# [FOM] Full reflection does not imply inaccessibility

Ali Enayat enayat at american.edu
Fri Mar 11 16:02:35 EST 2005

In his message of March 10, 2005, Nate Ackerman writes:

>So, natural models ofAckermann set theory are (V_\alpha, V_\beta) where
>V_\alpha is an
>elementary substructure of V_\beta (and hence \alpha is an inaccessible).

The parenthetical statement is false.  Indeed, if V(beta) is a model of ZFC
such that beta has uncountable cofinality, then the first alpha such that
V(alpha) is an elementary substructure of V(beta) has countable cofinality,
and is therefore not inaccessible.

To see this, let alpha_n be the first ordinal less than beta such that
V(alpha_n) is a Sigma_n elementary submodel of the universe (such an alpha_n
exists by Levy's version of the Montague-Vaught- reflection theorem). The
desired alpha is the supremum of the alpha_n's.

Regards,

Ali Enayat