FOM: Classes = sets + truth?
Jeffrey John Ketland
Jeffrey.Ketland at nottingham.ac.uk
Tue Feb 1 12:00:37 EST 2000
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>
More information about the FOM
mailing list