[FOM] Higher Order Set Theory

[Dmytro Taranovsky]

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