martin at eipye.com
Fri Jun 27 15:38:49 EDT 2003
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,
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
<<... 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.
of higher and higher types
is necessary for proving theorems even of a
relatively simple structure. >>
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