[FOM] The defence of well-founded set theory

[Aatu Koskensilta]

On Oct 5, 2005, at 10:11 AM, Roger Bishop Jones wrote:

> On Tuesday 04 October 2005  4:39 am, Aatu Koskensilta wrote:
>> NBG is consistent with the usual conception of set theoretic
>> hierarchy in a rather trivial sense: one can consider the
>> totality of classes to consist of properties or collections
>> definable in the language of set theory. The intelligibility
>> of the language of set theory itself seems to imply the
>> acceptability of such a totality of classes.
>
> I find this hard to swallow.
...
> However, as soon as we try to make the totality of
> pure well-founded sets into a collection we run into
> trouble, because of course it must be and cannot be
> a pure well-founded set.

Yes, which is why its a proper class of particularly simple sort: 
definable without parameters in the language of set theory. To 
understand claims such as "there is a set x in the cumulative hierarchy 
such that P(x)" one needs not assume more than that the claim "exists 
alpha, x in V_alpha" makes sense. Quantification over definable classes 
is reducible to quantification over formulas + a truth predicate. 
Quantification over formulas we already have, so all we need is a truth 
predicate. And we don't even need a full fledged truth predicate for 
which the Tarskian properties are provable, we need only one for which 
the T-schema holds. This we have in NBG. (In fact, NBG can be 
axiomatized by using just such a truth predicate).

> If indeed "The intelligibility of the language of
> set theory itself seems to imply the acceptability
> of such a totality of classes" we would have to
> conclude that set theory is unintelligible.

Hopefully my comment above clarified what I meant. Surely if we 
understand what set theoretic formulas mean we can quantify over them 
and make claims about their applicability to some set. This is all we 
need to make sense of the predicative classes of NBG. We would need to 
assume a totality of classes existing in some substantial sense only if 
we allowed classes defined by quantification over all classes.

> Of course, all this changes if we compromise by
> interpreting set theory in one or more V(alpha)
> rather than in V as a whole.

Yes, but to me this is rather uninteresting.

>
>> The only
>> controversial thing in NBG not motivable in this fashion is
>> the global axiom of choice for which you need to use forcing
>> to prove conservativity.
...
> Even if you interpret NBG in a V(alpha) rather than V
> (as I argued above one must) surely the choice sets must
> be there?

I was talking about the *global* axiom of choice, i.e. that there is a 
choice class for the universe. All other classes required to exist by 
the axioms of NBG are definable, but the global choice class (and the 
classes entailed to exist by its existence) is not. I too agree that 
choice naturally follows from the iterative conception.

>> NBG is of course only one first step in the sequence of
>> "predicative" class extensions of ZFC arguably as acceptable
>> as ZFC and the notion of set theoretic truth. These theories
>> are all weaker than ZFC+"There is an inaccessible" (in the
>> sense that ZFC+IA proves every set theoretic sentence provable
>> in any of these extensions of ZFC).
>
> Is there a rationale for sticking here to "predicative" class
> extensions?

Yes. They allow us to quantify over classes without any substantial 
ontological commitment to existence of non-set collections. I am, as 
you seem to be too, rather skeptical about higher order set theory, so 
its natural to me to see how much of it you can make sense without 
assuming existence of a determined totality of proper classes.

I have in mind the following picture. Take an inaccessible kappa, so 
that V_kappa is a model of ZFC. Consider then the structure V' = 
<V_kappa,L[V]_alpha,epsilon> where alpha is the least ordinal s.t.

  1) alpha>kappa
  2) if there is a well-ordering of type b in some L_c with c < alpha, 
then b < alpha

Now, someone "living" in V_kappa can make sense of assertions about V' 
if he understands the notion of set theoretic truth, i.e. has a truth 
predicate for sentences in the language of set theory and can iterate 
it along any well-ordering (class or set) he can come up with. 
Quantification over e.g. L[V]_1 is just quantification over collections 
definable with parameters in V_kappa and so forth. Any well-ordering 
required to iterate the truth predicate to make sense of assertions 
about L[V]_alpha already occur below alpha, i.e. are obtained by fewer 
iterations of the truth predicate.

Next, instead of V_kappa we take V, the universe. Exactly as in the 
above picture we can make sense of (some) assertions about collections 
anologous to "L[V]_alpha". Of course, statements quantifying over 
classes need to be interpreted so that we don't assume a determined 
totality of predicative classes (because if there was such 
(predicatively) determined totality, we could predicatively prove the 
existence of classes not in the totality). So e.g. "for all X P(X)" is 
taken to mean "no matter what classes turn out to exist, P must hold of 
them" and "exists X P(x)" is taken to mean "there is a (class) ordinal, 
namely A, such that there is a class X definable by means of iterating 
the truth predicate A times, such that P(x)". Formulating all this as a 
theory requires a bit of care and the use of inference rules, 
analogously to some systems of ordinary predicative analysis.

This is a natural extension of the reading of NBG I outlined above.

(Actually, I like to leave things a bit more open: instead of saying 
that the classes are all predicative in the sense outlined above, I 
just postulate that at least the predicative classes exist, suitably 
formulated.)

> This is just to keep the theory no more controversial than ZFC?
> So someone who accepts large cardinals would be rational to
> drop the predicativity?

No. Proper classes are collections (or properties or what ever) which 
are not sets. If there are large cardinals, there will still be 
collections of sets which are not sets, such as V, the class of 
ordinals or the class of all true sentences (with parameters). The same 
picture still applies even if there billions of Woodin-cardinals or 
what you have.

Using such a predicative extension we have, without recourse to actual 
higher order commitments, many theorems which are obviously true in V, 
but not expressible in the language of set theory. For example, we get 
the existence of a V_alpha reflecting V. Also, certain things which 
usually require some coding to express in ZFC become directly 
expressible.

My interest in this kind of predicative extensions of ZFC lies in the 
possibility of formulating formal analogues of the idea that "V should 
contain all possible mathematical structures". To make some sense of 
"possible mathematical structure", which seems as elusive as "set with 
structure", I want to see what happens if we take "possible 
mathematical structure" to mean "class with structure" and formulate 
suitable reflection principles. In particular I want to see how far in 
the hierarchy of large cardinals this takes us.

Aatu Koskensilta (aatu.koskensilta at xortec.fi)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
  - Ludwig Wittgenstein, Tractatus Logico-Philosophicus