FOM: second-order logic is a myth

[Randall Holmes]

Steve Simpson said (responding to Robert Black):

It's very odd, if not absurd, to say that first-order logic can't
express mathematical theories.  All of mathematics can be formalized
in ZFC.  This is very well-understood and well-established
conventional wisdom in f.o.m.  If you insist on contradicting this
conventional wisdom, you need to make a very strong and novel case
against it.  You haven't even tried to do that.

My comment:

I think it is easy to see what Black means.  I give an example.  The
axioms of PA capture much of what we believe about the natural
numbers, but they cannot express what we actually believe about the
natural numbers; they have nonstandard models.  If we use a theory T
stronger than PA with confidence, we believe Con(T), which is a
statement about the natural numbers that cannot be proved in T and so
cannot be proved in PA.  PA is at best a partial description of what
we mean by "the natural numbers".  The second-order theory of PA
expresses precisely what we mean by "the natural numbers" but has the
disadvantage that we cannot precisely define what constitutes valid
reasoning from this theory (though this certainly includes all proofs
from the first-order two-sorted theory commonly called second-order
PA).

And God posted an angel with a flaming sword at | Sincerely, M. Randall Holmes
the gates of Cantor's paradise, that the       | Boise State U. (disavows all) 
slow-witted and the deliberately obtuse might | holmes at math.idbsu.edu
not glimpse the wonders therein. | http://math.idbsu.edu/~holmes