FOM: Second order logic

[Martin Schlottmann]

Robert Black wrote:
> 
> I say that second-order logic is an essential tool for the expression of
> mathematical theories, because only in second-order logic can you have
> categorical theories with infinite models.
<snip>

I have seen this assertion a couple of times
in this interesting thread and I am wondering
why the discussion doesn't concentrate on
this what seems to be the key point in the
argument for 2nd order logics.

More specifically: Is it not true that the
alleged categoricity of certain 2nd order
theories is always only relative to a designated
range for the second order quantifiers?
Then, how is this "standard semantics"
distinguished from the "Henkin semantics"
mentioned by Dr. Simpson? To my under-
standing the categoricity of some 2nd order
theories is well-defined precisely to the
extend that we can exactly specify the
intended model for "properties", and it
seems to me that for the latter we can
do no better than 1st order set theory.


-- 
Martin Schlottmann
Sessional Lecturer
Department of Mathematical Sciences
University of Alberta, Edmonton AB T6G 2G1, Canada