[FOM] ZFC

[Martin Davis]

1. Axiom of Choice: I believe the right context in which to view its 
history (see Moore's excellent ZERMELO'S AXIOM OF CHOICE, Springer 1982, 
for a detailed blow-by-blow account) is a conflict for the "soul" of 
mathematics among three contender's:
1. a purely extensional mathematics (Cantor, Dedekind, Frege,Hilbert - but 
with a bow to 2)
2. a constructivist mathematics (Kronecker, Brouwer, Weyl)
3. a predicative mathematics (Poincare, Weyl -again but earlier, Lebesgue, 
Borel)
Although the battle was fought with loud philosophical proclamations on all 
sides, in my opinion what was really decisive was the question of the 
extent to which the widest view could be reliably used in mathematical 
practice. After being profoundly shaken by the paradoxes, calm and 
assurance was gradually restored, and, except on the fringes, view 1 has 
prevailed. Of course, 2 and 3 have each given rise to interesting and 
important foundational studies.

I do not believe it to be true that Cantor ever sought a proof of AC (as 
Harvey said); he simply assumed well ordering as an obvious consequence of 
his extensional viewpoint. It was part of Hilbert's 1st problem (along with 
CH). I believe that the first explicit statement of something like AC as a 
separate principle was by Peano, who regarded it as obviously false. It was 
Zermelo who isolated AC and showed how to prove well-ordering from it. 
Criticism of his proof led him to formulate his axioms for set theory to 
make it all explicit.

2. "Completeness" of ZFC: Harvey has proposed that in some sense ZFC is 
complete. Without an explicit indication, at least roughly, of what he has 
in mind, this is hard to discuss. But I do want to register my profound 
skepticism. In one sense, Harvey's own brilliant work on combinatorial 
results not obtainable from ZFC is an obvious obstacle such a result would 
need to overcome.

I see the ZF axioms as "closure" principles under suitable operations, as 
in algebraic structures. The axioms are of the two forms:
a) Such and such is a set.
b) If this and that is a set, then so is this other.
Such a situation leads one naturally to form the minimal structure closed 
under these operations.
In the case of ZF this leads one to accept (strongly) inaccessible 
cardinals. This is the sort of thing G\"odel had in mind when he said (in 
1933):

  <<... we are confronted by a strange situation. We set out to find a 
formal system [of axioms] for mathematics and instead of that found an 
infinity of systems, and whichever system you choose ..., there is one ... 
whose axioms are stronger. 
 

But ... this character of our systems ... is in perfect accord with certain 
facts which can be established quite independently ... For any formal 
system you can construct a proposition in fact a proposition of the 
arithmetic of integers which is certainly true if the given system is free 
from contradictions but cannot be proved in the given system. Now if the 
system under consideration (call it S) is based on the theory of types, it 
turns out that ... this proposition becomes a provable theorem if you add 
to S the next higher type and the axioms concerning it. 
 the construction 
of higher and higher types 
 is necessary for proving theorems even of a 
relatively simple structure. >>

Martin