FOM: Classes = sets + truth

[Volker Halbach]

Here are some comments and questions on Jeff Ketland's posting of 1 Feb.

Technical remarks:

PA(S) is PA plus "there is a full inductive satisfaction class"

1. PA(S) and ACA are L_{PA}-conservatively interpretable in each another.
The relation of the respective systems with arithmetical induction only is
more complicated. ACA_0 is easily shown to be conservative over PA (Harvey
Friedman provided a fairly general proof strategy for this and more
advanced results in a recent posting). 

John Burgess has emphasized that one wants to have conservativeness proof
that can be formalized in relatively weak systems. Till recently there was
only a complicated model-theoretic proof by Kotlarski, Krajewski and
Lachlan of the conservativeness of PA(S)_0 over PA. In particular, PA(S)_0
is not easly interpretable in ACA_0 (although it has been suggested in the
literature that arithmetical truth (satisfying the Tarski-clauses) is
definable in ACA_0). 

In fact, one can show that a truth predicate satisfying the Tarski clauses
can be defined in ACA_0 (this is an easy consequence of Lachlan's theorem),
though ACA_0 defines a truth predicate satisfying the T-sentences.

2. I am sorry that I caused the impression that DeVidi and Solomon
conjectured something like ZFC(S)=MK with respect to their set-theoretic
content. ZFC(S) looks much weaker than MK.

Jeff asked whether ZFC(S)_0= is equivalent to NBG. It seems that NBG can be
embedded in ZFC(S)_0 (in essentially the same way as ACA_0 can be embedded
in PA(S)_0). The other direction is harder. I doubt that one can define a
truth predicate in NBG that commutes with all connectives and quantifiers,
because something like Lachlan's theorem should show again that such a
truth definition in NBG is impossible. Again, this does not withstand
Mostowski's result that NBG proves the T-sentences (for sentences not
containing class variables).
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Volker Halbach
Universitaet Konstanz
Fachgruppe Philosophie
Postfach 5560
78434 Konstanz Germany
Office phone: 07531 88 3524
Fax: 07531 88 4121
Home phone: 07732 970863
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