Re: Question about Quine's New Foundations

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
.

Relevant Pages

  • Re: Question about Quines New Foundations
    ... that every wff phi is equivalent to the corresponding formula phi+ which ... It is very clear what the rank of a set is. ... This is not what is meant by a typical ambiguity scheme. ... Instead of considering TST which is an obvious typed ...
    (sci.logic)
  • Re: Question about Quines New Foundations
    ... that every wff phi is equivalent to the corresponding formula phi+ which ... It is very clear what the rank of a set is. ... This is not what is meant by a typical ambiguity scheme. ... Instead of considering TST which is an obvious typed ...
    (sci.logic)
  • Re: Question about Quines New Foundations
    ... that every wff phi is equivalent to the corresponding formula phi+ which results from ... It is very clear what the rank of a set is. ... This is not what is meant by a typical ambiguity scheme. ...
    (sci.logic)
  • Re: Question about Quines New Foundations
    ... that every wff phi is equivalent to the corresponding formula phi+ which results from ... Setting aside the fact that it's not clear what is meant by the "rank of a variable" in general, the formula "all sets of rank 0 are empty" is certainly not equivalent to "all sets of rank 1 are empty". ...
    (sci.logic)