Forcing (computability): Difference between revisions

Revision as of 21:03, 4 October 2007

Revision as of 20:28, 8 November 2007

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== Generic objects ==

The intuition behind forcing is that our conditions are finite approximations to some object we wish to build and that <math>\sigma</math> is stronger than <math>\tau</math> when <math>\sigma</math> agrees with everything <math>\tau</math> says about the object we are building and adds some information of it'sits own. For instance in Cohen forcing the conditions can be viewed as finite approximations to a real and if <math>\tau \succ_C \sigma</math> then <math>\sigma</math> tells us the value of the real on more places.

In a moment we will define a relation <math>\sigma \Vdash_P \psi</math> (read <math>\sigma</math> forces <math>\psi</math>) that holds between conditions (elements of <math>P</math>) and sentences but first we need to explain the [[language (mathematics)]] that <math>\psi</math> is a sentence for. However, forcing is a technique not a definition and the language for <math>\psi</math> will depend on the application one has in mind and the choice of <math>P</math>.