[FOM] Higher Order Set Theory [Ackermann Set Theory]

[Nate Ackerman]

I believe Ackermann set theory was an attempt to create a model of set 
theory which could deal with definitions of class of classes, class of 
class of classes, ect. As I understand it/think of it (and I am sure there 
are people on this list who know more about it than I do), the view was 
motivated by the idea that our class of all sets is just an initial 
segment of in the hierarchy of the universe of all classes.

And what is more (in some sense) any statement which is true in the 
universe should be true in the class of all sets. So, natural models of 
Ackermann set theory are (V_\alpha, V_\beta) where V_\alpha is an 
elementary substructure of V_\beta (and hence \alpha is an inaccessible). 
(It is also worth mentioning though that it has been shown that Ackermann 
set theory is equiconsistent with ZF)

Nate