Consistency

A con­sis­tent the­ory is one in which there are well-formed state­ments that you can­not prove from its ax­ioms; or equiv­a­lently, that there is no \(X\) such that \(T\vdash X\) and \(T\vdash \neg X\).

From the point of view of model the­ory, a con­sis­tent the­ory is one whose ax­ioms are satis­fi­able. Thus, to prove that a set of ax­ioms is con­sis­tent you can re­sort to con­struct­ing a model us­ing a for­mal sys­tem whose con­sis­tency you trust (nor­mally us­ing set the­ory) in which all the ax­ioms come true.

Arith­metic is ex­pres­sive enough to talk about con­sis­tency within it­self. If \(\square_{PA}\) rep­re­sents the stan­dard prov­abil­ity pred­i­cate in Peano Arith­metic then a sen­tence of the form \(\neg\square_{PA}(\ulcorner 0=1\urcorner)\) rep­re­sents the con­sis­tency of \(PA\), since it comes to say that there ex­ists a dis­prov­able sen­tence for which there is no proof. Gödel’s sec­ond in­com­plete­ness the­o­rem comes to say that such a sen­tence is not prov­able from the ax­ioms of \(PA\).

Parents:

  • Mathematics

    Math­e­mat­ics is the study of num­bers and other ideal ob­jects that can be de­scribed by ax­ioms.

  • Formal Logic

    For­mal logic stud­ies the form of cor­rect ar­gu­ments through rigor­ous and pre­cise math­e­mat­i­cal the­o­ries.