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

Nate Ackerman nate at math.mit.edu
Thu Mar 10 23:33:31 EST 2005

Hi Joe,
> 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" ?

I believe it suffices to have a single inaccessible, although I don't
remember the proof off the top of my head (but I don't think it is that
hard). I also remember seeing a proof of the other direction. That if you
have V_\alpha < V_\beta, then V_\alpha \models ZF, but once again I don't
remember the proof off the top of my head.

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

Nope, I am not related at all to the original Ackermann,

Nate