# [FOM] Antwort: Re: Higher Order Set Theory [Ackermann Set Theory]

Klaus-Georg Deck deck at ba-mosbach.de
Fri Mar 11 13:35:35 EST 2005

Hi Nate and Joe,

as far as I remember, the proof is due to "Levy & Vaught: Principles of
partial reflection in the set theories of Zermelo and Ackermann, PacJMath
11 (1961), 1045-62.

Klaus-Georg

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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

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