- Generic filter
In the mathematical field of
set theory , a generic filter is a kind of object used in the theory of forcing, a technology used for many purposes, but especially to establish the independence of certain propositions from certain formal theories, such asZFC . For example,Paul Cohen used the method to establish that ZFC, if consistent, cannot prove thecontinuum hypothesis , which states that there are exactlyaleph-one real number s. In the contemporary re-interpretation of Cohen's proof, it proceeds by constructing a generic filter that codes more than reals, without changing the value of .Formally, let "P" be a
poset (partially ordered set), and let "F" be a filter on "P"; that is, "F" is a subset of "P" such that:
#"F" is nonempty
#If "p","q"∈"P" and "p"≤"q" and "p" is an element of "F", then "q" is an element of "F" ("F" is "closed upward")
#If "p" and "q" are elements of "F", then there is an element "r" of "F" such that "r"≤"p" and "r"≤"q" (any two elements of "F" are "compatible")Now if "D" is a collection of dense open subsets of "P", in the topology whose basic open sets are all sets of the form {"q"|"q"≤"p"} for particular "p" in "P", then "F" is said to be "D"-generic if "F" meets all sets in "D"; that is,
: for all E ∈ D
Similarly, if "M" is a transitive model of ZFC (or some sufficient fragment thereof), with "P" an element of "M", then "F" is said to be "M"-generic, or sometimes generic over "M", if "F" meets all dense open subsets of "P" that are elements of "M".
References
* K. Ciesielski, "Set Theory for the Working Mathematician", London Mathematical Society
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