[FOM] The semantics of set theory
Ralf Schindler
rds at logic.univie.ac.at
Mon Oct 7 02:38:44 EDT 2002
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
1090 Wien, Austria URL: http://www.logic.univie.ac.at/~rds/
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