[FOM] What is second order ZFC?

Harvey Friedman hmflogic at gmail.com
Mon Sep 9 14:50:15 EDT 2013


If you think that there is an explainable foundational/philosophical
purpose to your work in 2nd order logic and related matters, it would be
nice for you to give a brief (or not so brief) explanation of that, rather
than refer to a publication - especially a publication without a link.
Experience shows that almost no one will go to the trouble of looking it up
(they need library priveledges), but would read what they see in front of


On Fri, Sep 6, 2013 at 4:03 AM, <jkennedy at mappi.helsinki.fi> wrote:

> 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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