[FOM] Extending Higher Order Set Theory

Dmytro Taranovsky dmytro at MIT.EDU
Sat Mar 19 14:21:36 EST 2005

The language of set theory with reflective ordinals satisfies current 
mathematical needs, and it is too early to accept further extensions.  
However, the following two related questions may make further study important:

Is there a definable set that is not ordinal definable?

Is it possible to define (the real number coding) first order set theory
locally (without invoking large cardinals)? 

An affirmative answer to the second question could radically revise our 
understanding of the mathematical universe. It would imply existence of 
properties inherently more expressive than those currently accepted.

A reflective ordinal kappa is an ordinal sufficiently large relative to 
smaller ordinals for the theory (V, \in, kappa) with parameters in V(kappa) to
be "correct".  The idea of reflective ordinals can be iterated and applied to
other extensions of set theory.  For example, an ordinal kappa is reflective 
in second degree iff the theory (V, \in, kappa, R) with parameters in V(kappa)
is correct, where R is the predicate for reflectiveness. In L, the
indiscernibles are reflective in every finite degree, but non-existence of 0#
in L prevents formalization except for fixed finite degrees.

Higher degree reflective ordinals can be axiomatized inductively using
elementary embeddings analogously to axiomatization of reflective ordinals. In
the models, the critical point kappa (representing Ord) can be reflective in
second degree in j(V(kappa)); j can extend first degree reflectiveness in
V(kappa) until j(kappa); and second degree reflectiveness of kappa can be used
to define such reflectiveness below kappa.  (However, it is unclear whether
using a single elementary embedding for all degrees of reflectiveness is

A different extension is through reflective sequences.  In the most general
form, the predicate (call it P) is defined by (if what is below counts as a
1.  Every set of ordinals that excludes its limit points and has sufficiently
strong reflection properties (for its order type) satisfies P.
2.  P(S) iff S is a set of ordinals and for every set of ordinals T of 
the same order type satisfying P, 
for all x \in V(min(S, T)), Th(V, \in, S, x) = Th(V, \in, T, x).

The predicate for reflective ordinals of finite degree n has slightly greater
expressive power than the predicate for reflective n-tuples of ordinals.  For
example, {reflective ordinal of second degree, a larger reflective ordinal} 
is a reflective pair.  However, infinite reflective sequences have greater
expressive power than iterations of reflectiveness:  If S is such a sequence,
then for every ordinal alpha, reflectiveness of degree alpha restricted to
min(S) is definable from S and alpha.  It is unclear if infinite reflective
sequences exist and how to axiomatize them.

For a fixed natural number n and ordinal alpha, reflectiveness for n-tuples
below alpha is ordinal definable (this also applies to L).  However, could it 
be that a real number is ordinal definable iff it is definable from a 
reflective n-tuple of ordinals?

The way to proceed may be study analogues of these notions for hereditarily
countable sets since the theory of such sets is well-understood.  Will such
studies lead to a local definition of first order set theory?

Additional background information can be found in my postings "Higher Order 
Set Theory" and "Axiomatizing Higher Order Set Theory", as well as my paper:

Dmytro Taranovsky

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