FOM: Sterility

[John Mayberry]

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



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John Mayberry
J.P.Mayberry at bristol.ac.uk
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