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

JoeShipman@aol.com JoeShipman at aol.com
Thu Mar 10 16:54:25 EST 2005

Nate, are you any relation to the original Ackermann, losing
a terminal "n" on Ellis Island?

You write:
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)

OK, so Ackermann set theory is equiconsistent with ZF, but 
what is the consistency strength of "there exists (V_\alpha, 
V_\beta) where \alpha is inaccessible and V_\alpha is an 
elementary substructure of V_\beta" ?

"There are more than continuum many inaccessible cardinals" 
certainly works, because then at least two have the same 
theory, and the corresponding ranks satisfy that the lower is 
an elementary substructure of the higher.  If there are 
exactly continuum many inaccessible cardinals, it seems one 
could apply Easton's theorem to construct a model with this 
many inaccessibles where each of the corresponding ranks has 
a different first-order theory, so maybe this is an exact 
answer, but I need to check that this actually works.

-- JS

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