# [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
Best wishes, Ralf

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Ralf Schindler                                Phone: +43-1-4277-50511
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