[FOM] Definability beyond HOD

Dmytro Taranovsky dmytro at mit.edu
Wed Mar 21 18:15:00 EDT 2012

Ordinal definability has traditionally been considered an outer limit to 
what is definable in V.  However, I believe that by extending the 
language of set theory through iterations of reflectiveness, we can 
transcend that limit.

Reflective cardinals were discussed and axiomatized in my last posting, 
and by themselves they are not sufficient to transcend HOD since the 
restriction of R (the predicate for reflectiveness) to an ordinal is 
ordinal definable. (Specifically, R|alpha is definable from alpha, beta, 
gamma where gamma > beta > alpha and R(beta) and R(gamma).)  However, 
the notion of R can be iterated.

An ordinal is 0-reflective iff it is regular uncountable.
An ordinal kappa is n+1 reflective iff the theory (V, in, kappa, R_n) 
with parameters in V_kappa is correct -- that is it agrees with theory 
of (V, in, lambda, R_n) for all lambda>kappa with sufficiently strong 
reflection properties.  Here R_n is the predicate for n-reflective 
An ordinal is omega-reflective iff it is n-reflective for every finite n.

Axiomatization proceeds similarly to axiomatization of reflective (that 
is 1-reflective) cardinals, and is given in my paper.  The consistency 
strength for n+1-reflective cardinals (n finite) is between n-ineffable 
and n+1-subtle.

To understand the relation between omega-reflective cardinals and HOD, 
consider what happens in a well-understood model of set theory -- the 
constructible universe L.  Assuming zero sharp exists, every uncountable 
cardinal in V has sufficiently strong reflection properties in L, which 
makes R_n (n<=omega) for L definable in V.  For all finite n, R_n (that 
is R_n for L) can be added as a predicate symbol to L without 
introducing nonconstructible sets.  However, R_omega for L consists 
precisely of the Silver indiscernibles for L and thus transcends L.
Moreover, the analogous relation appears to hold for other canonical 
inner models for which the large cardinal structure has not been 
iterated away to infinity.  For example, it holds for L[U] (the minimal 
inner model with a measurable cardinal) but not in K^{DJ} which is 
obtained from L[U] by iterating away U to infinity.

omega-reflective cardinals form indiscernibles for V, and it is likely 
that they can be used to transcend HOD.  Note that while an arbitrary 
set can be coded into HOD through forcing, V is canonical, so no such 
coding is implemented in V -- by maximality and symmetry of V, HOD is, 
in a sense, minimal.  Assuming that it is not ordinal definable (and 
that it can be used to enumerate all ordinal definable reals), we define 
HOD Sharp to be the real number coding:
{phi: V_kappa satisfies phi(kappa_1, ..., kappa_n) where kappa_1 < ... < 
kappa_n < kappa are omega-reflective}.

To extend HOD sharp and more beyond real numbers, we propose the 
following conjecture:
Conjecture:  For every nonempty set of ordinals S, every ordinal 
definable from S subset of sup(S) (supremum of S) is definable (in (V, 
in)) from S, an element of sup(S), and a finite set of omega-reflective 
cardinals above sup(S).

We continue our climb in the levels of expressiveness.  Since 
omega-reflective cardinals are indiscernibles for V at the level of 
finite sequences, the next big step is reflective omega-sequences:  A 
set S of ordinals of order type omega is reflective iff for every alpha 
< sup(S), the theory of (V, in, S\alpha) with parameters in V_alpha is 
correct:  That is it agrees with the theory of (V, in, T) for every set 
of ordinals T of order type omega with sufficiently strong reflection 
properties and min(T) > alpha.  (A weaker extension (consistent with 
V=HOD) is to continue iterating reflectiveness to ordinals >omega, which 
corresponds to weak versions of reflective omega-sequences.)  This 
formulation negates V=HOD outright:

Theorem:  There is a Sigma-V-2 formula phi with one free variable such 
that there is no ordinal definable set of ordinals S with phi(S) <==> 
phi(S\{S_0}) (S\{S_0} is S with the first element removed).

Given the vastness and non-arbitrariness of V, it is intuitively clear 
that given phi, there is S with phi(S) <==> phi(S\{S_0}).  We can simply 
pick infinitely many ordinals that are effectively indistinguishable, 
and since we control S, the ability of phi to reference S should not be 
a problem.  Infinite Ramsey theorem and Galvin-Prikry theorem 
intuitively appear much stronger than phi(S) <==> phi(S\{S_0}).  
Admittedly, however, an analogous argument exists against the axiom of 
choice, but the key difference here is that phi is a formula rather than 
an arbitrary set.

I have long held the negation of V=HOD to be intuitively true -- for 
example, if R includes all the reals, how can we possibly define a 
well-ordering of R?  However, if V=HOD false, then the question arises 
how do we define some sets that are not ordinal definable, and is there 
a natural extension to the language of set theory that as a concept 
refutes V=HOD?  My work here gives a preliminary answer to this question.

As in my last posting, the results -- and much more -- are in my paper:

Dmytro Taranovsky

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