Asymptotic Logical Uncertainty: Uniform Coherence

EDIT: This post is out of date, the new, bet­ter defi­ni­tion is here.

This post is part of the Asymp­totic Log­i­cal Uncer­tainty se­ries. Here, I give a con­crete pro­posal for a defi­ni­tion of Uniform Co­her­ence, as men­tioned here.

This is only a pro­posal for a defi­ni­tion. It may be that this defi­ni­tion is bad, and we would rather re­place it with some­thing else

Let be a Tur­ing ma­chine which on in­put runs for some amount of time then out­puts a prob­a­bil­ity, rep­re­sent­ing the prob­a­bil­ity as­signed to .

In the fol­low­ing defi­ni­tion, we fix a func­tion (e.g. ). We say that a se­quence is quickly com­putable if it is an in­creas­ing se­quence and there ex­ists a Tur­ing ma­chine which de­ter­mines whether or not an in­put is of the form in time .

We say that is Uniformly Co­her­ent if

  1. If is quickly com­putable and for all , then ex­ists.

  2. If , , and are quickly com­putable and for all , then

Open Ques­tion 1: Does there ex­ist a uniformly co­her­ent log­i­cal pre­dic­tor ?

Open Ques­tion 2: Does there ex­ist a uniformly co­her­ent log­i­cal pre­dic­tor which also passes the Gen­er­al­ized Ben­ford Test? (Here, we mean the gen­er­al­iza­tion of the Ben­ford Test to all ir­re­ducible pat­terns)

Theroem: If is uniformly co­her­ent, then if we define , then is defined for all and is a com­putably ap­prox­imable co­her­ent prob­a­bil­ity as­sign­ment. (see here for defi­ni­tions.)

Proof: Com­putable ap­prox­ima­bil­ity is clear. For co­her­ence, it suffices to show that is well defined, for prov­able , for dis­prov­able , and .

The fact that is well defined comes from ap­ply­ing 2 to the se­quence .

The fact that for prov­able , comes from ap­ply­ing 3 to , , and . The fact that for dis­prov­able , comes from ap­ply­ing 3 to , , and , since we already know that .

The fact that comes from ap­ply­ing 3 to , , and then ap­ply­ing 3 again to , , and to get that .

Uniform Co­her­ence is how­ever much stronger than co­her­ence. To see an ex­am­ple, con­sider an in­finite se­quence of sen­tences “PA is con­sis­tent on proofs of length up to .” Uniform co­her­ence can be shown to im­ply that . (Each of the sen­tences is prov­able, so you can ap­ply 3 to this se­quence and a pair of se­quences.)

The­o­rem: If we take a quickly com­putable se­quence of mu­tu­ally ex­clu­sive sen­tences, , then if is uniformly co­her­ent, we have .

Proof: Let be the dis­junc­tion of all the for . Let . By 2, con­verges to some . Ap­ply­ing 3 to , and shows that con­verges to . There­fore, ap­ply­ing 3 to , , and shows that con­verges to 0.(Note that I as­sumed here that was rea­son­able enough that and ane also quickly com­putable)