Forcing (computability): Difference between revisions

Revision as of 20:00, 14 February 2011

Revision as of 12:26, 4 February 2012

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;condition: An element in a notion of forcing. We say a condition <math>p</math> is stronger than a condition <math>q</math> just when <math>q \succ_P p</math>.

;<math>p\mid q</math> (<math>p</math> incompatible with <math>q</math>): Given conditions <math>p,q</math> say that <math>p</math> and <math>q</math> are incompatible (denoted <math>p\mid q</math>) if there is no condition <math>r</math> with <math> p \succ_P r</math> and <math>q \succ_P r</math>. <math>p</math> is compatible with <math>q</math> if they are not incompatible. <!-- Sometimes we will use '''consistent''' as a synonym for compatible. -->

;Filter : A subset <math>F</math> of a notion of forcing <math>P</math> is a filter if <math>p,q \in F \implies p \nmid q</math> and <math>p \in F \land q \succ_P p \implies q \in F</math>. In other words a filter is a compatible set of conditions closed under weakening of conditions.