# Generic filter

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In the mathematical field of [set theory](/source/set_theory), a '''generic filter''' is a kind of object used in the theory of [forcing](/source/forcing_(mathematics)), a technique used for many purposes, but especially to establish the [independence](/source/independence_(mathematical_logic)) of certain propositions from certain [formal theories](/source/mathematical_theory), such as [ZFC](/source/ZFC). For example, [Paul Cohen](/source/Paul_Cohen_(mathematician)) used forcing to establish that ZFC, if [consistent](/source/consistent_theory), cannot prove the [continuum hypothesis](/source/continuum_hypothesis), which states that there are exactly [<math>\aleph_1</math>](/source/aleph-one) [real number](/source/real_number)s. In the contemporary re-interpretation of Cohen's proof, it proceeds by constructing a generic filter that codes more than <math>\aleph_1</math> reals, without changing the value of <math>\aleph_1</math>.

Formally, let ''P'' be a [partially ordered set](/source/partially_ordered_set), and let ''F'' be a [filter](/source/filter_(mathematics)) on ''P''; that is, ''F'' is a subset of ''P'' such that:
#''F'' is nonempty
#If ''p'',&nbsp;''q''&nbsp;&isin;&nbsp;''P'' and ''p''&nbsp;&le;&nbsp;''q'' and ''p'' is an element of ''F'', then ''q'' is an element of ''F'' (''F'' is [closed upward](/source/upper_set))
#If ''p'' and ''q'' are elements of ''F'', then there is an element ''r'' of ''F'' such that ''r''&nbsp;&le;&nbsp;''p'' and ''r''&nbsp;&le;&nbsp;''q'' (''F'' is [downward directed](/source/directed_set))

Now if ''D'' is a collection of [dense](/source/dense_set) [open](/source/open_set) subsets of ''P'', in the topology whose basic open sets are all sets of the form {''q''∈''P''&nbsp;|&nbsp;''q''&nbsp;&le;&nbsp;''p''} for particular ''p'' in ''P'', then ''F'' is said to be '''''D''-generic''' if ''F'' meets all sets in ''D''; that is,

:<math>F\cap E \ne \varnothing,\,</math> for all ''E'' &isin; ''D''.

Similarly, if ''M'' is a [transitive](/source/transitive_set) [model](/source/model_theory) of ZFC (or some sufficient fragment of ZFC), with ''P'' an element of ''M'' (partially ordered by ∈), 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''.

==See also==
* {{annotated link|1-generic}} in [computability](/source/Computability_theory)
* {{annotated link|Rasiowa–Sikorski lemma}}

==References==
{{reflist}}
{{refbegin}}
* {{cite book|author=K. Ciesielski|year=1997|url=https://books.google.com/books?id=tTEaMFvzhDAC|title=Set Theory for the Working Mathematician|series=London Mathematical Society, Student Texts 39|publisher=Cambridge University Press|isbn=9780521594653 }}
{{refend}}

Category:Forcing (mathematics)

{{Mathematical logic}}
{{Set theory}}

{{settheory-stub}}

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