{ localUrl: '../page/well_ordered_set.html', arbitalUrl: 'https://arbital.com/p/well_ordered_set', rawJsonUrl: '../raw/55r.json', likeableId: '3098', likeableType: 'page', myLikeValue: '0', likeCount: '1', dislikeCount: '0', likeScore: '1', individualLikes: [ 'KevinClancy' ], pageId: 'well_ordered_set', edit: '6', editSummary: '', prevEdit: '5', currentEdit: '6', wasPublished: 'true', type: 'wiki', title: 'Well-ordered set', clickbait: 'An ordered set with an order that always has a "next element".', textLength: '1419', alias: 'well_ordered_set', externalUrl: '', sortChildrenBy: 'likes', hasVote: 'false', voteType: '', votesAnonymous: 'false', editCreatorId: 'DylanHendrickson', editCreatedAt: '2016-07-07 17:09:09', pageCreatorId: 'DylanHendrickson', pageCreatedAt: '2016-07-06 20:23:49', 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: '67', text: 'A **well-ordered set** is a [-540] $(S, \\leq)$, such that for any nonempty subset $U \\subset S$ there is some $x \\in U$ such that for every $y \\in U$, $x \\leq y$; that is, every nonempty subset of $S$ has a least element.\n\nAny finite totally ordered set is well-ordered. The simplest [infinity infinite] well-ordered set is [45h $\\mathbb N$], also called [ordinal_omega $\\omega$] in this context.\n\nEvery well-ordered set is [4f4 isomorphic] to a unique [-ordinal_number], and thus any two well-ordered sets are comparable.\n\nThe order $\\leq$ is called a "well-ordering," despite the fact that "well" is usually an adverb.\n\n#Induction on a well-ordered set\n\n[mathematical_induction] works on any well-ordered set. On well-ordered sets longer than $\\mathbb N$, this is called [-transfinite_induction]. \n\nInduction is a method of proving a statement $P(x)$ for all elements $x$ of a well-ordered set $S$. Instead of directly proving $P(x)$, you prove that if $P(y)$ holds for all $y < x$, then $P(x)$ is true. This suffices to prove $P(x)$ for all $x \\in S$.\n\n%%hidden(Show proof):\nLet $U = \\{x \\in S \\mid \\neg P(x) \\}$ be the set of elements of $S$ for which $P$ doesn't hold, and suppose $U$ is nonempty. Since $S$ is well-ordered, $U$ has a least element $x$. That means $P(y)$ is true for all $y < x$, which implies $P(x)$. So $x \\not\\in U$, which is a contradiction. Hence $U$ is empty, and $P$ holds on all of $S$.\n%%', metaText: '', isTextLoaded: 'true', isSubscribedToDiscussion: 'false', isSubscribedToUser: 'false', isSubscribedAsMaintainer: 'false', discussionSubscriberCount: '0', maintainerCount: '0', 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', 'JoeZeng' ], childIds: [], parentIds: [ 'totally_ordered_set' ], commentIds: [ '564' ], questionIds: [], tagIds: [], 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: '16000', pageId: 'well_ordered_set', userId: 'DylanHendrickson', edit: '6', type: 'newEdit', createdAt: '2016-07-07 17:09:09', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '15970', pageId: 'well_ordered_set', userId: 'DylanHendrickson', edit: '5', type: 'newEdit', createdAt: '2016-07-07 14:08:50', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '15853', pageId: 'well_ordered_set', userId: 'JoeZeng', edit: '4', type: 'newEdit', createdAt: '2016-07-06 23:16:27', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '15846', pageId: 'well_ordered_set', userId: 'JoeZeng', edit: '2', type: 'newEdit', createdAt: '2016-07-06 23:04:09', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '15771', pageId: 'well_ordered_set', userId: 'DylanHendrickson', edit: '0', type: 'newAlias', createdAt: '2016-07-06 20:24:19', auxPageId: '', oldSettingsValue: '55r', newSettingsValue: 'well_ordered_set' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '15770', pageId: 'well_ordered_set', userId: 'DylanHendrickson', edit: '0', type: 'newParent', createdAt: '2016-07-06 20:23:51', auxPageId: 'totally_ordered_set', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '15768', pageId: 'well_ordered_set', userId: 'DylanHendrickson', edit: '1', type: 'newEdit', createdAt: '2016-07-06 20:23:49', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' } ], feedSubmissions: [], searchStrings: {}, hasChildren: 'false', hasParents: 'true', redAliases: {}, improvementTagIds: [], nonMetaTagIds: [], todos: [], slowDownMap: 'null', speedUpMap: 'null', arcPageIds: 'null', contentRequests: {} }