FOM: Sterility
John Mayberry
J.P.Mayberry at bristol.ac.uk
Tue Mar 23 11:00:50 EST 1999
Steve Simpson really ought to cease his incessant repetition of the
mantra "Second order logic isn't logic because it lacks rules of
inference". Quite apart from being irrelevant to the point at issue,
it's not even true. The very same formal axioms and rules that are
complete for Henkin semantics are *sound* for standard semantics. What
second order logic lacks is not formal axioms and rules, but a *system*
of formal axioms and rules that is *complete* for the intended standard
semantics.
ZFC is not complete for its intended interpretation
either. Are we to rule it out of court on that basis? Of course ZFC is
not logic - or is it? Might it not be fruitful, for some purposes, to
think of ZFC with urelements, as logic of absolutely infinite order,
order On, over the set of urelements? But no, Simpson will not allow
it. Indeed, he insists on cramming the whole of mathematics onto the
procrustean bed of formal 1st order logic, which is the only logic he
is prepared to acknowledge. But surely it is obvious that no formal
1st order theory, *qua* formal first order theory, can serve as a
foundation for mathematics. Strictly speaking - and we ought to speak
strictly here - a formal 1st order theory is the theory of the class of
structures that satisfy it. Formal first order group theory is about
all groups, i.e., all those structures that satisfy the familiar axioms
of group theory; formal first order Zermelo-Fraenkel set theory is
about all those structures that satisfy the axioms of ZFC.
Simpson is muddled here. ZFC does not provide the foundation
for mathematics. *Set theory* provides the foundation for mathematics,
and ZFC is a first order formalisation of set theory. Since it is a
formal, first order theory it has all sorts of interpretations that are
radically different from its intended interpretation. That means that
we cannot simply *identify* set theory with ZFC as a formal 1st order
theory. But, as everybody knows, some of those non-intended
interpretations have important uses.
What makes the formal theory ZFC so important for foundations
is the fact that any proof in ordinary mathematics has a formal
counterpart in ZFC. (Naturally, this is not a mathematically exact
claim, since "ordinary mathematics" is not an exact concept.). It is
this fact that gives significance to theorems to the effect that such
and such a formal proposition (e.g., the formal sentence corresponding
to CH) is not among the formal theorems of ZFC. The point of setting up
that formal theory is NOT to provide a foundation for mathematics - set
theory performs that task. The point is rather to provide us with the
means for applying rigorous mathematical reasoning to questions about
what we can prove.
For example, opponents of set-theoretical foundations have
sometimes argued that the Axiom of Replacement is not relevant to
ordinary mathematics. But the results of Friedman and Martin on Borel
Determinacy have confounded them. Martin's proof of Borel Determinacy
establishes a deep and important fact in the theory of real numbers.
But Friedman's proof that any proof of Borel Determinacy must
essentially involve sets which are of stupendous size from the
perspective of ordinary arguments in analysis has profound and direct
application to foundations. For his result *proves* that Replacement is
relevant to ordinary mathematics, viz., to the study of Borel sets in
analysis. Similarly, his recent series of results shows that the study
of large cardinals is not just a "baroque" and highly specialised
branch of mathematical logic (as some opponents of set theory have
claimed) but is directly relevant to ordinary combinatorics. All of
these results involve proving that various propositions are NOT
provable in certain formal 1st order theories.
The categoricity theorems for Peano systems (models of second
order PA) and complete ordered fields are not even remotely "sterile".
They are an essential ingredient in the argument to establish that the
foundations of ordinary mathematics lie in the theory of sets. What is
sterile, and indeed even absurd (given what we now know), is the old
fashioned formalism that Simpson seems to be advocating. For the
supposition that it is ZFC *as a formal first order theory* that
provides the foundations for mathematics, rather than set theory
itself, of which ZFC is merely the formalisation, *that* supposition is
just old fashioned formalism. And it is utterly untenable.
John Mayberry
School of Mathematics
University of Bristol
-----------------------------
John Mayberry
J.P.Mayberry at bristol.ac.uk
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