[FOM] The semantics of set theory

[Ralf Schindler]

On Sun, 6 Oct 2002, Kanovei wrote:

> To define that a set theoretic formula A (with parameters or 
> even without parameters) "is true" one has to claim the 
> existence of a class satisfying certain known properties 
> and containing A. 
> Such a class itself cannot be definable, e.g. predicative, 
> if we want to treat A as a free variable. 

I guess I misunderstand you, as otherwise you'd be wrong.
It is possible to define "x is a true statement of set theory"
in a language which has class variables. The point is that we 
can intend the class variables to range just over predicative 
classes as we can prove all instances of the Tarski schema in
the theory BGC. (Generalizations of this are in an old paper
of mine.) The idea of course is simply that x is true iff 
it belongs to a class which contains only truths; however, if
x is \Sigma_n then the canonical recursively defined such class
is \Sigma_n as well (for n>0), hence predicative. --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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