[FOM] Projective Determinacy
dmytro at mit.edu
Fri Oct 10 20:18:22 EDT 2003
Projective determinacy is perhaps the best example of a statement that
(1) generally accepted as true,
(2) of fundamental importance in mathematics, and
(3) unprovable in ZFC.
Note: General acceptance only means general acceptance among those who
work in the relevant fields; most people are aware of neither ZFC nor of
projective determinacy. Projective determinacy is of fundamental
importance in second order arithmetic, that is in the study of real
numbers. It is probably not of fundamental importance to those who work
on finite structures. However, Boolean Relation Theory requires some
axioms beyond ZFC, and projective determinacy would do.
In this posting, I will concentrate on explaining why projective
determinacy is true.
First, for finite games, determinacy is obvious: Assuming flawless
play, either the first player can win, or he cannot, in which case the
second player wins. It is also provable: Final positions are
determined, a position is winnable if and only if there is a move that
makes the position losable for the other player; by induction from last
positions into previous ones, every position is winnable or losable.
Projective determinacy is simply the transfer of our proven intuitions
into the infinite. Such transfer is not new: ZFC itself is the
transfer of our basic intuitions about sets into the infinite. Of
course, one has to be careful in making such transfers: Since in all
finite games, the payoff set and each position are definable, projective
determinacy only asserts determinacy where positions can be coded by
integers and payoff set definable in a simple way from real numbers.
Second, projective determinacy is an existence axiom:
Theorem: Projective Determinacy is true if and only if for every real
number z and natural number n, there is an iterable transitive model
that contains z and n Woodin cardinals.
One can get more precise equivalence results as well:
Theorem: Under (boldface) Sigma - 1 - n determinacy, Sigma - 1 - n+1
(z) determinacy is equivalent to the existence of the sharp (a real
number that captures the statements that are true of the indiscernibles)
of an inner model that contains z and n Woodin cardinals.
(Does anyone know the exact equivalence under ZFC between Sigma - 1 -
n+1 (z) determinacy and models with large cardinals?)
Fact: Every statement in set theory that has a sufficiently strong set
existence component implies projective determinacy.
For example, if kappa is a successor of a singular strong limit cardinal
that does not violate the singular continuum hypothesis and every tree
of height kappa either has either a branch of length kappa or an
anti-chain of cardinality kappa, then projective determinacy holds.
Third, projective determinacy provides a very nice canonical theory of
second order arithmetic. Every projective set is measurable, has Baire
property, and has perfect subset property. The pointclasses Pi - 1 -
2n+1 and Sigma - 1 - 2n have the scale property, which is a very strong
structural property and implies the properties of prewellordering and of
uniformization. The proofs from determinacy are much more natural than
proofs from V=L. Finally, every nonrestrictive canonical theory of
countable sets implies projective determinacy.
As my last point, to counter the occasional proposals to replace the
axiom of choice with the axiom of determinacy, I note:
Theorem (ZF): The axiom of choice holds if and only if every perfect
information game of length 2 is determined.
In my paper,
the section "Philosophy of Determinacy Hypotheses", explains in detail
why (and which) determinacy hypotheses should be true.
A good introduction to projective determinacy can be found in Woodin's
paper "The Continuum Hypothesis, Part I".
More information about the FOM