[FOM] The semantics of set theory

[Ralf Schindler]

On Wed, 9 Oct 2002, Volker Halbach wrote:

> Ralf, how does one prove that we don't get the Tarski rules in BG?

   Let BG_j be BG as exposed in Jech, "set thy," p.76, say, and let
BG_c be BG as exposed in Cohen, "set thy + CH," p.74f., say. BG_j has
the comprehension schema, and BG_c has the (finitely many) axioms for
class formation. As we all know, ZF, BG_j, and BG_c all prove the very
same set theoretical assertions.
   A simple compactness argument shows BG_j doesn't prove the Tarski 
rules. The point is that for each finite subset T of BG_j there is   
some integer n such that T has a model in which the classes are exactly
the \Sigma_n definable ones.
   On the other hand, the axioms for class formation can easily be   
used to verify that BG_c proves that if there is a class containing
exactly the true set theoretical statements of rank \leq n then there 
is one containing exactly the true set theoretical statements of rank 
\leq n+1. BG_c + \Sigma^1_1 induction then proves the Tarski rules.
   One might add the remark that "all classes are predicative" (as 
being formalized in the language of class thy) is consistent with
BG_c + \Sigma^1_1 induction.
   --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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