{
  localUrl: '../page/reals_as_dedekind_cuts_form_a_field.html',
  arbitalUrl: 'https://arbital.com/p/reals_as_dedekind_cuts_form_a_field',
  rawJsonUrl: '../raw/53v.json',
  likeableId: '2954',
  likeableType: 'page',
  myLikeValue: '0',
  likeCount: '1',
  dislikeCount: '0',
  likeScore: '1',
  individualLikes: [
    'JoeZeng'
  ],
  pageId: 'reals_as_dedekind_cuts_form_a_field',
  edit: '4',
  editSummary: '',
  prevEdit: '3',
  currentEdit: '4',
  wasPublished: 'true',
  type: 'wiki',
  title: 'The reals (constructed as Dedekind cuts) form a field',
  clickbait: 'The reals are an archetypal example of a field, but if we are to construct them from simpler objects, we need to show that our construction does indeed have the right properties.',
  textLength: '2996',
  alias: 'reals_as_dedekind_cuts_form_a_field',
  externalUrl: '',
  sortChildrenBy: 'likes',
  hasVote: 'false',
  voteType: '',
  votesAnonymous: 'false',
  editCreatorId: 'DylanHendrickson',
  editCreatedAt: '2016-07-08 17:20:19',
  pageCreatorId: 'PatrickStevens',
  pageCreatedAt: '2016-07-05 22:03:55',
  seeDomainId: '0',
  editDomainId: 'AlexeiAndreev',
  submitToDomainId: '0',
  isAutosave: 'false',
  isSnapshot: 'false',
  isLiveEdit: 'true',
  isMinorEdit: 'false',
  indirectTeacher: 'false',
  todoCount: '9',
  isEditorComment: 'false',
  isApprovedComment: 'true',
  isResolved: 'false',
  snapshotText: '',
  anchorContext: '',
  anchorText: '',
  anchorOffset: '0',
  mergedInto: '',
  isDeleted: 'false',
  viewCount: '30',
  text: 'The real numbers, when [50g constructed as Dedekind cuts] over the [4zq rationals], form a [481 field].\n\nWe shall often write the one-sided [dedekind_cut Dedekind cut] $(A, B)$ %%note:Recall: "one-sided" means that $A$ has no greatest element.%% as simply $\\mathbf{A}$ (using bold face for Dedekind cuts); we can do this because if we already know $A$ then $B$ is completely determined.\nThis will make our notation less messy.\n\nThe field structure, together with the [total_order total ordering] on it, is as follows (where we write $\\mathbf{0}$ for the Dedekind cut $(\\{ r \\in \\mathbb{Q} \\mid r < 0\\}, \\{ r \\in \\mathbb{Q}  \\mid r \\geq 0 \\})$): \n\n- $(A, B) + (C, D) = (A+C, B+D)$\n- $\\mathbf{A} \\leq \\mathbf{C}$ if and only if everything in $A$ lies in $C$.\n- Multiplication is somewhat complicated.\n - If $\\mathbf{0} \\leq \\mathbf{A}$, then $\\mathbf{A} \\times \\mathbf{C} = \\{ a c \\mid a \\in A, a > 0, c \\in C \\}$. [todo: we've missed out the complement in this notation, and can't put set-builder sets in boldface]\n - If $\\mathbf{A} < \\mathbf{0}$ and $\\mathbf{0} \\leq \\mathbf{C}$, then $\\mathbf{A} \\times \\mathbf{C} = \\{ a c \\mid a \\in A, c \\in C, c > 0 \\}$.\n - If $\\mathbf{A} < \\mathbf{0}$ and $\\mathbf{C} < \\mathbf{0}$, then $\\mathbf{A} \\times \\mathbf{C} = \\{\\} $ [todo: write down the form of the set]\n\nwhere $(A, B)$ is a one-sided [dedekind_cut Dedekind cut] (so that $A$ has no greatest element).\n\n(Here, the "set sum" $A+C$ is defined as "everything that can be made by adding one thing from $A$ to one thing from $C$": namely, $\\{ a+c \\mid a \\in A, c \\in C \\}$ in [-3lj]; and $A \\times C$ is similarly $\\{ a \\times c \\mid a \\in A, c \\in C \\}$.)\n\n# Proof\n\n## Well-definedness\n\nWe need to show firstly that these operations do in fact produce [dedekind_cut Dedekind cuts].\n\n### Addition\nFirstly, we need everything in $A+C$ to be less than everything in $B+D$.\nThis is true: if $a+c \\in A+C$, and $b+d \\in B+D$, then since $a < b$ and $c < d$, we have $a+c < b+d$.\n\nNext, we need $A+C$ and $B+D$ together to contain all the rationals.\nThis is true: [todo: this, it's quite boring]\n\nFinally, we need $(A+C, B+D)$ to be one-sided: that is, $A+C$ needs to have no top element, or equivalently, if $a+c \\in A+C$ then we can find a bigger $a' + c'$ in $A+C$.\nThis is also true: if $a+c$ is an element of $A+C$, then we can find an element $a'$ of $A$ which is bigger than $a$, and an element $c'$ of $C$ which is bigger than $C$ (since both $A$ and $C$ have no top elements, because the respective Dedekind cuts are one-sided); then $a' + c'$ is in $A+C$ and is bigger than $a+c$.\n\n### Multiplication\n[todo: this section]\n\n### Ordering\n[todo: this section]\n\n## Additive [3jb commutative] [3gd group structure]\n\n[todo: identity, associativity, inverse, commutativity]\n\n## [3gq Ring structure]\n\n[todo: multiplicative identity, associativity, distributivity]\n\n## [481 Field structure]\n\n[todo: inverses]\n\n## Ordering on the field\n\n[todo: a <= b implies a+c <= b+c, and 0 <= a, 0 <= b implies 0 <= ab]',
  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: [
    'DylanHendrickson',
    'PatrickStevens'
  ],
  childIds: [],
  parentIds: [
    'real_number_as_dedekind_cut'
  ],
  commentIds: [],
  questionIds: [],
  tagIds: [
    'work_in_progress_meta_tag',
    'proof_meta_tag'
  ],
  relatedIds: [],
  markIds: [],
  explanations: [],
  learnMore: [],
  requirements: [
    {
      id: '4829',
      parentId: 'algebraic_field',
      childId: 'reals_as_dedekind_cuts_form_a_field',
      type: 'requirement',
      creatorId: 'PatrickStevens',
      createdAt: '2016-07-05 18:20:51',
      level: '1',
      isStrong: 'false',
      everPublished: 'true'
    },
    {
      id: '4830',
      parentId: 'real_number_as_dedekind_cut',
      childId: 'reals_as_dedekind_cuts_form_a_field',
      type: 'requirement',
      creatorId: 'PatrickStevens',
      createdAt: '2016-07-05 18:20:57',
      level: '1',
      isStrong: 'false',
      everPublished: 'true'
    }
  ],
  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: '16235',
      pageId: 'reals_as_dedekind_cuts_form_a_field',
      userId: 'DylanHendrickson',
      edit: '4',
      type: 'newEdit',
      createdAt: '2016-07-08 17:20:19',
      auxPageId: '',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '15968',
      pageId: 'reals_as_dedekind_cuts_form_a_field',
      userId: 'DylanHendrickson',
      edit: '3',
      type: 'newEdit',
      createdAt: '2016-07-07 14:01:42',
      auxPageId: '',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '15469',
      pageId: 'reals_as_dedekind_cuts_form_a_field',
      userId: 'PatrickStevens',
      edit: '0',
      type: 'newTag',
      createdAt: '2016-07-05 22:04:18',
      auxPageId: 'work_in_progress_meta_tag',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '15467',
      pageId: 'reals_as_dedekind_cuts_form_a_field',
      userId: 'PatrickStevens',
      edit: '0',
      type: 'newRequirement',
      createdAt: '2016-07-05 22:03:57',
      auxPageId: 'algebraic_field',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '15468',
      pageId: 'reals_as_dedekind_cuts_form_a_field',
      userId: 'PatrickStevens',
      edit: '0',
      type: 'newRequirement',
      createdAt: '2016-07-05 22:03:57',
      auxPageId: 'real_number_as_dedekind_cut',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '15465',
      pageId: 'reals_as_dedekind_cuts_form_a_field',
      userId: 'PatrickStevens',
      edit: '0',
      type: 'newParent',
      createdAt: '2016-07-05 22:03:56',
      auxPageId: 'real_number_as_dedekind_cut',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '15466',
      pageId: 'reals_as_dedekind_cuts_form_a_field',
      userId: 'PatrickStevens',
      edit: '0',
      type: 'newTag',
      createdAt: '2016-07-05 22:03:56',
      auxPageId: 'proof_meta_tag',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '15463',
      pageId: 'reals_as_dedekind_cuts_form_a_field',
      userId: 'PatrickStevens',
      edit: '1',
      type: 'newEdit',
      createdAt: '2016-07-05 22:03:55',
      auxPageId: '',
      oldSettingsValue: '',
      newSettingsValue: ''
    }
  ],
  feedSubmissions: [],
  searchStrings: {},
  hasChildren: 'false',
  hasParents: 'true',
  redAliases: {},
  improvementTagIds: [],
  nonMetaTagIds: [],
  todos: [],
  slowDownMap: 'null',
  speedUpMap: 'null',
  arcPageIds: 'null',
  contentRequests: {}
}