[FOM] The semantics of set theory
Ralf Schindler
rds at logic.univie.ac.at
Mon Oct 7 09:35:52 EDT 2002
Hi Vladimir, hi FOMers,
On Mon, 7 Oct 2002, Kanovei wrote:
> Yes you clearly misunderstand the point.
let me back up. For any fixed concrete integer n>0, the formula
"x is a true \Sigma_n sentence of set theory" can be written as a
\Sigma_n formula of set theory; ZF proves every relevant instance of
the Tarski schema. Therefore, the formula "x is a true sentence of
set theory" can be written as a \Sigma_1 (!) formula of class theory
(by saying "there is a class containing only truths to which x belongs");
BG proves every instance of the Tarski schema (of course, there is
no single class witnessing x is true for all true x; if x is \Sigma_n
we typically need a \Sigma_n definable class as a witness). It follows
that "predicative classes suffice for defining set theoretical truth."
My historical knowledge is pretty poor. I think all this appears
(first?) quite explicitly in a Fund.Math. paper of Mostowski's from 1950.
Its title is "Some impredicative definitions in axiomatic set theory."
Isn't life confusing?
I find it philosophically significant that we do *not* commit
ourselves to the existence of non-predicative classes if we define
set theoretical truth.
Best, Ralf
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Ralf Schindler Phone: +43-1-4277-50511
Institut fuer Formale Logik Fax: +43-1-4277-50599
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