# [FOM] Set Theory with Reflective Sequences

Dmytro Taranovsky dmytro at mit.edu
Sun Mar 1 12:45:59 EST 2015

Within the language of set theory, one reaches higher and higher
expressive power by climbing higher in the cumulative hierarchy of V.
But how can we go further once the language allows quantification over
the whole V?  Intuitively, we would want to continue the hierarchy above
V, except that all sets are already in V.  The solution is to label a
cardinal kappa such that V_kappa is sufficiently close to V, and
continue the hierarchy above V_kappa. V_kappa represents V, and with
kappa labeled in an extended language, hierarchy above V_kappa
corresponds to higher order set theory.

To go further, we can iterate higher order set theory by picking
lambda>kappa with V_lambda representing V, and then mu>lambda, and
continuing to longer sequences of ordinals.  Thus, continuing the
cumulative hierarchy above V corresponds to labeling certain ordinals
that are sufficiently similar to Ord -- in other words, certain ordinals
with sufficiently strong reflection properties -- while staying within V.

How do we choose the right kappa?  The answer is that we postulate a
certain degree of symmetry and reflection in V.
Convergence Hypothesis (general form): For an appropriate type of
objects, all objects of that type with sufficient reflection properties
are, in a certain sense, indistinguishable from each other.
Convergence Hypothesis (ordinals): If alpha and beta are ordinals with
sufficiently strong reflection properties, then phi(S, alpha) <==>
phi(S, beta) whenever rank(S) < min(alpha, beta) and phi is a first
order formula of set theory with two free variables.
Definition: kappa is a reflective cardinal, denoted by R(kappa), iff
(V,in,kappa) has the same theory with parameters in V_kappa as
(V,in,lambda) for every cardinal lambda>kappa with sufficiently strong
reflection properties.
Example:  In L (assuming zero sharp), every Silver indiscernible has
sufficient reflection properties, which allows us to define R^L in V.
Note:  A formalist can treat the Convergence Hypothesis as a guiding
principle to get good systems of axioms.

To the extent that it holds, the Convergence Hypothesis allows us to
define new reflective notions without ambiguity.  We specify a notion by
stating its type (example: a pair of ordinals) along with the class of
predicates for which it agrees with all objects of sufficient reflection
properties.  See "Reflective Cardinals" (2012,
http://arxiv.org/abs/1203.2270 ) for analysis, axiomatization, and
theory of reflective cardinals.  Here, I will introduce the theory of
infinite reflective sequences.

Definition: Given a length condition P (such as having length omega), an
increasing sequence of ordinals S satisfying P is reflective iff
(1) the theory of (V, in, S) with parameters in V_min(S) is correct,
that is it agrees with (V, in, T) for every T satisfying P and having
sufficient reflection properties and min(T)>min(S), and
(2) for every alpha<sup(S) S\alpha is reflective under the above
criterion (where the length condition is modified to correspond to
S\alpha if S\alpha is "shorter" than S).
A variation on the notion is to omit the second condition.

To understand the hierarchy of reflective sequences, imagine a process
of picking ordinals with sufficient reflection properties, and with the
process itself having sufficient reflection properties.  Let S be a
resulting sequence.  Here are some stages in the process:

1. n ordinals for finite n.  This is well-understood (see "Reflective
Cardinals" for theory).
2. omega ordinals.  See "Reflective Cardinals" for some results. This
notion contradicts V=HOD (assuming that removing the first element does
not change its theory), which makes analysis and finding of canonical
models difficult.  It may be related to Prikry forcing.
3. alpha ordinals for countable alpha.  We expect reflectiveness to be
closed under subsequences that preserve order type.
4. omega_1 ordinals.  Assuming the Continuum Hypothesis (CH), we cannot
require closure of reflectiveness under all subsequences of the same
length. Even without CH, this limitation holds at omega_2.
5. alpha ordinals for a fixed ordinal alpha > omega_1 (min(S)>alpha).
6. More complicated conditions about the length of S, such as
|S|=min(S). S excludes its limit points.
7. Length of S has sufficiently strong reflection properties relative to
S. S excludes its limit points.
8. To go beyond (7), we assume that S includes its limit points at
places with sufficient (relative to S) degree of reflectiveness. One may
then state conditions on the length of S such as "S includes exactly
omega limit points, and they are cofinal in S" or "S has maximum element
and it is the only element such that S below it is stationary".
9. Length of S has sufficiently strong reflection properties relative to S.

To go beyond what is expressible with S, we mark some points on S that
have sufficient (relative to S) degree of reflectiveness, analogously to
how S is obtained by marking some ordinals with sufficient
reflectiveness in V.  To go further, we then add a third type of
marking, and in general, for each ordinal in S, we can assign a degree
of reflectiveness through a function f:S→Ord\{0}. For a limit alpha,
f(x)=alpha corresponds to x receiving all markings <alpha.  In the
definition of reflectiveness, use f in place of S, min(dom(f)) in place
of min(S) (and min(dom(T)) in place of min(T)), sup(dom(f)) in place of
sup(S), and in place of S\alpha, use f restricted to Ord\alpha modified
to make f(min(dom(f)\alpha)) equal 1.  Here are some notions. (Here
S=dom(f).)

10. For a fixed ordinal alpha, f(sup(S))=alpha, and sup(S) is the only
ordinal s with f(s)=alpha (and min(dom(f))>alpha).
11. f(sup(S))=sup(S), and sup(S) is the only ordinal kappa with
f(kappa)=kappa.
12. To go beyond f(kappa)=kappa, we modify the condition -- that
f(kappa)=alpha+1 implies that kappa has sufficiently strong reflection
properties relative to f that is clipped to alpha -- by invoking an
ordinal notation system (for ordinals >=kappa) or just a comparison
method between f(lambda) and f(kappa), and considering f clipped below
f(kappa) as defined through the comparison method.  A reasonable
stopping point is the following: There is only one ordinal kappa with
f(kappa)=(kappa^+)^HOD, and it equals sup(dom(f)).

An axiomatization of notion 9 and notion 12 is in my paper (linked
below).  The axiomatizations avoid the inconsistency in my FOM posting
"Axiomatization of Reflective Sequences" (12 Feb 2015), but their
consistency remains unclear.  Here I will note a connection with inner
model theory.

The type of f and many of its basic properties can be formalized using
inner models.  For example, for notion 11 we can require that there is
an inner model M of ZFC such that
- M is an iterate of the minimal inner model with o(kappa)=kappa for
some kappa (o refers to Mitchell order)
- dom(f) is the set of measurable cardinals in M
- for every kappa in the domain of f, f(kappa)=o(kappa)^M

The possibilities for iteration appear sufficiently rich to get any
reasonable candidate for notion 11.  Also, one can similarly specify M
for other notions.

If a function f satisfies the condition on M, then the theory of
(L(f),in,f) is independent of f, which is one example of the Convergence
Hypothesis, and moreover, we can study this theory to get a better
understanding of f.

It is unclear how far the Convergence Hypothesis extends in V, and if it
reaches that far, how to handle M with overlapping extenders.

For more details, see my paper:
http://web.mit.edu/dmytro/www/ReflectiveSequences.htm

Sincerely,
Dmytro Taranovsky
http://web.mit.edu/dmytro/www/main.htm