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

Nate Ackerman nate at math.mit.edu
Thu Mar 10 01:34:09 EST 2005

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