[FOM] Axiomatization of Reflective Sequences

Dmytro Taranovsky dmytro at mit.edu
Thu Feb 12 18:49:58 EST 2015

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