<div class="xblock xblock-public_view xblock-public_view-vertical" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+24.118x+2T2020+type@vertical+block@cd9c29b7cddc4b07a87f020bef2d6cb4" data-init="VerticalStudentView" data-runtime-version="1" data-course-id="course-v1:MITx+24.118x+2T2020" data-block-type="vertical" data-has-score="False" data-graded="False" data-request-token="b19cf196fec611ee9bb616fff75c5923">
<h2 class="hd hd-2 unit-title">Living Large</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+24.118x+2T2020+type@html+block@445d19e8f8f0456f9aa13f6b555c1579">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+24.118x+2T2020+type@html+block@445d19e8f8f0456f9aa13f6b555c1579" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+24.118x+2T2020" data-block-type="html" data-has-score="False" data-graded="False" data-request-token="b19cf196fec611ee9bb616fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><span style="font-family: 'book antiqua', palatino;">There is an important strategy for building infinite sets of bigger and bigger cardinalities. It is based on deploying two basic resources: the power set operation, and the union operation. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">We’ll begin by reviewing these resources.</span></p>
<style type="text/css"><!--
p.p1 {margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px book antiqua}
--></style>
<style type="text/css"><!--
p.p1 {margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px book antiqua}
--></style>
<style type="text/css"><!--
p.p1 {margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px book antiqua}
--></style>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:MITx+24.118x+2T2020+type@html+block@e9e3463a450c465ba802e0ef81729b6d">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+24.118x+2T2020+type@html+block@e9e3463a450c465ba802e0ef81729b6d" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+24.118x+2T2020" data-block-type="html" data-has-score="False" data-graded="False" data-request-token="b19cf196fec611ee9bb616fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<h4><span style="font-family: 'book antiqua', palatino;">The power set operation</span></h4>
<p style="margin: 0px;"><span style="font-family: 'book antiqua', palatino;">In Lecture 1, we proved Cantor's Theorem, which shows that a set's power set always has more members than the set itself. This result entails that there is an infinite hierarchy of bigger and bigger infinities. More specifically, it entails: <span class="math display">\[|\mathbb{N}| < | \mathscr{P}^1(\mathbb{N})| < | \mathscr{P}^2(\mathbb{N})| < | \mathscr{P}^3(\mathbb{N})| < \ldots\]</span> where: <span class="math display">\[ \mathscr{P}^n(S) = \underbrace{ \mathscr{P}( \mathscr{P}(\ldots \mathscr{P}(S)\ldots))}_{\text{$n$ times}}\]</span></span></p>
<p><span style="font-family: 'book antiqua', palatino;">The sequence above includes some very big sets. For example, the set <span class="math inline">\( \mathscr{P}^{10^{10^{10}}}(\mathbb{N})\)</span> is fantastically bigger than any set we considered in Lecture 1. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">But could we go bigger still? Could we characterize a set that is bigger than <span class="math inline">\( \mathscr{P}^k(\mathbb{N})\)</span> for <em>every</em> natural number <span class="math inline">\(k\)</span>? </span></p>
<p><span style="font-family: 'book antiqua', palatino;">One might be tempted to introduce a set by way of the following definition: <span class="math display">\[ \mathscr{P}^\infty(\mathbb{N}) = \underbrace{ \mathscr{P}( \mathscr{P}(\ldots \mathscr{P}(\mathbb{N})\ldots))}_{\text{$\infty$ times}}\]</span> But it is not clear that such a definition makes sense. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">To see the problem, note that <span class="math inline">\( \mathscr{P}^\infty(\mathbb{N})\)</span> is supposed to be defined by iterating the ordinary power set operation, <span class="math inline">\( \mathscr{P}\)</span>. But each application of <span class="math inline">\( \mathscr{P}\)</span> requires a definite input. When <span class="math inline">\(k\)</span> is a positive integer, the input of <span class="math inline">\( \mathscr{P}^{k}(\mathbb{N})\)</span> is <span class="math inline">\( \mathscr{P}^{k-1}(\mathbb{N})\)</span>. But what input should one use at the supposed "infinite-th" stage of the process? </span></p>
<p><span style="font-family: 'book antiqua', palatino;">There is not a clear answer to this question. (Notice, in particular, that "<span class="math inline">\( \mathscr{P}^{\infty-1}(\mathbb{N})\)"</span> won’t do as an answer, since it is not clear what <span class="math inline">\( \mathscr{P}^{\infty-1}(\mathbb{N})\)</span> is supposed to be.)</span></p>
<p><span style="font-family: 'book antiqua', palatino;">The union operation will help us get around this problem.</span></p>
</div>
</div>
<div class="vert vert-2" data-id="block-v1:MITx+24.118x+2T2020+type@html+block@6c451c84d2364908802a360318e0a02a">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+24.118x+2T2020+type@html+block@6c451c84d2364908802a360318e0a02a" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+24.118x+2T2020" data-block-type="html" data-has-score="False" data-graded="False" data-request-token="b19cf196fec611ee9bb616fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<h4><span style="font-family: 'book antiqua', palatino;">The Union Operation</span></h4>
<p><span style="font-family: 'book antiqua', palatino;">In Lecture 1, we encountered a version of the union operation that takes finitely many sets as input, and delivers a single set as output. More specifically, we took <span class="math inline">\(A_1 \cup A_2 \cup \dots \cup A_n\)</span> to be the set of individuals <span class="math inline">\(x\)</span> such that <span class="math inline">\(x\)</span> is in at least one of <span class="math inline">\(A_1,A_2\dots,A_n\)</span>. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">We will now consider a variant of the union operation that that takes as input a set <span class="math inline">\(A\)</span> of arbitrarily many sets. Let <span class="math inline">\(A\)</span> be a set of sets; as it might be, <span class="math display">\[A = \{S_1, S_2, S_3, \dots\}\]</span> Then the union of <span class="math inline">\(A\)</span> (in symbols: <span class="math inline">\(\bigcup A\)</span>) is the result of pooling together the elements of each of the sets in <span class="math inline">\(A\)</span>. In other words: <span class="math inline">\(\bigcup A = S_1 \cup S_2 \cup S_3 \cup \dots\)</span>. (Formally, we define <span class="math inline">\(\bigcup A\)</span> as the set of individuals <span class="math inline">\(x\)</span> such that <span class="math inline">\(x\)</span> is a member of some member of <span class="math inline">\(A\)</span>.)</span></p>
<p><span style="font-family: 'book antiqua', palatino;">The key advantage of this new version of the union operation is that <span class="math inline">\(\bigcup A\)</span> is well-defined even if <span class="math inline">\(A\)</span> has infinitely many sets as members. This makes the union operation incredibly powerful. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">To illustrate this point, consider the set <span class="math inline">\(\{\mathbb{N},\mathscr{P}^1(\mathbb{N}), \mathscr{P}^2(\mathbb{N}), \dots\} \)</span><span style="font-family: 'book antiqua', palatino; font-size: 1em;">. Even though it only has as many elements as there are natural numbers, its union </span><span class="math display" style="font-family: 'book antiqua', palatino; font-size: 1em;">\[\bigcup \{\mathbb{N}, \mathscr{P}^1(\mathbb{N}), \mathscr{P}^2(\mathbb{N}), \dots \}\]</span><span style="font-family: 'book antiqua', palatino; font-size: 1em;"> is far bigger than the set of natural numbers. In fact, it is bigger than </span><span class="math inline" style="font-family: 'book antiqua', palatino; font-size: 1em;">\( \mathscr{P}^{k}(\mathbb{N})\)</span><span style="font-family: 'book antiqua', palatino; font-size: 1em;"> for each natural number </span><span class="math inline" style="font-family: 'book antiqua', palatino; font-size: 1em;">\(k\)</span><span style="font-family: 'book antiqua', palatino; font-size: 1em;">, since it includes everything in </span><span class="math inline" style="font-family: 'book antiqua', palatino; font-size: 1em;">\( \mathscr{P}^{k+1}(\mathbb{N})\)</span><span style="font-family: 'book antiqua', palatino; font-size: 1em;">.</span></span></p>
</div>
</div>
<div class="vert vert-3" data-id="block-v1:MITx+24.118x+2T2020+type@video+block@d410630789ea487bae4b33e385023901">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+24.118x+2T2020+type@video+block@d410630789ea487bae4b33e385023901" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+24.118x+2T2020" data-block-type="video" data-has-score="False" data-graded="False" data-request-token="b19cf196fec611ee9bb616fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">Video Review: Unions</h3>
<div
id="video_d410630789ea487bae4b33e385023901"
class="video closed"
data-metadata='{"speed": null, "publishCompletionUrl": "/courses/course-v1:MITx+24.118x+2T2020/xblock/block-v1:MITx+24.118x+2T2020+type@video+block@d410630789ea487bae4b33e385023901/handler/publish_completion", "streams": "1.00:Q97qm4eRu5I", "prioritizeHls": false, "completionEnabled": false, "autoAdvance": false, "captionDataDir": null, "transcriptLanguages": {"en": "English"}, "recordedYoutubeIsAvailable": true, "showCaptions": "true", "completionPercentage": 0.95, "saveStateUrl": "/courses/course-v1:MITx+24.118x+2T2020/xblock/block-v1:MITx+24.118x+2T2020+type@video+block@d410630789ea487bae4b33e385023901/handler/xmodule_handler/save_user_state", "ytMetadataEndpoint": "", "duration": 0.0, "autohideHtml5": false, "transcriptLanguage": "en", "ytApiUrl": "https://www.youtube.com/iframe_api", "lmsRootURL": "https://openlearninglibrary.mit.edu", "sources": [], "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+24.118x+2T2020/xblock/block-v1:MITx+24.118x+2T2020+type@video+block@d410630789ea487bae4b33e385023901/handler/transcript/available_translations", "savedVideoPosition": 0.0, "transcriptTranslationUrl": "/courses/course-v1:MITx+24.118x+2T2020/xblock/block-v1:MITx+24.118x+2T2020+type@video+block@d410630789ea487bae4b33e385023901/handler/transcript/translation/__lang__", "saveStateEnabled": false, "ytTestTimeout": 1500, "poster": null, "autoplay": false, "generalSpeed": 1.0, "start": 0.0, "end": 0.0}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="d410630789ea487bae4b33e385023901"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_d410630789ea487bae4b33e385023901">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_d410630789ea487bae4b33e385023901">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+24.118x+2T2020/xblock/block-v1:MITx+24.118x+2T2020+type@video+block@d410630789ea487bae4b33e385023901/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+24.118x+2T2020/xblock/block-v1:MITx+24.118x+2T2020+type@video+block@d410630789ea487bae4b33e385023901/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
<div class="vert vert-4" data-id="block-v1:MITx+24.118x+2T2020+type@problem+block@89abf68b605f49a8999791994cc69fba">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+24.118x+2T2020+type@problem+block@89abf68b605f49a8999791994cc69fba" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+24.118x+2T2020" data-block-type="problem" data-has-score="True" data-graded="False" data-request-token="b19cf196fec611ee9bb616fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_89abf68b605f49a8999791994cc69fba" class="problems-wrapper" role="group"
aria-labelledby="89abf68b605f49a8999791994cc69fba-problem-title"
data-problem-id="block-v1:MITx+24.118x+2T2020+type@problem+block@89abf68b605f49a8999791994cc69fba" data-url="/courses/course-v1:MITx+24.118x+2T2020/xblock/block-v1:MITx+24.118x+2T2020+type@problem+block@89abf68b605f49a8999791994cc69fba/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="1"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="89abf68b605f49a8999791994cc69fba-problem-title" aria-describedby="block-v1:MITx+24.118x+2T2020+type@problem+block@89abf68b605f49a8999791994cc69fba-problem-progress" tabindex="-1">
Problem 1
</h3>
<div class="problem-progress" id="block-v1:MITx+24.118x+2T2020+type@problem+block@89abf68b605f49a8999791994cc69fba-problem-progress"></div>
<div class="problem">
<div>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><p>
<span style="font-family: 'book antiqua', palatino;">Which of the following, if any, is a set <span class="math inline">\(A\)</span> such that <span class="math inline">\(|\bigcup A| &lt; |A|\)</span>?</span>
</p>
<div class="choicegroup capa_inputtype" id="inputtype_89abf68b605f49a8999791994cc69fba_2_1">
<fieldset aria-describedby="status_89abf68b605f49a8999791994cc69fba_2_1">
<div class="field">
<input type="radio" name="input_89abf68b605f49a8999791994cc69fba_2_1" id="input_89abf68b605f49a8999791994cc69fba_2_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="89abf68b605f49a8999791994cc69fba_2_1-choice_0-label" for="input_89abf68b605f49a8999791994cc69fba_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_89abf68b605f49a8999791994cc69fba_2_1">
<span style="font-family: 'book antiqua', palatino;">
<span class="math inline">\(A = \{\mathbb{N}\}\)</span>
</span>
</label>
</div>
<div class="field">
<input type="radio" name="input_89abf68b605f49a8999791994cc69fba_2_1" id="input_89abf68b605f49a8999791994cc69fba_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="89abf68b605f49a8999791994cc69fba_2_1-choice_1-label" for="input_89abf68b605f49a8999791994cc69fba_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_89abf68b605f49a8999791994cc69fba_2_1">
<span style="font-family: 'book antiqua', palatino;">
<span class="math inline">\(A = \mathscr{P}(\mathbb{N})\)</span>
</span>
</label>
</div>
<div class="field">
<input type="radio" name="input_89abf68b605f49a8999791994cc69fba_2_1" id="input_89abf68b605f49a8999791994cc69fba_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="89abf68b605f49a8999791994cc69fba_2_1-choice_2-label" for="input_89abf68b605f49a8999791994cc69fba_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_89abf68b605f49a8999791994cc69fba_2_1">
<span style="font-family: 'book antiqua', palatino;">\(A = \{\mathbb{N}, \mathscr{P}^1(\mathbb{N}), \mathscr{P}^2(\mathbb{N}), \dots \} \)</span>
</label>
</div>
<div class="field">
<input type="radio" name="input_89abf68b605f49a8999791994cc69fba_2_1" id="input_89abf68b605f49a8999791994cc69fba_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="89abf68b605f49a8999791994cc69fba_2_1-choice_3-label" for="input_89abf68b605f49a8999791994cc69fba_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_89abf68b605f49a8999791994cc69fba_2_1">
<span style="font-family: 'book antiqua', palatino;">None of the above.</span>
</label>
</div>
<span id="answer_89abf68b605f49a8999791994cc69fba_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_89abf68b605f49a8999791994cc69fba_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div><div class="solution-span">
<span id="solution_89abf68b605f49a8999791994cc69fba_solution_1"/>
</div></div>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Problem 1" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_89abf68b605f49a8999791994cc69fba" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_89abf68b605f49a8999791994cc69fba">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="89abf68b605f49a8999791994cc69fba-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="89abf68b605f49a8999791994cc69fba-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="89abf68b605f49a8999791994cc69fba-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="False">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
<div class="vert vert-5" data-id="block-v1:MITx+24.118x+2T2020+type@html+block@700fe4e4c6b84a0b9f27498dd41088a7">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+24.118x+2T2020+type@html+block@700fe4e4c6b84a0b9f27498dd41088a7" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+24.118x+2T2020" data-block-type="html" data-has-score="False" data-graded="False" data-request-token="b19cf196fec611ee9bb616fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<h4><span style="font-family: 'book antiqua', palatino;">Bringing the two operations together</span></h4>
<p><span style="font-family: 'book antiqua', palatino;">This is what we’ve learned so far: <span class="math display">\[|\mathbb{N}| < | \mathscr{P}^1(\mathbb{N})| < | \mathscr{P}^2(\mathbb{N})| < \dots < |\bigcup \{\mathbb{N}, \mathscr{P}^1(\mathbb{N}),\dots\}|\]</span> Can we construct sets that are bigger still? </span></p>
<p><span style="font-family: 'book antiqua', palatino;">Of course we can! </span></p>
<p><span style="font-family: 'book antiqua', palatino;">Let us use "<span class="math inline">\(\mathscr{U}\)"</span> to abbreviate "<span class="math inline">\(\bigcup \{\mathbb{N}, \mathscr{P}^1(\mathbb{N}),\dots\}\)"</span>. Then Cantor’s Theorem tells us that if we apply the powerset operation to <span class="math inline">\(\mathscr{U}\)</span>, we will get something even bigger. And each successive application of the powerset operation gives us something bigger still: <span class="math display">\[|\mathscr{U}| < \dots | \mathscr{P}^1\left(\mathscr{U}\right)| < | \mathscr{P}^2\left(\mathscr{U}\right)| < \dots\]</span> And we can keep going. We can apply the union operation to the set of everything we’ve built so far. And then apply further iterations of the powerset operation. And then apply the union operation to everything we’ve built so far. And then apply further iterations of the power sets. And so forth.</span></p>
<p><span style="font-family: 'book antiqua', palatino;">We now have a procedure for constructing sets of greater and greater cardinality. In rough outline the procedure is very simple: after applying the powerset operation countably many times, apply the union operation. And repeat. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">The main objective of this chapter is to show you how to develop the idea properly, by introducing you to one of the most beautiful tools in the whole of mathematics: the notion of an ordinal.</span></p>
<p></p>
<p><span style="font-size: 1em; font-family: 'book antiqua', palatino;"> </span></p>
<style type="text/css"><!--
p.p1 {margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Helvetica}
--></style>
<p></p>
<style type="text/css"><!--
p.p1 {margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Helvetica}
--></style>
</div>
</div>
<div class="vert vert-6" data-id="block-v1:MITx+24.118x+2T2020+type@video+block@0468139b50eb476b8ab1d2cc7739fd36">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+24.118x+2T2020+type@video+block@0468139b50eb476b8ab1d2cc7739fd36" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+24.118x+2T2020" data-block-type="video" data-has-score="False" data-graded="False" data-request-token="b19cf196fec611ee9bb616fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">Video Question: Clarifying the Union Operation</h3>
<div
id="video_0468139b50eb476b8ab1d2cc7739fd36"
class="video closed"
data-metadata='{"speed": null, "publishCompletionUrl": "/courses/course-v1:MITx+24.118x+2T2020/xblock/block-v1:MITx+24.118x+2T2020+type@video+block@0468139b50eb476b8ab1d2cc7739fd36/handler/publish_completion", "streams": "1.00:vD3cBKqfk3g", "prioritizeHls": false, "completionEnabled": false, "autoAdvance": false, "captionDataDir": null, "transcriptLanguages": {"en": "English"}, "recordedYoutubeIsAvailable": true, "showCaptions": "true", "completionPercentage": 0.95, "saveStateUrl": "/courses/course-v1:MITx+24.118x+2T2020/xblock/block-v1:MITx+24.118x+2T2020+type@video+block@0468139b50eb476b8ab1d2cc7739fd36/handler/xmodule_handler/save_user_state", "ytMetadataEndpoint": "", "duration": 0.0, "autohideHtml5": false, "transcriptLanguage": "en", "ytApiUrl": "https://www.youtube.com/iframe_api", "lmsRootURL": "https://openlearninglibrary.mit.edu", "sources": [], "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+24.118x+2T2020/xblock/block-v1:MITx+24.118x+2T2020+type@video+block@0468139b50eb476b8ab1d2cc7739fd36/handler/transcript/available_translations", "savedVideoPosition": 0.0, "transcriptTranslationUrl": "/courses/course-v1:MITx+24.118x+2T2020/xblock/block-v1:MITx+24.118x+2T2020+type@video+block@0468139b50eb476b8ab1d2cc7739fd36/handler/transcript/translation/__lang__", "saveStateEnabled": false, "ytTestTimeout": 1500, "poster": null, "autoplay": false, "generalSpeed": 1.0, "start": 0.0, "end": 66.0}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="0468139b50eb476b8ab1d2cc7739fd36"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_0468139b50eb476b8ab1d2cc7739fd36">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_0468139b50eb476b8ab1d2cc7739fd36">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+24.118x+2T2020/xblock/block-v1:MITx+24.118x+2T2020+type@video+block@0468139b50eb476b8ab1d2cc7739fd36/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+24.118x+2T2020/xblock/block-v1:MITx+24.118x+2T2020+type@video+block@0468139b50eb476b8ab1d2cc7739fd36/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
<div class="vert vert-7" data-id="block-v1:MITx+24.118x+2T2020+type@video+block@0fbdf7cfad6b4d728cc46f8a6039d0c7">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+24.118x+2T2020+type@video+block@0fbdf7cfad6b4d728cc46f8a6039d0c7" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+24.118x+2T2020" data-block-type="video" data-has-score="False" data-graded="False" data-request-token="b19cf196fec611ee9bb616fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">Video Question: Further Clarification</h3>
<div
id="video_0fbdf7cfad6b4d728cc46f8a6039d0c7"
class="video closed"
data-metadata='{"speed": null, "publishCompletionUrl": "/courses/course-v1:MITx+24.118x+2T2020/xblock/block-v1:MITx+24.118x+2T2020+type@video+block@0fbdf7cfad6b4d728cc46f8a6039d0c7/handler/publish_completion", "streams": "1.00:vD3cBKqfk3g", "prioritizeHls": false, "completionEnabled": false, "autoAdvance": false, "captionDataDir": null, "transcriptLanguages": {"en": "English"}, "recordedYoutubeIsAvailable": true, "showCaptions": "true", "completionPercentage": 0.95, "saveStateUrl": "/courses/course-v1:MITx+24.118x+2T2020/xblock/block-v1:MITx+24.118x+2T2020+type@video+block@0fbdf7cfad6b4d728cc46f8a6039d0c7/handler/xmodule_handler/save_user_state", "ytMetadataEndpoint": "", "duration": 0.0, "autohideHtml5": false, "transcriptLanguage": "en", "ytApiUrl": "https://www.youtube.com/iframe_api", "lmsRootURL": "https://openlearninglibrary.mit.edu", "sources": [], "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+24.118x+2T2020/xblock/block-v1:MITx+24.118x+2T2020+type@video+block@0fbdf7cfad6b4d728cc46f8a6039d0c7/handler/transcript/available_translations", "savedVideoPosition": 0.0, "transcriptTranslationUrl": "/courses/course-v1:MITx+24.118x+2T2020/xblock/block-v1:MITx+24.118x+2T2020+type@video+block@0fbdf7cfad6b4d728cc46f8a6039d0c7/handler/transcript/translation/__lang__", "saveStateEnabled": false, "ytTestTimeout": 1500, "poster": null, "autoplay": false, "generalSpeed": 1.0, "start": 164.0, "end": 0.0}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="0fbdf7cfad6b4d728cc46f8a6039d0c7"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_0fbdf7cfad6b4d728cc46f8a6039d0c7">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_0fbdf7cfad6b4d728cc46f8a6039d0c7">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+24.118x+2T2020/xblock/block-v1:MITx+24.118x+2T2020+type@video+block@0fbdf7cfad6b4d728cc46f8a6039d0c7/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+24.118x+2T2020/xblock/block-v1:MITx+24.118x+2T2020+type@video+block@0fbdf7cfad6b4d728cc46f8a6039d0c7/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
<div class="vert vert-8" data-id="block-v1:MITx+24.118x+2T2020+type@html+block@84ab4532f1c4490a8a068c8ea7cf08da">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+24.118x+2T2020+type@html+block@84ab4532f1c4490a8a068c8ea7cf08da" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+24.118x+2T2020" data-block-type="html" data-has-score="False" data-graded="False" data-request-token="b19cf196fec611ee9bb616fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><strong>Note:</strong> There's an important omission on 4:07 of the video. If I had written out a description of \(\mathscr{P}^2(\mathbb{N})\) on the board, it might have looked like this:</p>
<p>\[\mathscr{P}^2(\mathbb{N}) = {\Large\{}\{\}, \{\{\}\}, \{\{0\}\}, \{\{1\}\},\dots, \{\{\},\{0\}\}, \{\{\},\{1\}\}, \{\{0\},\{1\}\},\dots {\Large\}}\]</p>
<p>So although it's true that most members of \(\mathscr{P}^2(\mathbb{N})\) have two levels of nested brackets, there is one exception, since the empty set \(\{\}\) is a member of \(\mathscr{P}^2(\mathbb{N})\) (and of \(\mathscr{P}^n(\mathbb{N})\) for each \(n\)) but has only one level of nested brackets.</p>
</div>
</div>
</div>
</div>
© All Rights Reserved