{
  localUrl: '../page/59h.html',
  arbitalUrl: 'https://arbital.com/p/59h',
  rawJsonUrl: '../raw/59h.json',
  likeableId: '0',
  likeableType: 'page',
  myLikeValue: '0',
  likeCount: '0',
  dislikeCount: '0',
  likeScore: '0',
  individualLikes: [],
  pageId: '59h',
  edit: '2',
  editSummary: '',
  prevEdit: '1',
  currentEdit: '2',
  wasPublished: 'true',
  type: 'wiki',
  title: 'Proof of Gödel's first incompleteness theorem',
  clickbait: '',
  textLength: '1482',
  alias: '59h',
  externalUrl: '',
  sortChildrenBy: 'likes',
  hasVote: 'false',
  voteType: '',
  votesAnonymous: 'false',
  editCreatorId: 'JaimeSevillaMolina',
  editCreatedAt: '2016-10-11 20:24:50',
  pageCreatorId: 'JaimeSevillaMolina',
  pageCreatedAt: '2016-07-10 04:05:09',
  seeDomainId: '0',
  editDomainId: 'AlexeiAndreev',
  submitToDomainId: '0',
  isAutosave: 'false',
  isSnapshot: 'false',
  isLiveEdit: 'true',
  isMinorEdit: 'false',
  indirectTeacher: 'false',
  todoCount: '4',
  isEditorComment: 'false',
  isApprovedComment: 'true',
  isResolved: 'false',
  snapshotText: '',
  anchorContext: '',
  anchorText: '',
  anchorOffset: '0',
  mergedInto: '',
  isDeleted: 'false',
  viewCount: '52',
  text: '##Weak form\nThe weak Gödel's first incompleteness theorem states that every [ $\\omega$-consistent] [ axiomatizable] extension of minimal arithmetic is incomplete.\n\nLet $T$ extend [-minimal_arithmetic], and let $Prv_{T}$ be the [5gt standard provability predicate] of $T$. \n\nThen we apply the [59c diagonal lemma] to get $G$ such that $T\\vdash G\\iff \\neg Prv_{T}(G)$.\n\nWe assert that the sentence $G$ is undecidable in $T$. We prove it by contradiction:\n\nSuppose that $T\\vdash G$. Then $Prv_ {T}(G)$ is correct, and as it is a $\\exists$-rudimentary sentence then it is [every_true_e_rudimentary_sentence_is_provable_in_minimal_arithmetic provable in minimal arithmetic], and thus in $T$. So we have that $T\\vdash Prv_ {T}(G)$ and also by the construction of $G$ that $T\\vdash \\neg Prv_{T}(G)$, contradicting that $T$ is consistent.\n\nNow, suppose that $T\\vdash \\neg G$. Then $T\\vdash  Prv_{T}(G)$. But then as $T$ is consistent there cannot be a standard proof of $G$, so if $Prv_{T}(x)$ is of the form $\\exists y Proof_{T}(x,y)$ then for no natural number $n$ it can be that $T\\vdash Proof_ {T}(\\ulcorner G\\urcorner,n)$, so $T$ is $\\omega$-inconsistent, in contradiction with the hypothesis.\n\n##Strong form\n\n> Every [5km consistent] and [-axiomatizable] extension of [-minimal_arithmetic] is [complete incomplete].\n\nThis theorem follows as a consequence of the [ undecidability of arithmetic] combined with the lemma stating that [ any complete axiomatizable theory is undecidable]\n',
  metaText: '',
  isTextLoaded: 'true',
  isSubscribedToDiscussion: 'false',
  isSubscribedToUser: 'false',
  isSubscribedAsMaintainer: 'false',
  discussionSubscriberCount: '1',
  maintainerCount: '1',
  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'
  ],
  childIds: [],
  parentIds: [
    'godels_first_incompleteness_theorem'
  ],
  commentIds: [],
  questionIds: [],
  tagIds: [],
  relatedIds: [],
  markIds: [],
  explanations: [],
  learnMore: [],
  requirements: [],
  subjects: [],
  lenses: [],
  lensParentId: 'godels_first_incompleteness_theorem',
  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: '20114',
      pageId: '59h',
      userId: 'JaimeSevillaMolina',
      edit: '2',
      type: 'newEdit',
      createdAt: '2016-10-11 20:24:50',
      auxPageId: '',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '20113',
      pageId: '59h',
      userId: 'JaimeSevillaMolina',
      edit: '0',
      type: 'newParent',
      createdAt: '2016-10-11 20:16:21',
      auxPageId: 'godels_first_incompleteness_theorem',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '16343',
      pageId: '59h',
      userId: 'JaimeSevillaMolina',
      edit: '1',
      type: 'newEdit',
      createdAt: '2016-07-10 04:05:09',
      auxPageId: '',
      oldSettingsValue: '',
      newSettingsValue: ''
    }
  ],
  feedSubmissions: [],
  searchStrings: {},
  hasChildren: 'false',
  hasParents: 'true',
  redAliases: {},
  improvementTagIds: [],
  nonMetaTagIds: [],
  todos: [],
  slowDownMap: 'null',
  speedUpMap: 'null',
  arcPageIds: 'null',
  contentRequests: {}
}