Generic filter

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 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 aleph_1 reals, without changing the value of aleph_1.

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,

:Fcap E e varnothing,, 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


Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать курсовую

Look at other dictionaries:

  • Generic — means pertaining or appropriate to large classes or groups as opposed to specific members of the group. It may refer to:* Generic mood, a grammatical mood used to make generalized statements like Snow is white * Generic antecedents, referents in… …   Wikipedia

  • Mechanical filter — Figure 1. A mechanical filter made by the Kokusai Electric Company intended for selecting the narrow 2 kHz bandwidth signals in SSB radio receivers. It operates at 455 kHz, a common IF for these receivers, and is dimensioned 45×15×15 mm ( …   Wikipedia

  • Sallen–Key filter — A Sallen–Key filter is a type of active filter, particularly valued for its simplicity. The circuit produces a 2 pole (12 dB/octave) lowpass or highpass response using two resistors, two capacitors and (usually) a unity gain buffer amplifier.… …   Wikipedia

  • List of generic and genericized trademarks — Contents 1 List of former trademarks that have become generic terms 2 List of protected trademarks frequently used as generic terms 3 Notes 4 References …   Wikipedia

  • Distributed element filter — Figure 1. A circuit featuring many of the f …   Wikipedia

  • Band-stop filter — A generic ideal band stop filter, showing both positive and negative angular frequencies In signal processing, a band stop filter or band rejection filter is a filter that passes most frequencies unaltered, but attenuates those in a specific… …   Wikipedia

  • Deblocking filter (video) — A deblocking filter is a video filter applied to blocks in decoded video to improve visual quality and prediction performance by smoothing the sharp edges which can form between macroblocks when block coding techniques are used. The filter aims… …   Wikipedia

  • Deblocking filter — Un deblocking filter que l on peut traduire par un filtre anti blocs est appliqué dans la compression vidéo et plus spécialement sur l image décodée. Elle permet d améliorer la qualité visuelle de l image et les performances de prédiction en… …   Wikipédia en Français

  • Forcing (mathematics) — For the use of forcing in recursion theory, see Forcing (recursion theory). In the mathematical discipline of set theory, forcing is a technique invented by Paul Cohen for proving consistency and independence results. It was first used, in 1963,… …   Wikipedia

  • Rasiowa-Sikorski lemma — In axiomatic set theory, the Rasiowa Sikorski lemma is one of the most fundamental facts used in the technique of forcing. In the area of forcing, a subset D of a forcing notion ( P , ≤) is called dense in P if for any p ∈ P there is d ∈ D with d …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”