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

[Dmytro Taranovsky]

Joe Shipman wrote:
> 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" ?

The consistency strength of ZFC with that statement (and beta>alpha; Nate
Ackerman's answer is technically correct) is slightly above ZFC + {there is
Sigma-n correct inaccessible}_n (which is above ZFC + there is a proper class
of inaccessible cardinals), and is below ZFC minus power set plus there is a
Mahlo cardinal.  

Being an elementary substructure implies that V(alpha) satisfies correct
statements with parameters in V(alpha).  If we only allow formulas without
parameters, then the statement would be stronger than ZFC + there is an
(infinite) ordinal k such that there are at least k inaccessible cardinals and
L_k satisfies ZFC, but weaker than ZFC + there are omega_1 inaccessible
cardinals.

Dmytro Taranovsky