[FOM] On full reflection at an inaccessible
enayat at american.edu
Fri Mar 11 15:06:54 EST 2005
This is a reply to Shipman's recent query (March 16), who wrote:
>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" ?
1. Let T be the theory described by Shipman in the above paragraph. A strict
upper bound for the consistency strength of T is the consistency strength of
ZF + "there is a Mahlo cardinal". Even better: the latter theory implies the
former theory. This is because for an inaccessible beta, the set of alpha's
below beta such that V(alpha) is an elementary submodel of V(beta) form a
closed unbounded subset of beta [by a routine Skolem hull argument].
2. It is easy to see that T indeed proves the assertion "there is a proper
class of inaccessibles". For otherwise, there would be an upper bound theta
to the inaccessibles in V(alpha), and by elementarity, theta would also
serve as an upper bound to inaccessibles in V(beta), thus contradicting the
inaccessibility of alpha in V(beta).
3. This contradicts the claim of Shipman, where he writes:
>"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.
The resolution is as follows: if there are more than continuum many
inaccessibles, then two would surely have the same first order theory, but
this does not guarantee elementarity.
4. Indeed, even the assertion V(beta) is a model of ZFC + "there is a proper
class of inacessibles" is strictly weaker than "V(alpha) is an elementary
submodel of V(beta) and alpha is inaccessible", assuming ,say, that there
exists a Mahlo cardinal, kappa.
To see this, just look at the *first* ordinal beta such that V(beta) is a
model of ZFC + "there is a proper class of inaccessibles" [by reflection
over V(kappa), as in (1), such an ordinal exists]. There is no ordinal
alpha below beta such that V(alpha) is an elementary submodel of V(beta),
for if such an alpha existed, then it would also have to satisfy ZFC +
"there is a proper class of inaccessibles", thereby contradicting the
minimality of beta.
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