Paradox and Infinity

<div class="xblock xblock-public_view xblock-public_view-vertical" data-usage-id="block-v1:MITx+24.118x+2T2020+type@vertical+block@e129fd53df8d46ff937bd2f272f3da01" data-has-score="False" data-runtime-class="LmsRuntime" data-graded="False" data-course-id="course-v1:MITx+24.118x+2T2020" data-block-type="vertical" data-runtime-version="1" data-request-token="0c31962a2e3c11f088ab12f8c5a42579" data-init="VerticalStudentView"> <h2 class="hd hd-2 unit-title">The Philosophical Significance of Godel's Theorem</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+24.118x+2T2020+type@html+block@324c2bf0a25143f5a5c28e0ade801011"> <div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-usage-id="block-v1:MITx+24.118x+2T2020+type@html+block@324c2bf0a25143f5a5c28e0ade801011" data-has-score="False" data-runtime-class="LmsRuntime" data-graded="False" data-course-id="course-v1:MITx+24.118x+2T2020" data-block-type="html" data-runtime-version="1" data-request-token="0c31962a2e3c11f088ab12f8c5a42579" data-init="XBlockToXModuleShim"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "HTMLModule"} </script> <p><span style="font-family: 'book antiqua', palatino;">From a mathematical point of view, Gödel’s Theorem is an incredibly interesting result. </span></p> <p><span style="font-family: 'book antiqua', palatino;">But it also has far-reaching philosophical consequences. </span></p> <p><span style="font-family: 'book antiqua', palatino;">I’ll tell you about one of them in this section.</span></p> <h4><span style="font-family: 'book antiqua', palatino;">Certainty</span></h4> <p><span style="font-family: 'book antiqua', palatino;">Sometimes our best theories of the natural world turn out to be mistaken. The geocentric theory of the universe turned out to be mistaken. So did Newtonian physics. We now have different theories of the natural world. But how can we be sure that they won’t turn out to be mistaken too? Certainty would appear elusive.</span></p> <p><span style="font-family: 'book antiqua', palatino;">Many years ago, when I was young and reckless, I used to think that even if certainty remained forever elusive in our theories of the natural world, mathematical theories were different. I used to think that when it comes to mathematics, we really can aspire to absolute certainty. Alas, my youthful self was mistaken. It is a consequence of Gödel’s Theorem that absolute certainty is no more possible in our mathematical theories than it is in our theories of the natural world.</span></p> <p><span style="font-family: 'book antiqua', palatino;">There is an important disanalogy between physical theories and their mathematical counterparts. When Copernicus proposed the heliocentric theory of the universe, he defended his hypothesis by arguing that it was simpler than rival theories. But, of course, the fact that a theory is simple does not guarantee that the theory is true. Copernicus’s theory is false as he proposed it, since we now know that the planetary orbits are not circular, as he claimed, but rather elliptical. In contrast, when Euclid proposed that there are infinitely many prime numbers, he justified his hypothesis with a <em>proof</em> from basic principles. Since Euclid’s proof is valid, it <em>guarantees</em> that if his basic principles are correct, then his conclusion is correct as well. So it is natural to suppose—and so I assumed in my youth—that Euclid, unlike Copernicus, established his result <em>conclusively</em>.</span></p> <p><span style="font-family: 'book antiqua', palatino;">Unfortunately, there is a catch. A mathematical proof is always a <em>conditional</em> result: it shows that its conclusion is true <em>provided</em> that the basic principles on which the proof is based are true. </span></p> <p><span style="font-family: 'book antiqua', palatino;">This means that in order to show conclusively that a mathematical sentence is true, it is not enough to give a proof from basic principles. </span></p> <p><span style="font-family: 'book antiqua', palatino;">We must also show that our basic principles—our axioms—are true, and that our basic rules of inference are valid. </span></p> <p><span style="font-family: 'book antiqua', palatino;">How could one show that an axiom is true, or that an axiom is valid? It is tempting to answer that rules and axioms are principles so basic that they are absolutely obvious: their correctness is immediately apparent to us. There is, for example, an arithmetical axiom that says that different natural numbers must have different successors. What could possibly be more obvious?</span></p> <p><span style="font-family: 'book antiqua', palatino;">Sadly, the fact that an axiom seems obvious is not a guarantee of its truth. Some mathematical axioms that have seemed obviously true to great mathematicians have turned out to be false. Let me give you an example.</span></p> <p></p> </div> </div> </div> </div>

<div class="xblock xblock-public_view xblock-public_view-vertical" data-usage-id="block-v1:MITx+24.118x+2T2020+type@vertical+block@c6b596fe78a4441193e477d224fae181" data-has-score="False" data-runtime-class="LmsRuntime" data-graded="False" data-course-id="course-v1:MITx+24.118x+2T2020" data-block-type="vertical" data-runtime-version="1" data-request-token="0c31962a2e3c11f088ab12f8c5a42579" data-init="VerticalStudentView"> <h2 class="hd hd-2 unit-title">Russell's Paradox</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+24.118x+2T2020+type@html+block@5ed3f15e52334603ab5ba4fce6eedf68"> <div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-usage-id="block-v1:MITx+24.118x+2T2020+type@html+block@5ed3f15e52334603ab5ba4fce6eedf68" data-has-score="False" data-runtime-class="LmsRuntime" data-graded="False" data-course-id="course-v1:MITx+24.118x+2T2020" data-block-type="html" data-runtime-version="1" data-request-token="0c31962a2e3c11f088ab12f8c5a42579" data-init="XBlockToXModuleShim"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "HTMLModule"} </script> <p><span style="font-family: 'book antiqua', palatino;">Mathematicians sometimes talk about the <em>set</em> of prime numbers, or the <em>set</em> of functions from real numbers to real numbers. </span></p> <p><span style="font-family: 'book antiqua', palatino;">Suppose we wanted to formalize this talk of sets. What could we use as axioms? </span></p> <p><span style="font-family: 'book antiqua', palatino;">When the great German mathematician Gottlob Frege started thinking about sets, he proposed a version of the following axiom. (It seemed obviously true at the time.)</span></p> <p style="padding-left: 30px;"><span style="font-family: 'book antiqua', palatino;"><strong>Frege’s Axiom</strong></span><br /><span style="font-family: 'book antiqua', palatino;"> Select any objects you like (the prime numbers, for example, or the functions from real numbers to real numbers). There is a set whose members are all and only the objects you selected.</span></p> <p></p> <p><span style="font-family: 'book antiqua', palatino;">In fact, Frege’s Axiom is inconsistent. This was discovered by the British philosopher Bertrand Russell. Russell’s argument is devastatingly simple, and is sometimes known as <strong>Russell’s Paradox</strong>:</span></p> <blockquote> <p><span style="font-family: 'book antiqua', palatino;">Consider the objects that are not members of themselves. (The empty set has no members, for example, so it is an object that is not a member of itself.) An immediate consequence of Frege’s Axiom is that there is a set that has all and only the non-self-membered sets as members. Let us honor Russell by calling this set <span class="math inline">\(R\)</span>.</span></p> <p><span style="font-family: 'book antiqua', palatino;">Now consider the following question: is <span class="math inline">\(R\)</span> a member of <span class="math inline">\(R\)</span>? The definition of <span class="math inline">\(R\)</span> tells us that <span class="math inline">\(R\)</span>’s members are exactly the objects that have a certain property: the property of not being members of oneself. So we know that <span class="math inline">\(R\)</span> is a member of <span class="math inline">\(R\)</span> if and only if it has that property. In other words: <span class="math inline">\(R\)</span> is a member of itself if and only if it is not a member of itself.</span></p> <p><span style="font-family: 'book antiqua', palatino;">That’s a contradiction So <span class="math inline">\(R\)</span> can’t really exist. So Frege’s Axiom must be false.</span></p> </blockquote> </div> </div> </div> </div>

<div class="xblock xblock-public_view xblock-public_view-vertical" data-usage-id="block-v1:MITx+24.118x+2T2020+type@vertical+block@402afa577f5748d1875187483908044e" data-has-score="False" data-runtime-class="LmsRuntime" data-graded="False" data-course-id="course-v1:MITx+24.118x+2T2020" data-block-type="vertical" data-runtime-version="1" data-request-token="0c31962a2e3c11f088ab12f8c5a42579" data-init="VerticalStudentView"> <h2 class="hd hd-2 unit-title">A way out?</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+24.118x+2T2020+type@html+block@674fc157e4894513a29977bb0fe2cd9f"> <div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-usage-id="block-v1:MITx+24.118x+2T2020+type@html+block@674fc157e4894513a29977bb0fe2cd9f" data-has-score="False" data-runtime-class="LmsRuntime" data-graded="False" data-course-id="course-v1:MITx+24.118x+2T2020" data-block-type="html" data-runtime-version="1" data-request-token="0c31962a2e3c11f088ab12f8c5a42579" data-init="XBlockToXModuleShim"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "HTMLModule"} </script> <p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;"> I earlier told you about my youthful belief that mathematical proofs offer absolute certainty. We have now seen, however, that the situation is more complicated than it first appears. Mathematical proofs presuppose axioms and rules of inference. And, as we have seen, an axioms that appears obviously true can turn out to be false. </span></p> <p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;">At the beginning of the 20th century, the German mathematician David Hilbert suggested a program to overcome these difficulties. To a rough first approximation, the program was based on two ideas, a mathematical hypothesis and a philosophical hypothesis: </span></p> <p style="padding-left: 30px;"><span style="font-family: 'book antiqua', palatino;"><strong>Mathematical Hypothesis</strong></span><br /><span style="font-family: 'book antiqua', palatino;">There is an algorithmic method capable of establishing, once and for all, whether a set of axioms is consistent.</span></p> <p style="padding-left: 30px;"><span style="font-family: 'book antiqua', palatino;"><strong>Philosophical Hypothesis</strong></span><br /><span style="font-family: 'book antiqua', palatino;">All it takes for a set of axioms to count as a true description of some mathematical structure is for it to be consistent.</span></p> <p></p> <p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;">If Hilbert’s hypotheses had turned out to be true, it would have been possible, at least in principle, to establish conclusively whether a set of axioms is a true description of some mathematical structure: all we would have to do is apply our algorithm to test for consistency. And, of course, once we know which of our axiom systems are true, we can be sure that anything proved on the basis of those systems (on the basis of valid rules of inference) will be true as well. The dream of my youth!</span></p> </div> </div> </div> </div>

<div class="xblock xblock-public_view xblock-public_view-vertical" data-usage-id="block-v1:MITx+24.118x+2T2020+type@vertical+block@35cfc664ae9a48219ccd6e9aebb7d518" data-has-score="False" data-runtime-class="LmsRuntime" data-graded="False" data-course-id="course-v1:MITx+24.118x+2T2020" data-block-type="vertical" data-runtime-version="1" data-request-token="0c31962a2e3c11f088ab12f8c5a42579" data-init="VerticalStudentView"> <h2 class="hd hd-2 unit-title">Back to Godel</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+24.118x+2T2020+type@html+block@1b2d3bf4baf340c48ea29828cf26fc23"> <div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-usage-id="block-v1:MITx+24.118x+2T2020+type@html+block@1b2d3bf4baf340c48ea29828cf26fc23" data-has-score="False" data-runtime-class="LmsRuntime" data-graded="False" data-course-id="course-v1:MITx+24.118x+2T2020" data-block-type="html" data-runtime-version="1" data-request-token="0c31962a2e3c11f088ab12f8c5a42579" data-init="XBlockToXModuleShim"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "HTMLModule"} </script> <p><span style="font-family: 'book antiqua', palatino;">Sadly, Hilbert’s mathematical hypothesis is false. </span></p> <p><span style="font-family: 'book antiqua', palatino;">To see why this is so, let us bring together two different consequences of Gödel’s Theorem. </span></p> <p><span style="font-family: 'book antiqua', palatino;">Here is the first. As is shown in an appendix to Chapter 9, the behavior of Turing Machines can be described within the language of arithmetic. This means that if a problem can be solved by a Turing Machine, one can also solve it by figuring out whether a certain arithmetical sentence is true. </span></p> <p><span style="font-family: 'book antiqua', palatino;">In fact, Gödel showed something stronger: the relevant sentence can be proved to be true or false on the basis of the standard axiomatization of arithmetic. It follows that any problem that can be solved by a Turing Machine is settled by the standard axioms of arithmetic. </span></p> <p><span style="font-family: 'book antiqua', palatino;">Assuming the Curch-Turing Thesis (Chapter 9), this can be strengthened further: <em>any problem that can be solved by applying an algorithmic method is settled by the standard axioms of arithmetic</em>.</span></p> <p><span style="font-family: 'book antiqua', palatino;">The second consequence of Gödel’s Theorem is the following corollary of the Incompleteness Theorem:</span></p> <p style="padding-left: 30px;"><span style="font-family: 'book antiqua', palatino;"><strong>Gödel’s Second Theorem</strong></span><br /><span style="font-family: 'book antiqua', palatino;">If an axiomatic system is at least as strong as the standard axiomatization of arithmetic, it is unable prove its own consistency (unless it is inconsistent, in which case it can prove anything, including its own consistency).</span></p> <p></p> <p><span style="font-family: 'book antiqua', palatino;">Let us now bring the two ideas together to show that Hilbert’s mathematical hypothesis is false. </span></p> <p><span style="font-family: 'book antiqua', palatino;">We will proceed by <em>reductio</em>, and suppose that there is an algorithmic method that could establish the consistency of the standard axioms, assuming they are indeed consistent. By the first consequence of Gödel’s Theorem noted above, this means the standard axioms of arithmetic are able to prove their own consistency. But it follows from Gödel’s Second Theorem that the standard axioms can only prove their own consistency if they are inconsistent. So if the standard axioms are consistent, their consistency cannot be established by an algorithmic method. So Hilbert’s mathematical hypothesis must be false.</span></p> <p><span style="font-family: 'book antiqua', palatino;">A parallel argument can be used to show that there can be no algorithmic method for establishing the consistency of a <em>strengthening</em> of the standard axioms of arithmetic. Since the standard axiomatizations of real analysis and set-theory are both strengthenings of the standard arithmetical axioms, this means, in particular, that there can be no algorithmic method for establishing their consistency.</span></p> </div> </div> <div class="vert vert-1" data-id="block-v1:MITx+24.118x+2T2020+type@video+block@9a8009ee66d14e758221f13afef12990"> <div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-usage-id="block-v1:MITx+24.118x+2T2020+type@video+block@9a8009ee66d14e758221f13afef12990" data-has-score="False" data-runtime-class="LmsRuntime" data-graded="False" data-course-id="course-v1:MITx+24.118x+2T2020" data-block-type="video" data-runtime-version="1" data-request-token="0c31962a2e3c11f088ab12f8c5a42579" data-init="XBlockToXModuleShim"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Video"} </script> <h3 class="hd hd-2">Video Review: a Corollary of Gödel's Second Theorem</h3> <div id="video_9a8009ee66d14e758221f13afef12990" class="video closed" data-metadata='{"ytMetadataEndpoint": "", "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+24.118x+2T2020/xblock/block-v1:MITx+24.118x+2T2020+type@video+block@9a8009ee66d14e758221f13afef12990/handler/transcript/available_translations", "sources": [], "ytApiUrl": "https://www.youtube.com/iframe_api", "transcriptTranslationUrl": "/courses/course-v1:MITx+24.118x+2T2020/xblock/block-v1:MITx+24.118x+2T2020+type@video+block@9a8009ee66d14e758221f13afef12990/handler/transcript/translation/__lang__", "completionPercentage": 0.95, "saveStateEnabled": false, "prioritizeHls": false, "speed": null, "poster": null, "captionDataDir": null, "lmsRootURL": "https://openlearninglibrary.mit.edu", "showCaptions": "true", "recordedYoutubeIsAvailable": true, "savedVideoPosition": 0.0, "duration": 0.0, "completionEnabled": false, "end": 0.0, "autohideHtml5": false, "start": 0.0, "streams": "1.00:Gp6KdDqQA10", "ytTestTimeout": 1500, "transcriptLanguage": "en", "publishCompletionUrl": "/courses/course-v1:MITx+24.118x+2T2020/xblock/block-v1:MITx+24.118x+2T2020+type@video+block@9a8009ee66d14e758221f13afef12990/handler/publish_completion", "generalSpeed": 1.0, "transcriptLanguages": {"en": "English"}, "autoplay": false, "saveStateUrl": "/courses/course-v1:MITx+24.118x+2T2020/xblock/block-v1:MITx+24.118x+2T2020+type@video+block@9a8009ee66d14e758221f13afef12990/handler/xmodule_handler/save_user_state", "autoAdvance": false}' 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="9a8009ee66d14e758221f13afef12990"></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. 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<div class="xblock xblock-public_view xblock-public_view-vertical" data-usage-id="block-v1:MITx+24.118x+2T2020+type@vertical+block@d6c7f7ba1c144b5294ff4d13df184359" data-has-score="False" data-runtime-class="LmsRuntime" data-graded="False" data-course-id="course-v1:MITx+24.118x+2T2020" data-block-type="vertical" data-runtime-version="1" data-request-token="0c31962a2e3c11f088ab12f8c5a42579" data-init="VerticalStudentView"> <h2 class="hd hd-2 unit-title">Mathematics without Foundations</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+24.118x+2T2020+type@html+block@17f983c8bea64cb2ab585384b3541aca"> <div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-usage-id="block-v1:MITx+24.118x+2T2020+type@html+block@17f983c8bea64cb2ab585384b3541aca" data-has-score="False" data-runtime-class="LmsRuntime" data-graded="False" data-course-id="course-v1:MITx+24.118x+2T2020" data-block-type="html" data-runtime-version="1" data-request-token="0c31962a2e3c11f088ab12f8c5a42579" data-init="XBlockToXModuleShim"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "HTMLModule"} </script> <p><span style="font-family: 'book antiqua', palatino;">We have discussed a startling consequence of Gödel’s Theorem: when it comes to axiomatic systems strong enough to extend arithmetic, we cannot hope to establish their consistency using an algorithmic method.</span></p> <p><span style="font-family: 'book antiqua', palatino;">This not to say, however, that mathematicians should be worried that their preferred mathematical system might turn out to be inconsistent. When it comes to mathematical systems that are tried and true, there is no real reason to worry. </span></p> <p><span style="font-family: 'book antiqua', palatino;">But the upshot of our discussion is that this is not, in general, because we have a conclusive method for ruling out inconsistency. </span></p> <p><span style="font-family: 'book antiqua', palatino;">Rather, it’s because when it comes to tried and true mathematical systems, mathematicians have a good sense for how the system works and how it is connected to other systems, and this gives them a fairly good reason to think that there are no inconsistencies. </span></p> <p><span style="font-family: 'book antiqua', palatino;">But as Frege discovered, there is always the possibility of error. </span></p> <p><span style="font-family: 'book antiqua', palatino;">And there are mathematical systems which are not tried and true, and whose consistency is genuinely in doubt. One example is the brand of set theory developed by the American philosopher and logician W.V. Quine.</span></p> <p><span style="font-family: 'book antiqua', palatino;">The moral of our discussion is that my youthful views were mistaken. It was overly optimistic of me to think that we can be absolutely certain about the truth of our mathematical theories. </span></p> <p><span style="font-family: 'book antiqua', palatino;">As in the case of our best theories of the natural world, we have excellent reasons for thinking that our mathematical theories are true. But, as in the case of our best theories of the natural world, our reasons are not good enough for absolute certainty.</span></p> <p></p> </div> </div> <div class="vert vert-1" data-id="block-v1:MITx+24.118x+2T2020+type@video+block@bb51e10d0e73406c9ae33a07ad25bc15"> <div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-usage-id="block-v1:MITx+24.118x+2T2020+type@video+block@bb51e10d0e73406c9ae33a07ad25bc15" data-has-score="False" data-runtime-class="LmsRuntime" data-graded="False" data-course-id="course-v1:MITx+24.118x+2T2020" data-block-type="video" data-runtime-version="1" data-request-token="0c31962a2e3c11f088ab12f8c5a42579" data-init="XBlockToXModuleShim"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Video"} </script> <h3 class="hd hd-2">Video Review: Mathematics Without Foundations</h3> <div id="video_bb51e10d0e73406c9ae33a07ad25bc15" class="video closed" data-metadata='{"ytMetadataEndpoint": "", "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+24.118x+2T2020/xblock/block-v1:MITx+24.118x+2T2020+type@video+block@bb51e10d0e73406c9ae33a07ad25bc15/handler/transcript/available_translations", "sources": [], "ytApiUrl": "https://www.youtube.com/iframe_api", "transcriptTranslationUrl": "/courses/course-v1:MITx+24.118x+2T2020/xblock/block-v1:MITx+24.118x+2T2020+type@video+block@bb51e10d0e73406c9ae33a07ad25bc15/handler/transcript/translation/__lang__", "completionPercentage": 0.95, "saveStateEnabled": false, "prioritizeHls": false, "speed": null, "poster": null, "captionDataDir": null, "lmsRootURL": "https://openlearninglibrary.mit.edu", "showCaptions": "true", "recordedYoutubeIsAvailable": true, "savedVideoPosition": 0.0, "duration": 0.0, "completionEnabled": false, "end": 0.0, "autohideHtml5": false, "start": 0.0, "streams": "1.00:jzjg_AkWztE", "ytTestTimeout": 1500, "transcriptLanguage": "en", "publishCompletionUrl": "/courses/course-v1:MITx+24.118x+2T2020/xblock/block-v1:MITx+24.118x+2T2020+type@video+block@bb51e10d0e73406c9ae33a07ad25bc15/handler/publish_completion", "generalSpeed": 1.0, "transcriptLanguages": {"en": "English"}, "autoplay": false, "saveStateUrl": "/courses/course-v1:MITx+24.118x+2T2020/xblock/block-v1:MITx+24.118x+2T2020+type@video+block@bb51e10d0e73406c9ae33a07ad25bc15/handler/xmodule_handler/save_user_state", "autoAdvance": false}' 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="bb51e10d0e73406c9ae33a07ad25bc15"></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. 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