{
localUrl: '../page/rationals_are_countable.html',
arbitalUrl: 'https://arbital.com/p/rationals_are_countable',
rawJsonUrl: '../raw/511.json',
likeableId: '2933',
likeableType: 'page',
myLikeValue: '0',
likeCount: '1',
dislikeCount: '0',
likeScore: '1',
individualLikes: [
'EricBruylant'
],
pageId: 'rationals_are_countable',
edit: '1',
editSummary: '',
prevEdit: '0',
currentEdit: '1',
wasPublished: 'true',
type: 'wiki',
title: 'The set of rational numbers is countable',
clickbait: 'Although there are "lots and lots" of rational numbers, there are still only countably many of them.',
textLength: '1504',
alias: 'rationals_are_countable',
externalUrl: '',
sortChildrenBy: 'likes',
hasVote: 'false',
voteType: '',
votesAnonymous: 'false',
editCreatorId: 'PatrickStevens',
editCreatedAt: '2016-07-03 09:47:06',
pageCreatorId: 'PatrickStevens',
pageCreatedAt: '2016-07-03 09:47:07',
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: '27',
text: 'The set $\\mathbb{Q}$ of [4zq rational numbers] is countable: that is, there is a [499 bijection] between $\\mathbb{Q}$ and the set $\\mathbb{N}$ of [45h natural numbers].\n\n# Proof\n\nBy the [cantor_schroeder_bernstein_theorem Schröder-Bernstein theorem], it is enough to find an [4b7 injection] $\\mathbb{N} \\to \\mathbb{Q}$ and an injection $\\mathbb{Q} \\to \\mathbb{N}$.\n\nThe former is easy, because $\\mathbb{N}$ is a subset of $\\mathbb{Q}$ so the identity injection $n \\mapsto \\frac{n}{1}$ works.\n\nFor the latter, we may define a function $\\mathbb{Q} \\to \\mathbb{N}$ as follows.\nTake any rational in its lowest terms, as $\\frac{p}{q}$, say. %%note:That is, the [greatest_common_divisor GCD] of the numerator $p$ and denominator $q$ is $1$.%%\nAt most one of $p$ and $q$ is negative (if both are negative, we may just cancel $-1$ from the top and bottom of the fraction); by multiplying by $\\frac{-1}{-1}$ if necessary, assume without loss of generality that $q$ is positive.\nIf $p = 0$ then take $q = 1$.\n\nDefine $s$ to be $1$ if $p$ is positive, and $2$ if $p$ is negative.\n\nThen produce the natural number $2^p 3^q 5^s$.\n\nThe function $f: \\frac{p}{q} \\mapsto 2^p 3^q 5^s$ is injective, because [fundamental_theorem_of_arithmetic prime factorisations are unique] so if $f\\left(\\frac{p}{q}\\right) = f \\left(\\frac{a}{b} \\right)$ (with both fractions in their lowest terms, and $q$ positive) then $|p| = |a|, q=b$ and the sign of $p$ is equal to the sign of $a$.\nHence the two fractions were the same after all.',
metaText: '',
isTextLoaded: 'true',
isSubscribedToDiscussion: 'false',
isSubscribedToUser: 'false',
isSubscribedAsMaintainer: 'false',
discussionSubscriberCount: '2',
maintainerCount: '2',
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: [
'PatrickStevens'
],
childIds: [],
parentIds: [
'math'
],
commentIds: [],
questionIds: [],
tagIds: [
'proof_meta_tag'
],
relatedIds: [],
markIds: [],
explanations: [],
learnMore: [],
requirements: [
{
id: '4713',
parentId: 'rational_number',
childId: 'rationals_are_countable',
type: 'requirement',
creatorId: 'PatrickStevens',
createdAt: '2016-07-03 09:39:50',
level: '1',
isStrong: 'false',
everPublished: 'true'
},
{
id: '4714',
parentId: 'uncountable',
childId: 'rationals_are_countable',
type: 'requirement',
creatorId: 'PatrickStevens',
createdAt: '2016-07-03 09:39:54',
level: '1',
isStrong: 'false',
everPublished: 'true'
},
{
id: '4715',
parentId: 'math3',
childId: 'rationals_are_countable',
type: 'requirement',
creatorId: 'PatrickStevens',
createdAt: '2016-07-03 09:47:52',
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: '15292',
pageId: 'rationals_are_countable',
userId: 'EricBruylant',
edit: '0',
type: 'newParent',
createdAt: '2016-07-04 19:13:09',
auxPageId: 'math',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '15168',
pageId: 'rationals_are_countable',
userId: 'PatrickStevens',
edit: '0',
type: 'newRequirement',
createdAt: '2016-07-03 09:47:52',
auxPageId: 'math3',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '15165',
pageId: 'rationals_are_countable',
userId: 'PatrickStevens',
edit: '0',
type: 'newTag',
createdAt: '2016-07-03 09:47:08',
auxPageId: 'proof_meta_tag',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '15166',
pageId: 'rationals_are_countable',
userId: 'PatrickStevens',
edit: '0',
type: 'newRequirement',
createdAt: '2016-07-03 09:47:08',
auxPageId: 'rational_number',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '15167',
pageId: 'rationals_are_countable',
userId: 'PatrickStevens',
edit: '0',
type: 'newRequirement',
createdAt: '2016-07-03 09:47:08',
auxPageId: 'uncountable',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '15164',
pageId: 'rationals_are_countable',
userId: 'PatrickStevens',
edit: '1',
type: 'newEdit',
createdAt: '2016-07-03 09:47:07',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
}
],
feedSubmissions: [],
searchStrings: {},
hasChildren: 'false',
hasParents: 'true',
redAliases: {},
improvementTagIds: [],
nonMetaTagIds: [],
todos: [],
slowDownMap: 'null',
speedUpMap: 'null',
arcPageIds: 'null',
contentRequests: {}
}