[FOM] Extending the Language of Set Theory

[Aatu Koskensilta]

On Feb 27, 2005, at 8:07 PM, Dmytro Taranovsky wrote:

> Meaningfully extending the language of set theory opens new horizons 
> for
> mathematicians, but any such endeavor also raises a host of
> philosophical issues.  I plan to discuss some of the issues in future
> FOM postings.

Lately, I've been playing with ideas closely resembling yours, but from 
a slightly different angle. My emphasis has not really been on 
addressing the deficiencies of the language of set theory per se, but 
rather to see what we can get by trying to push two kinds of reflection 
- epistemological and set theoretical - as far as possible. By 
epistemological reflection I refer to the kind of reflection formalized 
by e.g. various proof theoretical reflection principles, iterated truth 
predicates and the like. By set theoretical reflection principles I 
refer to principles which are, in some sense, formalizations of the 
maxim UNIFY:

  Every possible mathematical structure should be exemplified as a set.

There are basically two kinds of attitudes one can adopt in study of 
extensions of ZFC by means of such reflection principles. One might be 
interested merely to provide an explication of what is "implicit in 
acceptance of ZFC" or some particular philosophical and mathematical 
position. On the other hand, one might be interested in actually 
establishing new mathematical results that are in some sense acceptable 
on basis of ZFC and intuitively plausible reasoning. The former leads 
to the kind of analysis exemplified in the classical results about 
predicative justifiability, the ordinal Gamma_0 and so forth. The 
latter is a more risky endeavor, but to me at least more interesting in 
the grand scheme of things.

When trying to formally capture UNIFY at least partially one hits the 
problem of defining what exactly is a possible mathematical structure. 
Any axiomatization of this concept seems to lead to just a new theory 
of sets so a more circumspect approach is called for. Luckily, as you 
have noted in your paper, various non-set collections - which can be 
seen as structures - make sense based on acceptance of ZFC. For 
example, the class of all true sentences of the language of set theory 
with a constant for every set makes sense, since the concept of set 
theoretic truth does. (If we don't accept that there is a determinate 
matter of fact as to the truth or falsity of set theoretic sentences 
what reason do we have to accept replacement for anything but upwards 
absolute formulae?).

Without further ado, let me present the formal system such musings 
naturally lead. The language is a two sorted language with a sort for 
sets and a sort for classes and a binary predicate for membership. As 
axioms for sets we have those of ZFC with replacement and separation as 
Pi^1_1 axioms. As to classes we have an axiom saying that V exists, for 
every set there corresponds a class of all classes corresponding to the 
members of the set and a rule of inference saying that if A is provably 
a class ordinal, then for all classes B, the class version of the 
constructive hierarchy relative to B up to A exists. This axiom 
basically says that if B makes sense, then anything constructible from 
it along a well-ordering which makes sense makes sense. As you note, 
the resulting theory is interpretable in ZFC+There is an inaccessible 
by taking V_kappa as V and the members of L[V_kappa]_alpha as classes, 
where alpha is the least ordinal, s.t.

  1. alpha > kappa
  2. if delta in L[V_kappa]_beta and beta < alpha, then delta < alpha

The advantage of this theory over ZFC+There is an inaccessible is that 
"in principle" all of its theorems are acceptable on basis of 
acceptance of ZFC. In addition, talk about Skolem functions of V, 
elementary substructures of V and so forth can be carried out. And 
since classes can be members of other classes, there is no need for 
tedious coding trickery.

We come now to set theoretical reflection. Since classes are certainly 
possible mathematical structures, there should be, for every class, a 
set that is in some sense structurally equivalent to the class. This I 
have, tentatively, formalized as follows. We add to the language a new 
binary function symbol c which takes a class structure A and a class 
ordinal B into a set structure of form <V_kappa, L[a]_alpha, a> with 
the following property

  for all x in V_kappa( <V,L[A]_B,A>|=phi(x) <=> 
<V_kappa,L[a]_alpha,a>|=phi(x))

Again, this is expressed as a rule of inference: if we have established 
that B is a class ordinal, then we can infer the above. In addition, we 
add the following rule of inference:

      Q_1A_1...Q_nA_n( <V,L[A]_B,A,A_1,...,A_n>|=phi)
  --------------------------------------------------------------
  Q_1a_1...Q_na_n( <V_kappa, L[a]_alpha, a, a_1,...,a_n> |= phi)

Where Q_i is a sequence of alternating quantifiers and A_i are class 
variables and a_i are set variables.

It's a rather trivial exercise to derive various small large cardinal 
axioms in this system. However, I don't know how far one can go. There 
are several directions for further extensions: the most obvious one is 
that the above analysis seems perfectly sensible and acceptable and 
hence there should be a natural model of the theory (by UNIFY). The 
problem is defining what exactly is a natural model of the theory, 
which is muddled by the presence of the special rules of inference.

I'd like to thank you for bringing up this interesting subject. Also, 
I'd be interested in the relation of the theory outlined above and the 
unfolding of set theory sketched by Solomon Feferman. I'll return to 
your paper in more detail as I've had a few moments to digest it all.

-- 
Aatu Koskensilta (aatu.koskensilta at xortec.fi)

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