[FOM] Set Theory with Reflective Sequences

[Dmytro Taranovsky]

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