# [FOM] What is second order ZFC?

Cole Leahy cleahy at mit.edu
Thu Sep 5 13:31:05 EDT 2013

On Tue, 3 Sep 2013, Martin Dowd wrote:

> **
> Suppose at stage alpha one adds the subsets which are second order
> definable. Call the result L_2.  Clearly L\subseteq L_2\subseteq V, so if
> V=L all three are equal. Obvious questions include the following. Is L_2 a
> model of ZFC? ... Is it consistent that L_2=neq V? ... Is L_2=L? ... What
> kind of sets might be in L_2-L? Is CH true in L_2?
>

If memory serves, Myhill and Scott showed in their "Ordinal Definability"
(1971) that L_2 = HOD follows from AC. We can therefore answer your
questions by noting that ZF + L != HOD != V and ZFC + V = HOD + ~CH are
consistent relative to ZF. (These facts are apparently due to McAloon.) I'm
not sure whether AC can be added in the first case.
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