[FOM] second-order ZF

[Robert Black]

Dear FOMers,

The following question only makes sense if you think (or are willing 
to pretend to think) that we have an unambiguous understanding of 
second-order logic in the sense that given an infinite set S, the 
notion of *every possible* subset of S has an absolute reference (and 
not just a reference relative to a particular model of first-order 
ZFC). For those of you who (like me or Zermelo: you're in good 
company!) think that:

Are there any considerations that rule out the following 
(undesirable!) possibility?

Perhaps first-order ZFC is consistent, but second-order ZF is 
unsatisfiable. In other words, first-order ZFC only manages to have 
models through having 'thin' models, where lots of the sets which 
ought to be there aren't. If they were all there, replacement would 
just keep on driving up the height of the universe so that one 
couldn't ever get a model. Or if you like: it's a truth of 
second-order logic that there's no genuinely inaccessible cardinal.

  I would like to know why this can't be (or, less ambitiously and 
more plausibly, why a modification of our reasons for thinking 
first-order ZFC consistent - those that go beyond the rather weak 
reason that we haven't found a contradiction yet - also lead to the 
conclusion that second-order ZF ought to be satisfiable).

Robert
-- 
PS I am at present in Berlin, which is why this message is coming 
from a gmx address. But you can reply to my usual 
<Robert.Black at nottingham.ac.uk>.