FOM: Classes = sets + truth?

[Jeffrey John Ketland]

Classes = sets + truth?

Given the recent discussion about class theories (set theories with 
extra axioms for proper classes), I wondered if I could interject 
some ideas (some old, some newer) about the relationship between

	(i) set theories (like Z, ZFC, etc.)
	(ii) axiomatic theories with a primitive truth predicate added
	(iii) class theories (like NBG and MK)

I. Arithmetic and truth predicates

One year ago I published my first (!) paper about what happens 
when you add a primitive truth predicate to an axiomatic theory 
(like PA). The paper is called “Deflationism and Tarski’s Paradise” 
and appeared in the philosophical journal Mind 108 (Jan 1999). At 
that time, I didn’t know about the large technical literature 
(especially on satisfaction classes) that already existed.

However, thanks to extremely valuable guidance from Volker 
Halbach, I now know much more about what happens when you 
truth predicates to arithmetic.

You can simply add the Tarski biconditionals to PA and that gives 
a conservative extension. You can add (what Volker calls) the 
uniform Tarski biconditionals to PA and that gives a conservative 
extension too. You can add Tarski’s inductive axioms for 
satisfaction and that gives a conservative extension if induction is 
not extended. If you add Tarski's axioms and expand the induction 
scheme, that gives a *non-conservative extension* (the latter case 
was the case I mentioned in my paper, for it impacts on some 
issues concerning the so-called “deflationary theory of truth”). The 
resulting system, PA + Tarskian truth (with induction expanded) is 
known as PA(S) by model theorists (Richard Kaye 1991: Models of 
Peano Arithmetic). Much of this has been studied by Sol Feferman 
before (e.g., “Reflecting on Incompleteness”, JSL 1991).

Volker has told me that PA(S) is bi-interpretable with ACA (and the 
restricted systems are similarly related). Volker also tells me that 
there are much more powerful truth-theoretic extensions (studied 
by Sol Feferman and Andrea Cantini) which reduce systems of 
predicative analysis. (I hope Volker can perhaps make some 
comments about what I say below).

So, in a sense, you can "eliminate" reference to arithmetically 
definable sets of numbers (those provided for by ACA) by 
introducing a primitive satisfaction/truth predicate and talking about 
which arithmetic formulas are *satisfied* by which (codes of 
(sequences of)) numbers.

II. Set theory and truth predicates

Here’s my question: How does this all work for set theories? And is 
there a close connection between set theories with a truth 
predicate and class theories?

Take ZFC and add a primitive satisfaction predicate S(x,y) 
[meaning "the formula coded by the set x is satisfied by the 
sequence y of sets"], with the usual Tarskian inductive axioms, 
and extend all the schemes to include formulas containing the 
satisfaction predicate. Call this theory ZFC(S). Volker tells me that 
Mostowski and others worked on this kind of thing back in the 40s 
and 50s. Volker also mentions to me that it has been conjectured 
(by Solomon and de Vidi in the Journal of Philosophical Logic 
1999) that this theory is equivalent to Morse-Kelly class theory.

So, here’s a nice technical question:

	(i) Is ZFC(S) equivalent to MK?

(where “equivalent” means “relatively interpretable, both ways”).
Now suppose that we add a satisfaction predicate, but don’t extend 
any axiom schemes in ZFC. Call the system ZFC(S)_0. We know 
that the class theory NBG is a conservative extension of ZFC. So:

	(ii) Is ZFC(S)_0 equivalent to NBG?

(If true, is it obviously true? If true, it partially settles a current 
debate involving Stewart Shapiro, Hartry Field and me about truth-
theoretic extensions).

More generally, is it true that for each set theory S, there is some 
canonical class theory C such that S + Truth = C? I.e., for each 
set theory, adding a truth predicate yields a class theory. Which 
class theory do you get? Can we invert the truth-extension 
operation: given a class theory C, what is the set theory S such 
that S + Truth = C? (Obviously this will depend upon the axioms for 
the primitive satisfaction predicate). Is there a class theory which is 
stronger than any truth-theoretic extension of any set theory?

Even more generally: what is the correspondence between set 
theories, class theories and the concept of (set-theoretical) truth?


Dr Jeffrey Ketland
Department of Philosophy, C15 Trent Building
University of Nottingham, University Park,
Nottingham, NG7 2RD. UNITED KINGDOM.
Tel:    0115 951 5843
Fax:    0115 951 5840
E-mail: <Jeffrey.Ketland at nottingham.ac.uk>