15/06/28 19:23:52.06 pEaR/2gu.net
Forcing (mathematics)
URLﾘﾝｸ(en.wikipedia.org)
In the mathematical discipline of set theory, forcing is a technique discovered by Paul Cohen for proving consistency and independence results.
It was first used, in 1963, to prove the independence of the axiom of choice and the continuum hypothesis from Zermelo?Fraenkel set theory.
Forcing was considerably reworked and simplified in the following years, and has since served as a powerful technique both in set theory and in areas of mathematical logic such as recursion theory.
Descriptive set theory uses the notion of forcing from both recursion theory and set theory. Forcing has also been used in model theory but it is common in model theory to define genericity directly without mention of forcing.
Contents
1 Intuitions
2 Forcing posets
2.1 P-names
2.2 Interpretation
2.3 Example
3 Countable transitive models and generic filters
4 Forcing
5 Consistency
6 Cohen forcing
7 The countable chain condition
8 Easton forcing
9 Random reals
10 Boolean-valued models
11 Meta-mathematical explanation
12 Logical explanation
13 See also
14 References
15 External links