{ localUrl: '../page/provability_predicate.html', arbitalUrl: 'https://arbital.com/p/provability_predicate', rawJsonUrl: '../raw/5j7.json', likeableId: '3200', likeableType: 'page', myLikeValue: '0', likeCount: '1', dislikeCount: '0', likeScore: '1', individualLikes: [ 'EricBruylant' ], pageId: 'provability_predicate', edit: '3', editSummary: '', prevEdit: '2', currentEdit: '3', wasPublished: 'true', type: 'wiki', title: 'Provability predicate', clickbait: '', textLength: '1622', alias: 'provability_predicate', externalUrl: '', sortChildrenBy: 'likes', hasVote: 'false', voteType: '', votesAnonymous: 'false', editCreatorId: 'JaimeSevillaMolina', editCreatedAt: '2016-08-19 11:11:55', pageCreatorId: 'JaimeSevillaMolina', pageCreatedAt: '2016-07-22 08:30:52', seeDomainId: '0', editDomainId: 'AlexeiAndreev', submitToDomainId: '0', isAutosave: 'false', isSnapshot: 'false', isLiveEdit: 'true', isMinorEdit: 'false', indirectTeacher: 'false', todoCount: '2', isEditorComment: 'false', isApprovedComment: 'true', isResolved: 'false', snapshotText: '', anchorContext: '', anchorText: '', anchorOffset: '0', mergedInto: '', isDeleted: 'false', viewCount: '34', text: '[summary: \nA provability predicate of a theory $T$ is a formula $P(x)$ with one free variable $x$ such that:\n\n1. If $T\\vdash S$, then $T\\vdash P(\\ulcorner S \\urcorner)$\n2. $T\\vdash P(\\ulcorner A\\rightarrow B \\urcorner)\\rightarrow (P(\\ulcorner A \\urcorner)\\rightarrow P(\\ulcorner B \\urcorner))$\n3. $T\\vdash P(\\ulcorner S \\urcorner)\\rightarrow P(\\ulcorner P(\\ulcorner S \\urcorner) \\urcorner)$\n]\n\nA provability predicate is a formula $P$ of a theory satisfying the Hilbert-Bernais derivability conditions. If the diagonal theorem is applicable in the theory as well, then [55w] and [ Gödel's second incompleteness theorem] are provable for $P$.\n\nThe Hilbert-Bernais derivability conditions are as follows:\n\n1. (**Necessitation**) If $T\\vdash S$, then $T\\vdash P(\\ulcorner S \\urcorner)$\n2. (**Provability of modus ponens / distributive axioms**) $T\\vdash P(\\ulcorner A\\rightarrow B \\urcorner)\\rightarrow (P(\\ulcorner A \\urcorner)\\rightarrow P(\\ulcorner B \\urcorner))$\n3. (**Provability of renecessitation**) $T\\vdash P(\\ulcorner S \\urcorner)\\rightarrow P(\\ulcorner P(\\ulcorner S \\urcorner) \\urcorner)$\n\nThe derivability conditions are tighlty related to the axioms and rules of inference of [-534]. In fact, the normal systems of provability are defined as those that have necessitation as a rule and the distributive axioms. %%note:They also have to be closed under substitution%% On the other hand, D3 is the axiom that defines the system [ K4], and it is also satisfied by [GL].\n\n##Examples\n\nThe **verum** formula $x=x$ trivially satisfies the derivability conditions.\n\nThe [-5gt] of arithmetic is a provability predicate.', metaText: '', isTextLoaded: 'true', isSubscribedToDiscussion: 'false', isSubscribedToUser: 'false', isSubscribedAsMaintainer: 'false', discussionSubscriberCount: '2', maintainerCount: '2', userSubscriberCount: '0', lastVisit: '', hasDraft: 'false', votes: [], voteSummary: 'null', muVoteSummary: '0', voteScaling: '0', currentUserVote: '-2', voteCount: '0', lockedVoteType: '', maxEditEver: '0', redLinkCount: '0', lockedBy: '', lockedUntil: '', nextPageId: '', prevPageId: '', usedAsMastery: 'false', proposalEditNum: '0', permissions: { edit: { has: 'false', reason: 'You don't have domain permission to edit this page' }, proposeEdit: { has: 'true', reason: '' }, delete: { has: 'false', reason: 'You don't have domain permission to delete this page' }, comment: { has: 'false', reason: 'You can't comment in this domain because you are not a member' }, proposeComment: { has: 'true', reason: '' } }, summaries: {}, creatorIds: [ 'JaimeSevillaMolina', 'EricRogstad' ], childIds: [], parentIds: [ 'math' ], commentIds: [], questionIds: [], tagIds: [ 'needs_parent_meta_tag' ], relatedIds: [], markIds: [], explanations: [], learnMore: [], requirements: [], subjects: [], lenses: [], lensParentId: '', pathPages: [], learnMoreTaughtMap: {}, learnMoreCoveredMap: {}, learnMoreRequiredMap: {}, editHistory: {}, domainSubmissions: {}, answers: [], answerCount: '0', commentCount: '0', newCommentCount: '0', linkedMarkCount: '0', changeLogs: [ { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '18831', pageId: 'provability_predicate', userId: 'JaimeSevillaMolina', edit: '3', type: 'newEdit', createdAt: '2016-08-19 11:11:55', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '17381', pageId: 'provability_predicate', userId: 'EricBruylant', edit: '0', type: 'newTag', createdAt: '2016-07-23 05:38:58', auxPageId: 'needs_parent_meta_tag', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '17320', pageId: 'provability_predicate', userId: 'EricRogstad', edit: '0', type: 'newParent', createdAt: '2016-07-22 17:04:29', auxPageId: 'math', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '3188', likeableType: 'changeLog', myLikeValue: '0', likeCount: '2', dislikeCount: '0', likeScore: '2', individualLikes: [], id: '17318', pageId: 'provability_predicate', userId: 'EricRogstad', edit: '2', type: 'newEdit', createdAt: '2016-07-22 17:04:25', auxPageId: '', oldSettingsValue: '', newSettingsValue: 'moved note after period; fixed a typo' }, { likeableId: '3182', likeableType: 'changeLog', myLikeValue: '0', likeCount: '1', dislikeCount: '0', likeScore: '1', individualLikes: [], id: '17283', pageId: 'provability_predicate', userId: 'JaimeSevillaMolina', edit: '1', type: 'newEdit', createdAt: '2016-07-22 08:30:52', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' } ], feedSubmissions: [], searchStrings: {}, hasChildren: 'false', hasParents: 'true', redAliases: {}, improvementTagIds: [], nonMetaTagIds: [], todos: [], slowDownMap: 'null', speedUpMap: 'null', arcPageIds: 'null', contentRequests: {} }