[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.
Definition:
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
cardinals.
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:
http://web.mit.edu/dmytro/www/ReflectiveCardinals.htm
Sincerely,
Dmytro Taranovsky
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