[FOM] Higher Order Set Theory
Dmytro Taranovsky
dmytro at MIT.EDU
Mon Mar 7 10:39:20 EST 2005
Since the universe includes all sets, second order statements about V appear
doubtful or meaningless. However, there is a way to make them meaningful, and
to get a reasonable axiomatization of theory, thus resolving how to deal with
proper classes, "collections" of proper classes, and proper class categories
such as the category of all groups.
The key idea is set theoretical reflection. An ordinal kappa has high
reflection properties by being "large" relative to lower ordinals and by
V(kappa) resembling V. The largeness is best asserted by saying that there are
ordinals below kappa resembling kappa. The universe is limitless, so large
that there are ordinals arbitrarily similar to it with respect to lesser
ordinals.
It is likely that as we reach ordinals of higher and higher reflection
levels, their properties converge. As this happen, ordinary language (first
order logic with membership relation) begins to fail in identifying them: How
does one say that (V(kappa), in) is an elementary substructure of (V, in)? For
any two such ordinals, the theory of (V(kappa), in) is the same.
Ultimately, one reaches the level at which ordinary language cannot
tell the difference in their properties with using a parameter at least as large
as one of the ordinals. At that point, a new word is needed: kappa is a
reflective ordinal, denoted by R(kappa), iff (V, kappa, in) has the same theory
with parameters in V(kappa) as (V, lambda, in) where lambda is any ordinal
>kappa with sufficiently strong reflection properties.
For example, in L every Silver indiscernible has sufficiently strong
reflection properties, so the notion of reflective ordinals for L makes sense.
Reflective ordinals satisfy all large cardinal properties (such as being
inaccessible) that are expressible in ordinary language and are realized in the
universe.
By the reflection principle, in so far as higher order set theory is
meaningful, a higher order statement with parameter x is true about V iff it is
true about V(kappa) with kappa reflective and x in V(kappa). For example, an
infinitary statement phi (can use infinite disjunctions such as x=1 or x=2 or
..., but not infinite strings of quantifiers (except with disjoint scope)) in L
is true about L iff it is true in L_kappa for an indiscernible kappa with phi
in L_kappa. (Full second order logic about L is problematic since
constructibility is not closed under subsets.)
The potential meaning of higher order set theory is thus uniquely
fixed. Therefore, we can officially define higher order semantics (with set
parameters) through reflective ordinals.
I will describe an axiomatization of reflective ordinals in my next posting
(not counting replies). Meanwhile, more information can be found in my paper:
http://web.mit.edu/dmytro/www/NewSetTheory.htm
Dmytro Taranovsky
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