[FOM] The semantics of set theory

[Ralf Schindler]

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
Universitaet Wien                      E-mail: rds at logic.univie.ac.at
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