[FOM] Higher Order Set Theory
Dmytro Taranovsky
dmytro at MIT.EDU
Wed Mar 9 16:25:41 EST 2005
Roger Bishop Jones wrote:
>Higher order set theory is a great deal less problematic than
>you make it appear.
The problem is whether higher order statements have any meaning or
are just symbols on paper. It is possible to axiomatize something
that looks like higher order set theory, and severals proposals
have been made. However, without semantics or guiding ideas, we
have no way to choose the formalization. For example, we would not
know whether to include the axiom of global choice.
Second order logic about, say, integers is meaningful because a
predicate on integers can be treated/defined as a set of integers.
By contrast, the universe includes every set, that is every
collection of objects.
If all predicates on sets had their own independent existence, then
we could make sets of proper class predicates, contradicting the
totality of the universe. Since they do not, we have to explain
what does it mean that there is a predicate satisfying such and
such conditions. (We can still talk about particular predicates.)
Third order set theory is still more problematic.
Fortunately, we do not have to debate metaphysical meaningfulness
of the notion of existence of a property. By using reflective
ordinals, we can recharacterize questions about predicates as
questions about sets.
Also, for ordinary set theory, we do not have to claim that V
exists. We could say that the universe is a convenient figure of
speech, and, for example, translate "large cardinals properties
realized in V" as "large cardinal properties for which there is a
set satisfying the property".
Dmytro Taranovsky
http://web.mit.edu/dmytro/www/main.htm
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