daruber muss man schweigen"
> - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
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