[FOM] What is second order ZFC?

[jkennedy at mappi.helsinki.fi]

My paper in the current September issue of the BSL puts second order L  
into a wider context and discusses examples of L constructed with  
different logics, some of which are fragments of second order logic.  
(This work is joint with Magidor and Vaananen).


Quoting Cole Leahy <cleahy at mit.edu>:

> 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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