[FOM] n-th order ZFC

[meskew at math.uci.edu]

> For many purposes (including the natural numbers and many
> other structures definable by induction) the constraint to
> well-founded models would suffice, though this does not suffice
> for cardinal arithmetic and hence problems like CH and GCH,
> for which a constraint equivalent to that in second order
> set theory, i.e. insisting on "full" power sets, is better
> and would achieve the same quasi-categoricity for a "first
> order" set theory as in second order set theory (though
> there would still be more models of the first order than of
> the second order system).

Can you name any mathematical problem that can be solved by appealing to
the categoricity of second order set theory?  It seems that the power of
first order set theory actually lies in its ambiguity, because without
that we would not have the techniques of downward Lowheim-Skolem,
reflection, ultraproducts, elementary embeddings, inner models, forcing.

Best,
Monroe