At 2:19 -0400 22/09/2002, Wei Dai wrote:

>This needs to be qualified a bit. Mathematical objects are more than the
>formal (i.e., deductive) consequences of their axioms. However, an axiom
>system can capture a mathematical structure, if it's second-order, and you
>consider the semantic consequences of the axioms instead of just the
>deductive consequences.

You are right. But this is a reason for not considering classical *second*
order logic as logic. Higher order logic remains "logic" when some
constructive assumption are made, like working in intuitionist logic.
A second order classical logic "captures" a mathematical structure in a very
weak sense. My opinion is that the "second order *classical* logics" are
misleading when seen as logical system. Why not taking at once as axioms
the set of all true sentences in the standard model of Zermelo Fraenkel (ZF)
set theory, and throw away all rules of inference. This "captures", even
categorically, the set universe. But it is only in a highly technical sense
that such a set can be seen as a theory.
"Logically" you are right, and what you said to Brent is correct.
I just point here that the use of second order classical logic can be
misleading especially for those who doesn't have a good idea of what is
a *first order* theory.


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