Consequences
Woodin cardinals are important in descriptive set theory. By a result of Martin and Steel, existence of infinitely many Woodin cardinals implies projective determinacy, which in turn implies that every projective set is measurable, has the Baire property (differs from an open set by a meager set, that is, a set which is a countable union of nowhere dense sets), and the perfect set property (is either countable or contains a perfect subset).
The consistency of the existence of Woodin cardinals can be proved using determinacy hypotheses. Working in ZF+AD+DC one can prove that is Woodin in the class of hereditarily ordinal-definable sets. is the first ordinal onto which the continuum cannot be mapped by an ordinal-definable surjection (see Θ (set theory)).
Shelah proved that if the existence of a Woodin cardinal is consistent then it is consistent that the nonstationary ideal on ω1 is -saturated. Woodin also proved the equiconsistency of the existence of infinitely many Woodin cardinals and the existence of an -dense ideal over .
Read more about this topic: Woodin Cardinal
Famous quotes containing the word consequences:
“If you are prepared to accept the consequences of your dreams ... then you must still regard America today with the same naive enthusiasm as the generations that discovered the New World.”
—Jean Baudrillard (b. 1929)
“The consequences of our actions grab us by the scruff of our necks, quite indifferent to our claim that we have “gotten better” in the meantime.”
—Friedrich Nietzsche (1844–1900)
“There is a delicate balance of putting yourself last and not being a doormat and thinking of yourself first and not coming off as selfish, arrogant, or bossy. We spend the majority of our lives attempting to perfect this balance. When we are successful, we have many close, healthy relationships. When we are unsuccessful, we suffer the natural consequences of damaged and sometimes broken relationships. Children are just beginning their journey on this important life lesson.”
—Cindy L. Teachey. “Building Lifelong Relationships—School Age Programs at Work,” Child Care Exchange (January 1994)