[FOM] A Case for the Axiom of Determinacy

[Dmytro Taranovsky]

The axiom of determinacy corresponds to the geometric point of view, which
emphasizes the primacy of the continuum.  The continuum is treated as something
flowing and inexhaustible, and not as a sequence of atoms.  In geometry, one
studies squares, circles, and other figures, but not uncountable sequences of
distinct reals.  One relies on geometric intuitions, such as the fact that
every body in space has a volume.

The notion of an arbitrary set of real numbers is too general, so one chooses to
study only nice sets of real numbers.  Set theory is an important tool for the
study of continuum, so ZF is accepted.  However (under the geometric point of
view), the axiom of choice is unacceptable because its implications--such as
the Banach-Tarski decomposition of the sphere--contradict our geometric
intuitions.  

The question arises as to what is the correct axiom for the geometric point of
view.  The theory of the continuum should be reasonably complete.  Sets of real
numbers should conform to our basic intuitions; for example, they should be
measurable.  The axiom should not be unduly restrictive.  Because higher set
theory is used only as a tool, such peculiarities as singular successor
cardinals are acceptable.

Studies have shown that the axiom of determinacy is clearly best suited for the
requirements above.  The axiom of determinacy states that every (perfect
information) game of length omega on integers is determined, that is one of the
players has a winning strategy.  Under the axiom of determinacy, every set of
reals is measurable, has Baire property, perfect subset property, and so on. 
One obtains a rich structural theory of sets of reals.  For example, in ZF+AD,
every set of real numbers has a Wadge rank corresponding to the complexity of
the set.  The continuum is very large; under ZF+AD, there are definable
surjections of the continuum onto weakly Mahlo cardinals.

Because one works without the axiom of choice, one needs every relevant bit of
the axiom of choice that one can get.  Fortunately, the two principles below
are consistent with the axiom of determinacy and fully satisfy the requirements
of ordinary mathematical analysis:
Dependent Choice:  If R is a binary relation on a nonempty set X and for every x
in X there is y in X such that x R y, then there is an infinite sequence (x(0),
x(1), x(2), ...) such that for all natural numbers n, x(n) R x(n+1).
Axiom of Choice for Sets Parameterized by Reals:  If f is a function that for
every real number assigns a nonempty set, then there is a function g such that
for every real number r, g(r) is a member of f(r).

Over ZF, the axiom of determinacy (AD) implies that dependent choice (DC) holds
in L(R).  The other principle fails in L(R), but for most purposes, V=L(power
set of R) can be used instead to get a canonical theory.  Over ZF+AD, the axiom
of choice for sets parameterized by reals implies the axiom of real determinacy
AD_R (all games on reals of length omega are determined) and that Theta, which
is the supremum of projections of sets of real numbers onto ordinals, is
regular.  The converse holds if V=L(P(R)).  Also, in ZF + AD + (at present) DC,
determinacy of games on reals of length 2 is equivalent to determinacy of games
on reals for every countable length.  Over ZF+AD_R, dependent choice in L(P(R))
is equivalent to cofinality of Theta being >omega (which is implied by Theta is
regular).

I do not subscribe to the geometric point of view, and I do not think that the
notion of an arbitrary set of real numbers is too general to admit true
canonical theory, but many mathematicians do.

Dmytro Taranovsky
http://web.mit.edu/dmytro/www/main.htm