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<h2 class="hd hd-2 unit-title">Preface</h2>
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<p><span style="font-family: book antiqua, palatino;"></span><span style="font-family: book antiqua, palatino;">Consider the square numbers: \(1^2,2^2, 3^2, 4^2, \ldots\), and their (positive) roots: \(1, 2, 3, 4, \ldots\). What are there more of: roots or squares? </span></p>
<p><span style="font-family: book antiqua, palatino;">It is tempting to think that there are more roots. After all, every square is the root of some positive integer, but not vice-versa. (The number \(2\), for instance, is the root of \(4\), since </span><span style="font-family: book antiqua, palatino;"><span style="font-family: book antiqua, palatino;">\(4 = 2^2\)</span>. But \(2\) is not the square of a positive integer.) </span></p>
<p><span style="font-family: book antiqua, palatino;">One might also think, however, that there are just as many roots as perfect squares. After all, as Galileo Galilei put it </span><span style="font-family: book antiqua, palatino;"><span style="font-family: book antiqua, palatino;">in his 1638 masterpiece, <em>Dialogues Concerning Two New Sciences</em></span>, "every square has its own root and every root has its own square, while no square has more than one root and no root has more than one square." (p. 78) In other words, we can <em>pair </em>each root with its square, with no overlap and no remainders: </span><span style="font-family: 'book antiqua', palatino;"><span class="math display">\[\begin{array}{ccccl} 1 & 2 & 3 & 4 & \dots\\ \downarrow & \downarrow & \downarrow & \downarrow \\ 1^2 & 2^2 & 3^2 & 4^2 &\text{\(\dots\)} \end{array}\]</span> <span class="math display"></span></span><br /><span style="font-family: book antiqua, palatino;">Galileo's observation leaves us with a paradox. We have seemingly decisive reasons for thinking that there are more roots than squares, but also seemingly decisive reasons for thinking that there are just as many roots as squares.</span></p>
<p><span style="font-family: book antiqua, palatino;">Galileo's Paradox is a paradox of the most interesting sort. A boring paradox is a paradox that leads nowhere. It is due to a superficial mistake and is no more than nuisance. An interesting paradox, on the other hand, is a paradox that reveals a genuine problem in our understanding of its subject-matter. The most interesting paradoxes of all are those that reveal a problem interesting enough to lead to the development of an improved theory. </span><br /><br /><span style="font-family: book antiqua, palatino;">Such is the case of Galileo's infinitary paradox. In 1874, almost two and a half centuries after <em>Two New Sciences</em>, the German mathematician Georg Cantor published an article that describes a rigorous methodology for comparing the sizes of infinite sets and yields the arresting conclusion that <em>there are different sizes of infinity</em>. </span></p>
<p><span style="font-family: book antiqua, palatino;">Cantor's work was the turning point in our understanding of infinity. By treading on the brink of paradox, he managed to replace a muddled pre-theoretic notion of infinite size with a rigorous and fruitful notion, which has become one of the essential tools of contemporary mathematics.</span></p>
<p><span style="font-family: book antiqua, palatino;">This is a class about insights like Cantor's. More generally, it is a class about some of the highlights from the intersection of philosophy and mathematics.<br /></span></p>
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<h2 class="hd hd-2 unit-title">Class Schedule</h2>
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<p><span style="font-family: book antiqua, palatino;">This class consists of ten lectures, divided into three modules:</span></p>
<p><span style="font-family: book antiqua, palatino;"></span></p>
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<h4>Module 1: INFINITY</h4>
<p><span style="font-family: book antiqua, palatino;">Week 1 Infinite Cardinalities</span></p>
<p><span style="font-family: book antiqua, palatino;">Week 2 The Higher Infinite</span></p>
<p><span style="font-family: book antiqua, palatino;">Week 3 Omega-Sequence Paradoxes</span></p>
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<h4>Module 2: DECISIONS, PROBABILITIES AND MEASURES</h4>
<p><span style="font-family: book antiqua, palatino;">Week 4 Time Travel</span></p>
<p><span style="font-family: book antiqua, palatino;">Week 5 Newcomb’s Problem</span></p>
<p><span style="font-family: book antiqua, palatino;">Week 6 Probability</span></p>
<p><span style="font-family: book antiqua, palatino;">Week 7 Non-Measurable Sets</span></p>
<p><span style="font-family: book antiqua, palatino;">Week 8 The Banach-Tarski Theorem</span></p>
<p><span style="font-family: book antiqua, palatino;"></span></p>
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<h4>Module 3: COMPUTABILITY AND GÖDEL’S THEOREM</h4>
<p><span style="font-family: book antiqua, palatino;">Week 9 Computability</span></p>
<p><span style="font-family: book antiqua, palatino;">Week 10 Gödel’s Theorem </span></p>
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<h2 class="hd hd-2 unit-title">What to Expect</h2>
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<p><span style="font-family: book antiqua, palatino;">Although many of topics we'll be exploring are closely related to paradox, this is not a class about paradoxes. It is a class about awe-inspiring issues at the intersection between philosophy and mathematics.</span><br /><br /><span style="font-family: book antiqua, palatino;">It is also worth keeping in mind that my aim is not to be comprehensive. It is to introduce you to some exceptionally beautiful ideas. I'll give you enough detail to put you in a position to understand the ideas themselves, rather than watered-down approximations, but not so much detail that it would dampen your enthusiasm. For learners who would like to go further, I offer a list of recommended readings.</span></p>
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<h3 class="hd hd-2">Video: Philosophy and Fruit</h3>
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<h2 class="hd hd-2 unit-title">Difficulty Level</h2>
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<p><span style="font-family: book antiqua, palatino;">Here are the levels of philosophical and mathematical demandingness of each lecture in the class:</span></p>
<p><span style="font-family: book antiqua, palatino;"><img src="/assets/courseware/v1/0678d1df5491cf59c11f2d3cdd25ab2b/asset-v1:MITx+24.118x+2T2020+type@asset+block/demandingness.png" alt="Levels of philosophical and mathematical demandingness corresponding to each lecture.." type="saveimage" target="[object Object]" preventdefault="function(){r.isDefaultPrevented=n}" stoppropagation="function(){r.isPropagationStopped=n}" stopimmediatepropagation="function(){r.isImmediatePropagationStopped=n}" isdefaultprevented="function t(){return!1}" ispropagationstopped="function t(){return!1}" isimmediatepropagationstopped="function t(){return!1}" height="509" width="862" /></span></p>
<p><span style="font-family: book antiqua, palatino;"> On the <strong>philosophical</strong> side, a demandingness level of 100% means that some of the ideas we'll be discussing are rather subtle; you won't need philosophical training to understand them, but you'll have to think about them very carefully.</span></p>
<p><span style="font-family: book antiqua, palatino;">On the <strong>mathematical</strong> side, a demandingness level of 100% means that the lecture is designed for someone who is familiar with college-level mathematics, or is otherwise experienced in mathematical proof.</span></p>
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