Generic filter

In the mathematical field of set theory, a generic filter is a kind of object used in the theory of forcing, a technique used for many purposes, but especially to establish the independence of certain propositions from certain formal theories, such as ZFC. For example, Paul Cohen used the method to establish that ZFC, if consistent, cannot prove the continuum hypothesis, which states that there are exactly aleph-one real numbers. In the contemporary re-interpretation of Cohen's proof, it proceeds by constructing a generic filter that codes more than ${\displaystyle \aleph _{1}}$ reals, without changing the value of ${\displaystyle \aleph _{1}}$.

Formally, let P be a partially ordered set, and let F be a filter on P; that is, F is a subset of P such that:

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