[FOM] Axiomatization of Reflective Sequences

[Dmytro Taranovsky]

Previously, I introduced the notion of reflective cardinals (cardinals 
that are sufficiently similar to the class of ordinals Ord), developed 
their theory and used them to provide semantics to higher order theory. 
A key intuition is that for certain types of objects, all objects of 
that type with sufficiently strong reflection properties are, in a 
certain sense, indistinguishable from each other.  For example, all 
Silver indiscernibles are indistinguishable in L.  One then uses this 
intuition to develop an axiomatization for objects (of an appropriate 
particular type) of sufficiently high reflection properties.  See my 
2012 paper
http://arxiv.org/abs/1203.2270
and related FOM postings.

Here, I will attempt an axiomatization for infinite reflective sequences 
of ordinals; its self-consistency is not clear.  To accomplish that it 
is actually sufficient to have a single proper class reflective sequence 
R since using R we can define set-sized reflective sequences. All 
ordinals in R have sufficiently high reflection properties and R itself 
has sufficient reflection properties.

Language: The language of set theory plus unary predicate R.
Axioms:
1. ZFC
2. forall x  x intersect R exists (as a set)
3. forall a (R(a) ==> a is an ordinal)
4. (schema, n is a natural number) there is kappa and U such that
     - V_kappa is a Sigma^R_n elementary substructure of V
     - U is a kappa-complete normal ultrafilter on kappa
     - R(kappa) holds and
     - forall S in U forall T in U the Theory(V_kappa, in, S intersect 
R) with parameters in V_min(S union T) agrees with Theory(V_kappa, in, T 
intersect R).

Explanation of 4:
* kappa represents Ord (hence the need for Sigma^R_n schema) and is used 
since the language does not directly allow quantification over proper 
classes.
* U, roughly speaking, concentrates on ordinals with sufficiently high 
reflection properties.
* R(kappa) ensures that members of R are measurable (and more); since we 
postulate many measurable cardinals, it would be odd if members of R are 
not measurable.
* The last clause of 4 is a very strong homogeneity condition on 
elements of R.  Its consistency is unclear.  A trivial implication is 
that R intersect kappa is in U, which implies reflection properties for 
R as a whole such as existence of x in R with {y<x: R(y)} stationary.

Let us call a predicate P (expressible in (V, in, R)) invariant if it 
can be expressed as
P(x) <==> for all sufficiently large ordinals alpha phi(x, R\alpha) (phi 
does not use R except as shown).
One can easily prove that phi(x,R\alpha) does not depend on alpha 
provided that x and other parameters of phi (not counting R\alpha) are 
in V_alpha.  These predicates are also invariant in that given kappa and 
U for axiom 4, the predicate (with x and parameters of phi in V_kappa) 
is independent of what R is.  All predicates on integers (or on V_kappa 
where kappa is definable in (V,in)) are invariant.
Philosophically, one argues that all candidate R with sufficient 
reflection properties will agree about invariant predicates, and thus 
invariant predicates are unambiguous in V (given the intended semantics 
of R having sufficient reflection properties).  If there is maximal 
canonical R that is correct about invariant predicates in V, then one 
option is to identify 'R' with that R.  Otherwise (if we do not want 
ambiguity), we can restrict the language to invariant predicates by 
replacing R with S_n (one for each n) such that S_n(x) holds iff 
(V_sup(x), in, x) agrees with (V, in, R) with regard to invariant 
predicates that have Sigma^R_n phi with parameters in V_sup(x).

If the axioms are inconsistent, the natural approach is to find 
consistent weakenings of the schema that still capture the homogeneity 
of R.  One weakening, which may be a good starting point for analysis, 
is to let S and T have order omega, perhaps weakening 4 as follows:
(schema) there is regular kappa such that R(kappa) and kappa is a 
Sigma^R_n elementary substructure of V
(schema) forall S subset R forall T subset R forall a in V_min(S union 
T) phi(S,a)<==>phi(T,a) (S and T have order type omega; phi does not use 
R and has only parameters shown)

Besides inconsistency, one should watch out for axioms implying an 
anti-large-cardinal assumption, either in general or about R.  One 
should be able to strengthen the axioms by imposing stronger large 
cardinal properties on kappa and U.  One can also increase expressive 
power by adding more structure to R, such as having R' as a subclass of 
R with strong reflection properties, but the axiomatization is unclear.

A key question related to consistency of the axioms is:
Is it consistent that there is a cardinal kappa and a normal kappa 
complete ultrafilter U on kappa such that for every ordinal definable 
function f:P(kappa)->{0,1} there is X in U such that for all Y subset X 
with Y in U  f(X) = f(Y)?
Consistency results for weaker infinite exponent definable partition 
relations would also be helpful.

Sincerely,
Dmytro Taranovsky