FOM: second-order logic is a sterile myth

[Stephen G Simpson]

Randall Holmes 22 Mar 1999 16:45:50 says:

 > I hardly think that it is a sterile concern for the foundations of
 > mathematics whether we can actually refer to such familiar
 > structures as "the natural numbers".  Second-order logic allows us
 > to do this, because it allows a categorical axiomatization of N and
 > other familiar structures; first order logic, even formalization in
 > a first-order theory as strong as ZFC, does not.

This formulation is very misleading.

The truth is that there is an extremely important sense in which ZFC
*does* allow us to refer to the natural numbers and other familiar
structures and *does* contain a categorical axiomatization of those
structures.  Namely, the isomorphism types of these structures are
straightforwardly definable in ZFC.

The above-mentioned fact about ZFC, which Holmes disregards, is
absolutely crucial for f.o.m., because it plays a key role in
formalizing mathematics within ZFC and similar systems.  This is
*much* more important for f.o.m. than any known fact about
second-order logic.  Indeed, it is *much* more important for
f.o.m. than everything that is currently known about second-order
logic put together.

[ Throughout this posting, when I say second-order logic, I am
referring to second-order logic with `standard' rather than Henkin
semantics.  My remarks do not apply to second-order logic with Henkin
semantics, which is really a system of first-order logic. ]

 > "What we can say" is a philosophical concern necessarily prior to
 > "what we can prove".  If I asked a philosopher what branch of
 > philosophy this question belonged to, I think the answer might be
 > ... logic.

So what?  Regardless of what a hypothetical philosopher might say,
logic is still the science of correct inference.  Second-order logic
is not logic in this sense, because it lacks rules of inference.  Why
deny this obvious point?

 > Formal logics for the purpose of expression or definition may annoy
 > Simpson

Not at all.  Where did you get that idea?

 > one may note correctly that [second-order logic] does not allow one
 > to prove any theorems about the natural numbers that cannot be
 > proved ... in ZFC,

Actually, second-order logic does not allow one to prove anything at
all, because it has no rules of inference.

 > but that is beside the point.

No, it's not beside the point.  It goes directly to my point that any
proposed approach to f.o.m. via second-order logic is useless and
sterile.  

All of the most important f.o.m. advances (G"odel's incompleteness
theorem, the work of G"odel and Cohen on the continuum hypothesis, etc
etc, not to mention reverse mathematics, etc) are concerned with
inference and therefore require a first-order approach and could not
have happened under a second-order approach.

I fervently hope that most FOM readers understand the above key point
concerning the role of first-order logic.  This is absolutely basic
for f.o.m.

A comment for Holmes: I am beginning to think that your remarks may
indicate some fundamental misunderstanding or off-beat assumption
about current research in f.o.m.  But I am having a hard time placing
my finger on exactly where you are going wrong.  Tell me, is your
advocacy of second-order logic somehow bound up with your advocacy of
NF?  NF strikes most people as a rather off-beat and unintuitive kind
of set theory ....

-- Steve