---
tag: WEDP73Y7GK
kind: concept
name: Generic filter
msc: ["03E40"]
sources:
  - url: https://en.wikipedia.org/wiki/Generic_filter
same_as:
  wikidata: Q5532675
  wikipedia: Generic filter
---

## Summary

A generic filter is a filter in a [[partial order|0QV2JS9RJ3]] that meets every dense subset belonging to a given model.
It is the new object that the method of forcing adds to the model.
Because it meets every dense set of the model, it has every property that the model can see as typical.
In nontrivial cases it cannot be an element of the model.
When the model is countable, generic filters exist by the Rasiowa–Sikorski lemma.

## Explanations

### L1 human

Think of the elements of the partial order as finite pieces of information about an unknown object.
A filter is a consistent collection of such pieces.
A dense set is a requirement that can always be met by adding more information.
A generic filter meets all the requirements that the old model can write down.
For example, in a generic sequence of zeros and ones, every finite pattern appears, because "the pattern appears" is a requirement that can always be met.