Dictionary of Meaning
<<Back
Please select a letter:
A |
B |
C |
D |
E |
F |
G |
H |
I |
J |
K |
L |
M |
N |
O |
P |
Q |
R |
S |
T |
U |
V |
W |
X |
Y |
Z |
0-9
Click here for Shopping
Axiom of projective determinacy
*** Shopping-Tip: Axiom of projective determinacy
In
mathematical logic, '''projective determinacy''' is the special case of the
axiom of determinacy applying only to
projective sets.
The '''axiom of projective determinacy''', abbreviated '''PD''', states that for any two-player game of perfect information of length ω in which the players play
natural numbers, if the victory set (for either player, since the projective sets are closed under complementation) is projective, then one player or the other has a
Determinacy#Winning strategies winning strategy.
The axiom is undecidable in
ZFC, unlike the full axiom of determinacy (AD), which contradicts the
Axiom of Choice. PD follows from certain
large cardinal axioms, such as the existence of infinitely many
Woodin cardinals.
PD implies that all projective sets are
Lebesgue measurable (in fact,
universally measurable) and have the
perfect set property and the
property of Baire. It also implies that every projective
binary relation may be
Uniformization (set theory) uniformized by a projective set.
References
* {{cite journal|author=Martin, Donald A. and John R. Steel|year=Jan., 1989|title=A Proof of Projective Determinacy|journal=Journal of the American Mathematical Society|volume=2|issue=1|pages=71-125}}
* {{cite book | author=Moschovakis, Yiannis N. | title=Descriptive Set Theory | publisher=North Holland | year=1980 |id=ISBN 0-444-70199-0}}
Category:Game theory
Category:Descriptive set theory
Category:Determinacy
{{mathlogic-stub}}
