[FOM] Extending the Language of Set Theory
Aatu Koskensilta
aatu.koskensilta at xortec.fi
Tue Mar 1 02:34:56 EST 2005
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
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