{ localUrl: '../page/real_numbers_uncountable.html', arbitalUrl: 'https://arbital.com/p/real_numbers_uncountable', rawJsonUrl: '../raw/6fk.json', likeableId: '3637', likeableType: 'page', myLikeValue: '0', likeCount: '1', dislikeCount: '0', likeScore: '1', individualLikes: [ 'EricBruylant' ], pageId: 'real_numbers_uncountable', edit: '2', editSummary: '', prevEdit: '1', currentEdit: '2', wasPublished: 'true', type: 'wiki', title: 'Real numbers are uncountable', clickbait: 'The real numbers are uncountable.', textLength: '1622', alias: 'real_numbers_uncountable', externalUrl: '', sortChildrenBy: 'likes', hasVote: 'false', voteType: '', votesAnonymous: 'false', editCreatorId: 'EricBruylant', editCreatedAt: '2016-10-21 11:16:58', pageCreatorId: 'EricBruylant', pageCreatedAt: '2016-10-21 11:05:05', seeDomainId: '0', editDomainId: 'AlexeiAndreev', submitToDomainId: '0', isAutosave: 'false', isSnapshot: 'false', isLiveEdit: 'true', isMinorEdit: 'false', indirectTeacher: 'false', todoCount: '0', isEditorComment: 'false', isApprovedComment: 'true', isResolved: 'false', snapshotText: '', anchorContext: '', anchorText: '', anchorOffset: '0', mergedInto: '', isDeleted: 'false', viewCount: '58', text: 'We present a variant of Cantor's diagonalization argument to prove the [4bc real numbers] are [2w0 uncountable]. This [constructive_proof constructively proves] that there exist [uncountable_set uncountable sets] %%note: Since the real numbers are an example of one.%%.\n\nWe use the decimal representation of the real numbers. An overline ( $\\bar{\\phantom{9}}$ ) is used to mean that the digit(s) under it are repeated forever. Note that $a.bcd\\cdots z\\overline{9} = a.bcd\\cdots (z+1)\\overline{0}$ (if $z < 9$; otherwise, we need to continue carrying the one); $\\sum_{i=k}^\\infty 10^{-k} \\cdot 9 = 1 \\cdot 10^{-k + 1} + \\sum_{i=k}^\\infty 10^{-k} \\cdot 0$. Furthermore, these are the only equivalences between decimal representations; there are no other real numbers with multiple representations, and these real numbers have only these two decimal representations.\n\n**Theorem** The real numbers are uncountable.\n\n**Proof** Suppose, for [46z contradiction], that the real numbers are [-6f8 countable]; suppose that $f: \\mathbb Z^+ \\twoheadrightarrow \\mathbb R$ is a surjection. Let $r_n$ denote the $n^\\text{th}$ decimal digit of $r$, so that the fractional part of $r$ is $r_1r_2r_3r_4r_5\\ldots$ Then define a real number $r'$ with $0 \\le r' < 1$ so that $r'_n$ is 5 if $(f(n))_n \\ne 5$, and 6 if $(f(n))_n = 5$. Then there can be no $n$ such that $r' = f(n)$ since $r'_n \\ne (f(n))_n$. Thus $f$ is not surjective, contradicting our assumption, and $\\mathbb R$ is uncountable. $\\square$\n\n\nNote that choosing 5 and 6 as our allowable digits for $r'$ side-steps the issue that $0.\\overline{9} = 1.\\overline{0}$. %%', metaText: '', isTextLoaded: 'true', isSubscribedToDiscussion: 'false', isSubscribedToUser: 'false', isSubscribedAsMaintainer: 'false', discussionSubscriberCount: '1', maintainerCount: '1', userSubscriberCount: '0', lastVisit: '', hasDraft: 'false', votes: [], voteSummary: [ '0', '0', '0', '0', '0', '0', '0', '0', '0', '0' ], 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: { Summary: 'We present a variant of Cantor's diagonalization argument to prove the [4bc real numbers] are [2w0 uncountable]. This [constructive_proof constructively proves] that there exist [uncountable_set uncountable sets] %%note: Since the real numbers are an example of one.%%.' }, creatorIds: [ 'EricBruylant' ], childIds: [], parentIds: [ 'uncountable', 'real_number' ], commentIds: [], questionIds: [], tagIds: [ 'c_class_meta_tag', 'proof_meta_tag' ], relatedIds: [], markIds: [], explanations: [], learnMore: [], requirements: [ { id: '6600', parentId: 'uncountable', childId: 'real_numbers_uncountable', type: 'requirement', creatorId: 'EricBruylant', createdAt: '2016-10-21 11:03:54', level: '2', isStrong: 'true', everPublished: 'true' }, { id: '6601', parentId: 'real_number', childId: 'real_numbers_uncountable', type: 'requirement', creatorId: 'EricBruylant', createdAt: '2016-10-21 11:04:10', level: '2', isStrong: 'true', everPublished: 'true' } ], subjects: [ { id: '6605', parentId: 'uncountable', childId: 'real_numbers_uncountable', type: 'subject', creatorId: 'EricBruylant', createdAt: '2016-10-21 11:10:35', level: '3', isStrong: 'false', everPublished: 'true' } ], lenses: [], lensParentId: 'uncountable', 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: '20219', pageId: 'real_numbers_uncountable', userId: 'EricBruylant', edit: '0', type: 'newTag', createdAt: '2016-10-21 11:17:38', auxPageId: 'c_class_meta_tag', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '20218', pageId: 'real_numbers_uncountable', userId: 'EricBruylant', edit: '2', type: 'newEdit', createdAt: '2016-10-21 11:16:58', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '20217', pageId: 'real_numbers_uncountable', userId: 'EricBruylant', edit: '0', type: 'newSubject', createdAt: '2016-10-21 11:10:36', auxPageId: 'uncountable', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '20214', pageId: 'real_numbers_uncountable', userId: 'EricBruylant', edit: '0', type: 'newParent', createdAt: '2016-10-21 11:05:35', auxPageId: 'uncountable', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '20212', pageId: 'real_numbers_uncountable', userId: 'EricBruylant', edit: '0', type: 'newParent', createdAt: '2016-10-21 11:05:26', auxPageId: 'real_number', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '20208', pageId: 'real_numbers_uncountable', userId: 'EricBruylant', edit: '0', type: 'newRequirement', createdAt: '2016-10-21 11:05:07', auxPageId: 'uncountable', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '20209', pageId: 'real_numbers_uncountable', userId: 'EricBruylant', edit: '0', type: 'newRequirement', createdAt: '2016-10-21 11:05:07', auxPageId: 'real_number', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '20210', pageId: 'real_numbers_uncountable', userId: 'EricBruylant', edit: '0', type: 'newTag', createdAt: '2016-10-21 11:05:07', auxPageId: 'proof_meta_tag', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '20207', pageId: 'real_numbers_uncountable', userId: 'EricBruylant', edit: '1', type: 'newEdit', createdAt: '2016-10-21 11:05:05', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' } ], feedSubmissions: [], searchStrings: {}, hasChildren: 'false', hasParents: 'true', redAliases: {}, improvementTagIds: [], nonMetaTagIds: [], todos: [], slowDownMap: 'null', speedUpMap: 'null', arcPageIds: 'null', contentRequests: {} }