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