[FOM] What is second order ZFC?

[MartDowd at aol.com]

Harvey,
 
I found this question confusing, but it prompted me to consider V=L and  
second order set theory.  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 (i'm pretty sure it is)
 Is it consistent that L_2=neq V? (I think the generic model over L_2  by a 
Cohen real shows that it is)
 Is L_2=L? (I couldn't see an obvious proof)
 What kind of sets might be in L_2-L?
 Is CH true in L_2?
 
- Martin Dowd
 
 
In a message dated 9/2/2013 9:00:06 A.M. Pacific Daylight Time,  
hmflogic at gmail.com writes:



I will jump start the ensuing discussion by asking these questions:


1. Is CH an axiom of ZFC?
2. Is CH an "axiom" of second order ZFC"?
3. Is CH a theorem of ZFC?
4. Is CH an "axiom of second order ZFC"?



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