Forcing (computability): Difference between revisions

Revision as of 21:54, 16 February 2008

Revision as of 05:45, 16 March 2008

Line 1:

{{references|date=January 2008}}

{{merge|Forcing (mathematics)|date=March 2008}}}

'''Forcing''' in [[recursion theory]] is a modification of [[Paul Cohen (mathematician)|Paul Cohen's]] original [[set theory|set theoretic]] technique of [[forcing (set theory)|forcing]] to deal with the effective concerns in [[recursion theory]]. Conceptually the two techniques are quite similar, in both one attempts to build [[generic set|generic]] objects (intuitively objects that are somehow 'typical') by meeting dense sets. Also both techniques are elegantly described as a relation (customarily denoted <math>\Vdash</math>) between 'conditions' and sentences. However, where set theoretic forcing is usually interested in creating objects that meet every dense set of conditions in the ground model, recursion theoretic forcing only aims to meet dense sets that are arithmetically or hyperarithmetically definable. Therefore some of the more difficult machinery used in set theoretic forcing can be eliminated or substantially simplified when defining forcing in recursion theory. But while the machinery may be somewhat different recursion theoretic and set theoretic forcing are properly regarded as an application of the same technique to different classes of formulas.