Forcing (computability): Difference between revisions

Revision as of 16:07, 21 October 2019

Revision as of 17:48, 28 November 2019

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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 its 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 at 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)|langugaelanguage]] 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>.

The idea is that our language should express facts about the object we wish to build with our forcing construction.