[FOM] The semantics of set theory

Ralf Schindler rds at logic.univie.ac.at
Wed Oct 9 06:05:34 EDT 2002

On Wed, 9 Oct 2002, Richard Heck wrote:

> So how deep does the analogy go?

   It should go pretty deep. We can define the truth predicate for set 
theory by a \Sigma^1_1 fmla of class theory and we can prove the Tarski 
schema in BG. However, we (provably) need more than BG in order to prove 
the Tarski rules (for instance, to prove that "if \phi(x) is true for all 
x then \forall v \phi(v) is true"); MK suffices, though. (This settles my 
debate with Vladimir; I was happy with proving the Tarski schema, he wanted 
to have the Tarski rules proven.) I think a natural thy which would prove 
the Tarski rules is BG + \Sigma^1_1 comprehension + \Pi^1_2 separation, 
which also proves the consistency of ZF. (I'm assuming we have \Sigma^1_1 
replacement at hand by how replacement is formulated in BG.)  
   I'm entirely ignorant about what can be found in the literature but 
would be interested in getting information about this.
   Best wishes, Ralf

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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