{ localUrl: '../page/only_empty_set_satisfies_up_of_emptyset.html', arbitalUrl: 'https://arbital.com/p/only_empty_set_satisfies_up_of_emptyset', rawJsonUrl: '../raw/603.json', likeableId: '3445', likeableType: 'page', myLikeValue: '0', likeCount: '2', dislikeCount: '0', likeScore: '2', individualLikes: [ 'EricBruylant', 'JaimeSevillaMolina' ], pageId: 'only_empty_set_satisfies_up_of_emptyset', edit: '2', editSummary: '', prevEdit: '1', currentEdit: '2', wasPublished: 'true', type: 'wiki', title: 'The empty set is the only set which satisfies the universal property of the empty set', clickbait: 'This theorem tells us that the universal property provides a sensible way to define the empty set uniquely.', textLength: '3950', alias: 'only_empty_set_satisfies_up_of_emptyset', externalUrl: '', sortChildrenBy: 'likes', hasVote: 'false', voteType: '', votesAnonymous: 'false', editCreatorId: 'PatrickStevens', editCreatedAt: '2016-08-26 17:16:09', pageCreatorId: 'PatrickStevens', pageCreatedAt: '2016-08-26 17:12:03', 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: '59', text: '[summary: Here we give three proofs that the only [3jz set] which satisfies the [-5zr] is the [-5zc] itself.]\n\nHere, we will prove that the only [3jz set] which satisfies the [-5zr] is the [-5zc] itself.\nThis will tell us that defining the empty set by this [-600] is actually a coherent thing to do, because it's not ambiguous as a definition.\n\nThere are three ways to prove this fact: one way looks at the objects themselves, one way takes a more maps-oriented approach, and one way is sort of a mixture of the two.\nAll of the proofs are enlightening in different ways.\n\nRecall first that the universal property of the empty set is as follows:\n\n> The empty set is the unique set $X$ such that for every set $A$, there is a unique function from $X$ to $A$.\n(To bring this property in line with our usual definition, we denote that unique set $X$ by the symbol $\\emptyset$.)\n\n# The "objects" way\n\nSuppose we have a set $X$ which is not empty.\nThen it has an element, $x$ say.\nNow, consider maps from $X$ to $\\{ 1, 2 \\}$.\n\nWe will show that there cannot be a unique [-3jy] from $X$ to $\\{ 1, 2 \\}$.\nIndeed, suppose $f: X \\to \\{ 1, 2 \\}$.\nThen $f(x) = 1$ or $f(x) = 2$.\nBut we can now define a new function $g: X \\to \\{1,2\\}$ which is given by setting $g(x)$ to be the *other* one of $1$ or $2$ to $f(x)$, and by letting $g(y) = f(y)$ for all $y \\not = x$.\n\nThis shows that the universal property of the empty set fails for $X$: we have shown that there is no unique function from $X$ to the specific set $\\{1,2\\}$.\n\n# The "maps" ways\n\nWe'll approach this in a slightly sneaky way: we will show that if two sets have the universal property, then there is a [499 bijection] between them. %%note: The most useful way to think of "bijection" in this context is "function with an inverse".%%\nOnce we have this fact, we're instantly done: the only set which bijects with $\\emptyset$ is $\\emptyset$ itself.\n\nSuppose we have two sets, $\\emptyset$ and $X$, both of which have the universal property of the empty set.\nThen, in particular (using the UP of $\\emptyset$) there is a unique map $f: \\emptyset \\to X$, and (using the UP of $X$) there is a unique map $g: X \\to \\emptyset$.\nAlso there is a unique map $\\mathrm{id}: \\emptyset \\to \\emptyset$. %%note: We use "id" for "identity", because as well as being the empty function, it happens to be the identity on $\\emptyset$.%%\n\nThe maps $f$ and $g$ are inverse to each other. Indeed, if we do $f$ and then $g$, we obtain a map from $\\emptyset$ (being the domain of $f$) to $\\emptyset$ (being the image of $g$); but we know there's a *unique* map $\\emptyset \\to \\emptyset$, so we must have the composition $g \\circ f$ being equal to $\\mathrm{id}$.\n\nWe've checked half of "$f$ and $g$ are inverse"; we still need to check that $f \\circ g$ is equal to the identity on $X$.\nThis follows by identical reasoning: there is a *unique* map $\\mathrm{id}_X : X \\to X$ by the fact that $X$ satisfies the universal property %%note: And we know that this map is the identity, because there's always an identity function from any set $Y$ to itself.%%, but $f \\circ g$ is a map from $X$ to $X$, so it must be $\\mathrm{id}_X$.\n\nSo $f$ and $g$ are bijections from $\\emptyset \\to X$ and $X \\to \\emptyset$ respectively.\n\n# The mixture\n\nThis time, let us suppose $X$ is a set which satisfies the universal property of the empty set.\nThen, in particular, there is a (unique) map $f: X \\to \\emptyset$.\n\nIf we pick any element $x \\in X$, what is $f(x)$?\nIt has to be a member of the empty set $\\emptyset$, because that's the codomain of $f$.\nBut there aren't any members of the empty set!\n\nSo there is no such $f$ after all, and so $X$ can't actually satisfy the universal property after all: we have found a set $Y = \\emptyset$ for which there is no map (and hence certainly no *unique* map) from $X$ to $Y$.\n\nThis method was a bit of a mixture of the two ways: it shows that a certain map can't exist if we specify a certain object.', 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: {}, creatorIds: [ 'PatrickStevens' ], childIds: [], parentIds: [ 'empty_set_universal_property' ], commentIds: [], questionIds: [], tagIds: [], relatedIds: [], markIds: [], explanations: [], learnMore: [], requirements: [], subjects: [], lenses: [], lensParentId: 'empty_set_universal_property', 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: '19186', pageId: 'only_empty_set_satisfies_up_of_emptyset', userId: 'PatrickStevens', edit: '2', type: 'newEdit', createdAt: '2016-08-26 17:16:09', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '19184', pageId: 'only_empty_set_satisfies_up_of_emptyset', userId: 'PatrickStevens', edit: '0', type: 'newParent', createdAt: '2016-08-26 17:12:04', auxPageId: 'empty_set_universal_property', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '19182', pageId: 'only_empty_set_satisfies_up_of_emptyset', userId: 'PatrickStevens', edit: '1', type: 'newEdit', createdAt: '2016-08-26 17:12:03', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' } ], feedSubmissions: [], searchStrings: {}, hasChildren: 'false', hasParents: 'true', redAliases: {}, improvementTagIds: [], nonMetaTagIds: [], todos: [], slowDownMap: 'null', speedUpMap: 'null', arcPageIds: 'null', contentRequests: {} }