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<h2 class="hd hd-2 unit-title">Omega-Sequence Paradoxes</h2>
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<p><span style="font-family: 'book antiqua', palatino;">The paradoxes we’ll be talking in this lecture all involve "omega-sequences" or "reverse omega-sequences". </span></p>
<p><span style="font-family: 'book antiqua', palatino;">An <strong>omega-sequnece</strong> (or <span class="math inline">\(\omega\)</span>-sequence) is a sequence of items which is ordered like the natural numbers <span class="math inline">\(\langle 0,1,2,3,4,\dots\rangle \)</span>, under the natural ordering. In other words: it is an ordering with shape "<span class="math inline">\(| | | | | \dots\)"</span>. An <span class="math inline">\(\omega\)</span>-sequence has a first member, and a second member, and a third member, and so forth—but no last member. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">A <strong>reverse omega-sequence</strong> (or reverse <span class="math inline">\(\omega\)</span>-sequence) is a sequence of items which is ordered like the natural numbers, under a <em>reverse</em> ordering: <span class="math inline">\(\dots 4,3,2,1,0\)</span>. In other words: it is an ordering with shape, "<span class="math inline">\(\dots | | | | |\)"</span>. A reverse <span class="math inline">\(\omega\)</span>-sequence has a last member, and a next-to-last member, and a next-to-next-to last member, and so forth—but no first member.</span></p>
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<h3 class="hd hd-2">Video Review: Introducing Omega Sequences</h3>
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<h2 class="hd hd-2 unit-title">Zeno's Paradox</h2>
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<p><span style="font-size: 12pt;"><span style="font-size: 12pt;"> <img src="FiguresDiagramsParadoxInfinity-05final.png" alt="" 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}" /><img src="FiguresDiagramsParadoxInfinity-05final.png" alt="" 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}" /><span style="font-family: 'book antiqua', palatino;">The following is a variant of one of the paradoxes attributed to ancient philosopher Zeno of Elea, who lived in the 5th Century BC.</span></span></span></p>
<blockquote>
<p><span style="font-family: 'book antiqua', palatino;">You wish to walk from point <span class="math inline">\(A\)</span> to point <span class="math inline">\(B\)</span>:</span></p>
<p><span style="font-family: 'book antiqua', palatino;"><img src="https://studio.edx.org/asset-v1:MITx+24.118x+1T2018+type@asset+block@FiguresDiagramsParadoxInfinity-5.png" width="784" height="321" type="saveimage" target="[object Object]" /></span></p>
<p><span style="font-family: 'book antiqua', palatino;">In order to do so, you must carry out an <span class="math inline">\(\omega\)</span>-sequence of tasks:</span></p>
<p><span style="font-family: 'book antiqua', palatino;"><img src="https://studio.edx.org/asset-v1:MITx+24.118x+1T2018+type@asset+block@FiguresDiagramsParadoxInfinity-5FINAL.png" width="776" height="388" type="saveimage" target="[object Object]" /></span></p>
<p><span style="font-family: 'book antiqua', palatino;">But nobody can complete infinitely many tasks in a finite amount of time. So it is impossible to get from point <span class="math inline">\(A\)</span> to point <span class="math inline">\(B\)</span>. More generally: movement is impossible.</span></p>
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<h2 class="hd hd-2 unit-title">Assessment: Zeno's Paradox</h2>
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<h3>Zeno’s Paradox<br />Paradox Grade: 2</h3>
<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;"> It seems to me that from a contemporary perspective, Zeno’s paradox deserves a lowish paradoxicality grade. This is because nowadays the mathematical notion of a limit is commonplace. And limits can be used to show that—contrary to what is claimed above—it is possible to complete infinitely many tasks in a finite amount of time, if the time required to complete successive tasks deceases quickly enough. </span></p>
<p><span style="font-family: 'book antiqua', palatino;"><span style="font-size: 12pt;">Here is an example. Tortoises of the genus <em>Gopherus</em> have been clocked walking at speeds of about </span>[mathjaxinline]0.1 m/s[/mathjaxinline]. Suppose the distance from point [mathjaxinline]A[/mathjaxinline] to point [mathjaxinline]B[/mathjaxinline] is [mathjaxinline]100m[/mathjaxinline]. This is how long it would take Marty the Tortoise to complete each of Zeno’s infinitely many tasks:</span></p>
<p><span style="font-family: 'book antiqua', palatino;"><img src="https://studio.edx.org/asset-v1:MITx+24.118x+1T2018+type@asset+block@FiguresDiagramsParadoxInfinity-5final2.png" width="775" height="403" type="saveimage" target="[object Object]" /></span></p>
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<p><span style="font-family: 'book antiqua', palatino;">But by using the mathematical notion of a limit, one can show that</span></p>
<center><span style="font-family: 'book antiqua', palatino;">[mathjaxinline]500 + 250 + 125 + \ldots + \frac{1000}{2^n} + \ldots = 1000[/mathjaxinline]</span></center>
<p><span style="font-family: 'book antiqua', palatino;">So even though Marty must complete infinitely many tasks to get from point [mathjaxinline]A[/mathjaxinline] to point [mathjaxinline]B[/mathjaxinline], he should be able to do so in 1000[mathjaxinline]s[/mathjaxinline].</span></p>
<p><span style="font-family: 'book antiqua', palatino;">I certainly don’t want to suggest that Zeno was a fool—he was not. For someone in Zeno’s position, the paradox deserves a grade much higher than 2.</span></p>
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<span style="font-family: 'book antiqua', palatino;">Is the following true or false?</span>
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<span class="math inline">\(\lim_{n \to \infty}\left(500 + 250 + 125 + \ldots + \dfrac{1000}{2^n} \right) = 1000\)</span>
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<span style="font-family: 'book antiqua', palatino;">Suppose Marty slows down as he approaches the 100m mark: <span class="math display">\[\begin{array}{ccc}
\text{Task number} &amp; \text{Task description} &amp; \text{ Speed}\\
\text{ Task 1:} &amp; \text{ travel $50m$ to reach $50m$ mark} &amp; \text{$\frac{1}{10 \cdot 2^1}m/s$}\\
\text{ Task 2:} &amp; \text{ travel $25m$ to reach $75m$ mark} &amp; \text{ $\frac{1}{10\cdot 2^2}m/s$} \\
\text{ Task 3:} &amp; \text{ travel $12.5m$ to reach $87.5m$ mark} &amp; \text{ $\frac{1}{10\cdot 2^3}m/s$} \\
\text{ $\vdots$} &amp; \text{ $\vdots$} &amp; \text{ $\vdots$}\\
\text{ Task $n$:} &amp; \text{ travel $\frac{100m}{2^n}$ to reach $\frac{(2^n-1)100m}{2^n}$ mark}&amp; \text{ \(\frac{1}{10 \cdot 2^n} m/s\)} \\
\text{$\vdots$} &amp; \text{$\vdots$}&amp; \text{$\vdots$}\\
\end{array}\]</span> On these revised assumptions, is it possible for Marty to make it to the 100m mark in a finite amount of time?</span>
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<h2 class="hd hd-2 unit-title">Thomson's Lamp</h2>
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<p><span style="font-family: 'book antiqua', palatino;">I would like to consider another example of running through an <span class="math inline">\(\omega\)</span>-sequence in finite time with seemingly paradoxical results. It was devised by the late James Thomson, who was a professor of philosophy at MIT (and was married to the great philosopher Judith Jarvis Thomson).</span></p>
<p><span style="font-family: 'book antiqua', palatino;">Suppose you have a lamp with a toggle button: press the button once and the lamp goes on, press it again and the lamp goes off. It is one minute to midnight and the lamp is off; at 30 seconds before midnight, you press the button; at <span class="math inline">\(15\)</span> seconds before midnight, you press it again; at <span class="math inline">\(7.5\)</span> seconds before midnight, you press it again; and so forth:</span></p>
<center>
<p><span class="math display" style="font-family: 'book antiqua', palatino;">\[\begin{array}{cc} \text{Time to midnight} & \text{Status of lamp shortly thereafter} \\ \text{60$s$} & \text{off} \\ \text{30$s$} & \text{on}\\ \text{15$s$} & \text{off}\\ \text{7.5$s$} & \text{on}\\ \text{$\vdots$} & \text{$\vdots$}\\ \text{$\dfrac{60}{2^{2n}}s$} & \text{off}\\ \text{$\dfrac{60}{2^{2n+1}}s$} & \text{on}\\ \text{$\vdots$} & \text{$\vdots$}\\ \end{array}\]</span></p>
</center>
<p><span style="font-family: 'book antiqua', palatino;"><em>Question:</em> At midnight, is the lamp on or off? Here are two arguments:</span></p>
<p style="padding-left: 30px;"><span style="font-family: 'book antiqua', palatino;"><strong>Argument 1</strong></span><br /><span style="font-family: 'book antiqua', palatino;">For every time the lamp gets turned off before midnight, there is a later time before midnight when it gets turned on. So the lamp can’t be off at midnight.</span></p>
<p style="padding-left: 30px;"><span style="font-family: 'book antiqua', palatino;"><strong>Argument 2</strong></span><br /><span style="font-family: 'book antiqua', palatino;">For every time the lamp gets turned on before midnight, there is a later time before midnight when it gets turned off. So the lamp can’t be on at midnight.</span></p>
<p><span style="font-family: 'book antiqua', palatino;">If both these arguments are right, the lamp is not on at midnight and not off at midnight. What would that even mean?</span></p>
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<h4><span style="font-family: 'book antiqua', palatino;">Logical Impossibility</span></h4>
<p><span style="font-family: 'book antiqua', palatino;">It is natural to answer the paradox by arguing that there could be no such thing as a lamp that behaves in the way the argument demands. But I don’t think such a response would take us to the heart of the matter.</span></p>
<p><span style="font-family: 'book antiqua', palatino;">It is certainly true that we are not in a position to carry out the experiment in practice. To use a phrase of Bertrand Russell’s, it is "medically impossible" for a human to carry out each of the infinitely many required tasks. The experiment is also barred by the laws of physics. For example, the switch would eventually have to travel faster than the speed of light. If this is all we had to say about the paradox, however, we would fail to appreciate what is most interesting about it. For even if the relevant setup is not physically possible, Thomson’s Lamp can teach us something interesting about the limits of <em>logical</em> possibility.</span></p>
<p><span style="font-family: 'book antiqua', palatino;">For a hypothesis to be <strong>logically impossible</strong> is for it to contain an absurdity. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">Consider, for example, the hypothesis that Smith raises her arm without moving it. This hypothesis contains an absurdity, because part of what it is to raise an arm is to move it. In contrast, the hypothesis that a button travels faster than the speed of light is not logically impossible. It involves a violation of the laws of physics, but there is no <em>absurdity</em> in hypothesizing different physical laws. It is for this reason that we can make sense of a world in which Newtonian physics is true (even though it’s actually false), and we can make sense of science fiction about faster-than-light travel (even though it’s actually barred by the laws of physics). In contrast, there is no sense to be made of a story in which Smith raises her arm without moving it. The hypothesis that someone raises an arm without moving it isn’t just contrary to physical law: it is absurd.</span></p>
<p><span style="font-family: 'book antiqua', palatino;">When we assess the paradoxes in this lecture, we will be interested in the limits of logical possibility. We will try to figure out whether the various scenarios we will be describing involve a genuine absurdity—and we will try to understand why the absurdity arises when it does. The more interesting the paradox, the more we stand to learn about the subtle absurdities lurking in our understanding of its premises.</span></p>
<p><span style="font-family: 'book antiqua', palatino;">Thomson’s Lamp is interesting because it seemingly derives an absurdity—that the lamp is neither on nor off at midnight—from seemingly innocuous premises. To defuse the paradox, we need to understand whether the absurdity is genuine; and if it is genuine we need to understand how it arises, by identifying a flaw in our reasoning, or a false premise.</span></p>
<p><span style="font-family: 'book antiqua', palatino;">I think the absurdity is genuine: there is no sense to be made of the hypothesis that it is neither on nor off at midnight (at least on the assumption that the lamp exists at midnight). </span></p>
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<h2 class="hd hd-2 unit-title">Assessment: Thomson's Lamp</h2>
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<h3><span style="font-family: 'book antiqua', palatino;">Thomson’s Lamp</span><br /><span style="font-family: 'book antiqua', palatino;">Paradox Grade: 3</span></h3>
<p><span style="font-family: 'book antiqua', palatino;">I would like to suggest that Arguments 1 and 2 are both invalid:</span></p>
<p style="padding-left: 30px;"><span style="font-family: 'book antiqua', palatino;"><strong>Argument 1</strong></span><br /><span style="font-family: 'book antiqua', palatino;">For every time the lamp gets turned off before midnight, there is a later time before midnight when it gets turned on. So the lamp can’t be off at midnight.</span></p>
<p style="padding-left: 30px;"><span style="font-family: 'book antiqua', palatino;"><strong>Argument 2</strong></span><br /><span style="font-family: 'book antiqua', palatino;">For every time the lamp gets turned on before midnight, there is a later time before midnight when it gets turned off. So the lamp can’t be on at midnight.</span></p>
<p></p>
<p></p>
<p><span style="font-family: 'book antiqua', palatino;">I certainly agree that the setup of the case entails each of the following:</span></p>
<ul>
<li>
<p><span style="font-family: 'book antiqua', palatino;">For every time before midnight the lamp is on, there is a later time before midnight that the lamp is off.</span></p>
</li>
<li>
<p><span style="font-family: 'book antiqua', palatino;">For every time before midnight the lamp is off, there is a later time before midnight that the lamp is on.</span></p>
</li>
</ul>
<p><span style="font-family: 'book antiqua', palatino;">But neither of these claims addresses the question of what happens at midnight. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">The sequence of button-pushings constrains what the Thomson Lamp must be like at every moment before midnight. But it does not give us any information about what the status of the lamp at midnight. The story is consistent with the lamp’s being on or off at midnight—or with there being no lamp at all.</span></p>
<p><span style="font-family: 'book antiqua', palatino;">What makes the Thompson Lamp case strange is not that there is no consistent way of specifying what happens at midnight, but rather that there is no way of specifying what happens at midnight <em>without introducing a discontinuity</em>. Let me explain.</span></p>
<p><span style="font-family: 'book antiqua', palatino;">We tend to think of the world as a place in which macroscopic change is always <strong>continuous</strong>. In other words: if a macroscopic object is in state <span class="math inline">\(s\)</span> at a given time <span class="math inline">\(t\)</span>, then it must be in states that are arbitrarily similar to <span class="math inline">\(s\)</span> at times sufficiently close to <span class="math inline">\(t\)</span>. For instance, if you walk to the grocery store and reach your destination at noon, then you must be arbitrarily close to the grocery store at times sufficiently close to noon. (Things work differently at quantum scales.)</span></p>
<p><span style="font-family: 'book antiqua', palatino;">The problem with Thomson’s Lamp is that you get a discontinuity both by assuming that the lamp is on at midnight and by assuming that the lamp is off at midnight. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">So you can’t consistently assume that Thomson’s Lamp exists, if you assume that there are no discontinuities.</span></p>
<p><span style="font-family: 'book antiqua', palatino;">Argument 1 and Argument 2 above both tacitly assume that there are no discontinuities. So even if they constitute valid forms of reasoning in a world without discontinuity, they cannot be used to assess the question of whether a Thomson Lamp is logically consistent. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">What I think is interesting about Thomson’s Lamp, and the reason I think it deserves a grade of 3, is that it shows that a situation can be logically consistent even if it is inconsistent with deeply held assumptions about the way the physical world works—assumptions such as continuity.</span></p>
<p><span style="font-family: 'book antiqua', palatino;">Princeton University philosopher Paul Benacerraf once suggested a clever modification of the case that does not require discontinuities. Suppose that each time you push the button, the lamp shrinks to half of its previous size. Then, at midnight, there will be no lamp (because there can be no lamp of size zero). So, at midnight, it’s not the case that the lamp is on and it’s not the case that the lamp is off!</span></p>
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<h3 class="hd hd-2">Video Review: Thomson's Lamp</h3>
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<span style="font-family: 'book antiqua', palatino;">Suppose the lamp&#8217;s button must travel a fixed distance <span class="math inline">\(d\)</span> each time it is pressed. Then for each natural number <span class="math inline">\(n\)</span> the lamp&#8217;s button must eventually travel faster than <span class="math inline">\(n\)</span> meters per second. For, as we approach midnight, the time available for the button to traverse <span class="math inline">\(d\)</span> will approach zero.</span>
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<span style="font-family: 'book antiqua', palatino;">Is there some point before midnight at which the button must travel infinitely fast?</span>
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<h2 class="hd hd-2 unit-title">Another Version of Thomson's Lamp</h2>
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<h3 class="hd hd-2">Video: A Reverse Version of Thomson's Lamp</h3>
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