Information About

Diamondsuit
  CATEGORIES ABOUT DIAMONDSUIT
   
set theory
   
APPAREL
BABY
BEAUTY
BOOKS
CAR TOYS
CELL PHONES
DVD'S
ELECTRONICS
GOURMET FOOD
GROCERIES
HEALTH & PERSONAL
HOME & GARDEN
JEWELRY
MUSIC
MUSIC INSTRUMENTS
OFFICE PRODUCTS
SOFTWARE
SPORTING GOODS
TOOLS & HARDWARE
TOYS
VIDEO GAMES
SHOPPING HOME
MORE SHOPPING...




DEFINITION


For a given Cardinal Number \kappa and a Stationary Set S\subseteq\kappa , the statement \Diamond_\kappa (S) is the statement that there is a Sequence \langle A_\alpha: \alpha \in S angle such that

  • each A_\alpha \subseteq \alpha

  • for every A \subseteq \kappa, \{\alpha \in S: A \cap \alpha = A_\alpha\} is stationary in \kappa


When S = \kappa, \Diamond_\kappa (S) is written \Diamond_\kappa , and \Diamond_{\omega_1} is written \Diamond


PROPERTIES AND USE


It can be shown that ◊ ⇒ CH ; also, ♣ + CH ⇒ ◊, but there also exist models of ♣ + ¬ CH, so ◊ and ♣ are not equivalent (rather, ♣ is weaker than ◊).


For all cardinals \kappa and stationary subsets S \subseteq \kappa^+ , \Diamond_{\kappa^+} (S) holds in the Constructible Universe .


REFERENCES


  • Charles Akemann, Nik Weaver, ''Consistency of a counterexample to Naimark's problem'', online



SEE ALSO