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<h2 class="hd hd-2 unit-title">Introduction to Ordinals</h2>
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<p><span style="font-family: 'book antiqua', palatino;">We’ll build up to the notion of an ordinal in stages, by transitioning through each of the following notions: <span class="math display">\[\text{Ordering} \rightarrow \text{Total Ordering} \rightarrow \text{Well-Ordering} \rightarrow \text{Well-Order Type} \rightarrow \text{Ordinal}\]</span> In a nutshell (and taking the notions above in inverse order): an <strong>ordinal</strong> is a representative of a <strong>well-order type</strong>, which corresponds to a class of <strong>well-orderings</strong>, which are special kind of <strong>total ordering</strong>, which is a special kind of <strong>ordering</strong>. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">This may not make much sense to you now, but it will soon!</span></p>
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<h3 class="hd hd-2">Ordinals as a tool to keep track of infinite size</h3>
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<h2 class="hd hd-2 unit-title">Orderings</h2>
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<p><span style="font-family: 'book antiqua', palatino;">To order a set is to establish a relation of <em>precedence</em> between members of the set. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">When I order my family members by birth date, for example, I take family member <span class="math inline">\(a\)</span> to <em>precede</em> family member <span class="math inline">\(b\)</span> (in symbols: <span class="math inline">\(a < b\)</span>) if and only if <span class="math inline">\(a\)</span> was born before <span class="math inline">\(b\)</span>. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">In general, we will say that a precedence relation <span class="math inline">\(<\)</span> counts as an <strong>ordering</strong> of a set <span class="math inline">\(A\)</span> if it satisfies the following two conditions for any <span class="math inline">\(a,b,c\)</span> in <span class="math inline">\(A\)</span>:</span></p>
<p style="padding-left: 30px;"><span style="font-family: 'book antiqua', palatino;"><strong>Anti-Symmetry</strong></span><br /><span style="font-family: 'book antiqua', palatino;">If <span class="math inline">\(a<b\)</span>, then <span class="math inline">\(b \nless a\)</span>.</span></p>
<p style="padding-left: 30px;"><span style="font-family: 'book antiqua', palatino;"><strong>Transitivity</strong></span><br /><span style="font-family: 'book antiqua', palatino;">If <span class="math inline">\(a<b\)</span> and <span class="math inline">\(b<c\)</span>, then <span class="math inline">\(a<c\)</span>.</span></p>
<p><span style="font-family: 'book antiqua', palatino;">These conditions are to some extent implicit in our informal notion of precedence: I don’t precede those who precede me (Anti-Symmetry), and I am preceded by those who precede my predecessors (Transitivity).</span></p>
<p><span style="font-family: 'book antiqua', palatino;">Taken together, our two conditions rule out precedence loops. For example, if a group of people are sitting around a table, Anti-Symmetry and Transitivity rule out an ordering where everyone precedes the person to their left.</span></p>
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<span style="font-family: 'book antiqua', palatino;">Consider the following condition:</span>
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<strong>Anti-Reflexivity</strong>
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<span style="font-family: 'book antiqua', palatino;"><span class="math inline">\(a \nless a\)</span>.</span>
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<span style="font-family: 'book antiqua', palatino;">Does the Anti-Symmetry condition above entail Anti-Reflexivity?</span>
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<h2 class="hd hd-2 unit-title">Total Orderings</h2>
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<p><span style="font-family: 'book antiqua', palatino;">Even though Anti-Symmetry and Transitivity are non-trivial, they are relatively weak ordering conditions. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">For example, they are both satisfied by an "empty" precedence relation <span class="math inline">\(<\)</span>, according to which <span class="math inline">\(a < b\)</span> is never the case. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">They are also compatible with orderings that have tree-like structures. Consider, for example, the set of Queen Victoria’s descendants, ordered as follows: <span class="math inline">\(a\)</span> precedes <span class="math inline">\(b\)</span> if and only if <span class="math inline">\(a\)</span> is <span class="math inline">\(b\)'</span>s (direct-line) ancestor. This ordering can be represented as a tree, with Victoria at the base, each of her children branching off from the base, each of her grandchildren branching off from their parents nodes, and so forth.</span></p>
<p><span style="font-family: 'book antiqua', palatino;">A feature of tree-like orderings is that they allow for individuals who don’t bear the precedence relation to one another. This is true, for example, of any two of Victoria's children. Since neither is an ancestor of the other, neither of them precedes the other in our ancestry-based ordering. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">Here we will restrict our attention to orderings on which any two objects are such that one precedes the other. What this means, formally speaking, is that we will restrict our attention to total orderings. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">A <strong>total ordering</strong> <span class="math inline">\(<\)</span> on <span class="math inline">\(A\)</span> is an ordering that satisfies the following condition whenever <span class="math inline">\(a\)</span> and <span class="math inline">\(b\)</span> are distinct elements of <span class="math inline">\(A\)</span>:</span></p>
<p style="padding-left: 30px;"><span style="font-family: 'book antiqua', palatino;"><strong>Totality</strong></span><br /><span style="font-family: 'book antiqua', palatino;"><span class="math inline">\(a < b\)</span> or <span class="math inline">\(b < a\)</span>.</span></p>
<p><span style="font-family: 'book antiqua', palatino;">Here is an example of a total ordering on the set of Victoria’s descendants: <span class="math inline">\(a\)</span> precedes <span class="math inline">\(b\)</span> if and only if <span class="math inline">\(a\)</span> was born before <span class="math inline">\(b\)</span>. Since no two of Victoria’s descendants were born at exactly the same time, any two of them will be such that one of them precedes the other. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">Another example of a total ordering is the standard ordering of the integers, <span class="math inline">\(<_\mathbb{Z}\)</span>: <span class="math display">\[\ldots\ <_\mathbb{Z} -2 <_\mathbb{Z} -1 <_\mathbb{Z} 0 <_\mathbb{Z} 1 <_\mathbb{Z} 2 <_\mathbb{Z} 3 <_\mathbb{Z} \ldots\]</span> Integer <span class="math inline">\(a\)</span> precedes integer <span class="math inline">\(b\)</span> on this ordering if and only if <span class="math inline">\(b = a +n\)</span>, for <span class="math inline">\(n\)</span> a positive integer. Since any two integers are such that there is some positive difference between them, any two of them are such that one of them <span class="math inline">\(<_\mathbb{Z}\)</span>-precedes the other.</span></p>
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<h4><span style="font-family: book antiqua,palatino;">Notation</span></h4>
<p><span style="font-family: book antiqua,palatino;">Total orderings go by different names. In the videos I call them <em>linear orderings</em>. Sorry about that!</span></p>
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<h3 class="hd hd-2">Video Review: Total Orderings (i.e. Linear Orderings)</h3>
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Problem 1
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<span style="font-family: 'book antiqua', palatino;">Is the following ordering a total ordering?</span>
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<span style="font-family: 'book antiqua', palatino;">The integers <span class="math inline">\(\dots, -2, -1, 0, 1, 2,\dots\)</span>, under an "inverse" ordering <span class="math inline">\(&lt;^{-1}\)</span> such that <span class="math inline">\(a&lt;^{-1}b\)</span> if and only if <span class="math inline">\(b &lt;_\mathbb{Z} a\)</span>, where <span class="math inline">\(&lt;_\mathbb{Z}\)</span> is the standard ordering of the integers, as characterized in the main text.</span>
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<span style="font-family: 'book antiqua', palatino;">Yes. It is a total ordering.</span>
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<span style="font-family: 'book antiqua', palatino;">No. It is not a total ordering.</span>
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Problem 2
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<span style="font-family: 'book antiqua', palatino;">Is the following ordering a total ordering?</span>
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<p>
<span style="font-family: 'book antiqua', palatino;">The power set of <span class="math inline">\(\{0,1\}\)</span>, under an ordering <span class="math inline">\(\subset\)</span> such that <span class="math inline">\(a \subset b\)</span> if and only if <span class="math inline">\(a\)</span> is a proper subset of <span class="math inline">\(b\)</span>. (<span class="math inline">\(a\)</span> is a proper subset of <span class="math inline">\(b\)</span> if and only if <span class="math inline">\(a \neq b\)</span> and every element of <span class="math inline">\(a\)</span> is an element of <span class="math inline">\(b\)</span>.)</span>
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<span style="font-family: 'book antiqua', palatino;">Yes. It is a total ordering.</span>
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<span style="font-family: 'book antiqua', palatino;">No. It is not a total ordering.</span>
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Problem 3
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<span style="font-family: 'book antiqua', palatino;">Is the following ordering a total ordering?</span>
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<span style="font-family: 'book antiqua', palatino;">The real numbers in the interval <span class="math inline">\([0,1]\)</span>, under the standard ordering <span class="math inline">\(&lt;_\mathbb{R}\)</span>, which is such that <span class="math inline">\(a &lt;_\mathbb{R} b\)</span> if and only if <span class="math inline">\(b = r+a\)</span> for <span class="math inline">\(r\)</span> a positive real number.</span>
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<span style="font-family: 'book antiqua', palatino;">Yes. It is a total ordering.</span>
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<span style="font-family: 'book antiqua', palatino;">The set of countably infinite sequences <span class="math inline">\(\langle d_0,d_1,\ldots \rangle \)</span> (where each <span class="math inline">\(d_i\)</span> is a digit <span class="math inline">\(0,1,\ldots, 9\)</span>), under an ordering <span class="math inline">\(&lt;\)</span> such that <span class="math inline">\(\langle d^1_0,d^1_1,\ldots\rangle &lt; \langle d^2_0,d^2_1,\ldots\rangle \)</span> if and only if <span class="math inline">\(0.d^1_0d^1_1,\ldots &lt;_\mathbb{R} 0.d^2_0d^2_1,\ldots\)</span> (where <span class="math inline">\(&lt;_\mathbb{R}\)</span> is defined as in the previous question).</span>
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<span style="font-family: 'book antiqua', palatino;">No. It is not a total ordering.</span>
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Problem 5
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<span style="font-family: 'book antiqua', palatino;">Is the following ordering a total ordering?</span>
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<span style="font-family: 'book antiqua', palatino;">The set of soldiers, under an ordering <span class="math inline">\(&lt;\)</span> such that <span class="math inline">\(x&lt;y\)</span> if and only if <span class="math inline">\(x\)</span> outranks <span class="math inline">\(y\)</span>.</span>
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<h2 class="hd hd-2 unit-title">Well-Orderings</h2>
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<p><span style="font-family: 'book antiqua', palatino;">The standard ordering of the natural numbers, <span class="math inline">\(<_\mathbb{N}\)</span>, is a total ordering: <span class="math display">\[0 <_\mathbb{N} 1 <_\mathbb{N}, 2 <_\mathbb{N} \dots\]</span> But it is a total ordering with an important special feature: it is a <em>well-ordering</em>.</span></p>
<p><span style="font-family: 'book antiqua', palatino;"> What this means is that any non-empty set of natural numbers has a <span class="math inline">\(<_\mathbb{N}\)</span>-smallest element: an element that precedes all others according to <span class="math inline">\(<_\mathbb{N}\)</span>. (The set of prime numbers, for example, has 2 as its <span class="math inline">\(<_\mathbb{N}\)</span>-smallest element, and the set of perfect numbers has 6.)</span></p>
<p><span style="font-family: 'book antiqua', palatino;">Not every total ordering is a well-ordering. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">For example, the standard ordering of the integers, <span class="math inline">\(<_\mathbb{Z}\)</span>, is not a well-ordering, since there there are non-empty subsets of <span class="math inline">\(\mathbb{Z}\)</span> with no <span class="math inline">\(<_\mathbb{Z}\)</span>-smallest integer. One example of such a subset is <span class="math inline">\(\mathbb{Z}\)</span> itself: <span class="math display">\[\ldots\ <_\mathbb{Z} -2 <_\mathbb{Z} -1 <_\mathbb{Z} 0 <_\mathbb{Z} 1 <_\mathbb{Z} 2 <_\mathbb{Z} 3 <_\mathbb{Z} \ldots\]</span> The set <span class="math inline">\([0,1]\)</span> under its standard ordering, <span class="math inline">\(<_\mathbb{R}\)</span>, also fails to be well-ordered. For even though the entire set <span class="math inline">\([0,1]\)</span> has 0 as its <span class="math inline">\(<_\mathbb{R}\)</span>-smallest element, <span class="math inline">\([0,1]\)</span> has subsets with no <span class="math inline">\(<_\mathbb{R}\)</span>-smallest element. One example of such a subset is the set <span class="math inline">\((0,1] = [0,1] - \{0\}\)</span>.</span></p>
<p><span style="font-family: 'book antiqua', palatino;">Formally, we shall say that a set <span class="math inline">\(A\)</span> is <strong>well-ordered</strong> by <span class="math inline">\(<\)</span> if <span class="math inline">\(A\)</span> is totally ordered by <span class="math inline">\(<\)</span> and satisfies the following additional condition:</span></p>
<p style="padding-left: 30px;"><span style="font-family: 'book antiqua', palatino;"><strong>Well-Ordering</strong></span><br /><span style="font-family: 'book antiqua', palatino;">Every non-empty subset <span class="math inline">\(S\)</span> of <span class="math inline">\(A\)</span> has a <span class="math inline">\(<\)</span>-smallest member (that is, a member <span class="math inline">\(x\)</span> such that <span class="math inline">\(x < y\)</span> for every <span class="math inline">\(y\)</span> in <span class="math inline">\(S\)</span> other than <span class="math inline">\(x\)</span>).</span></p>
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<h3 class="hd hd-2">Video Review: Well-Orderings</h3>
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<span style="font-family: 'book antiqua', palatino;">Is the following ordering a well-ordering?</span>
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<p>
<span style="font-family: 'book antiqua', palatino;">The positive rational numbers, under the standard ordering <span class="math inline">\(&lt;_\mathbb{Q}\)</span>, which is such that <span class="math inline">\(a &lt;_\mathbb{Q} b\)</span> if and only if <span class="math inline">\(b = q+a\)</span> for <span class="math inline">\(q\)</span> a positive rational number.</span>
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<span style="font-family: 'book antiqua', palatino;">Yes. It is a well-ordering.</span>
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<span style="font-family: 'book antiqua', palatino;">No. It is not a well-ordering.</span>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><p>
<span style="font-family: 'book antiqua', palatino;">Is the following ordering a well-ordering?</span>
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<p>
<span style="font-family: 'book antiqua', palatino;">The natural numbers, under an unusual ordering, <span class="math inline">\(&lt;_\star\)</span>, which is just like the standard ordering, except that the order of 0 and 1 is reversed: <span class="math display">\[1&lt;_\star0&lt;_\star2&lt;_\star3&lt;_\star4 &lt;_\star5&lt;_\star \dots\]</span></span>
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<span style="font-family: 'book antiqua', palatino;">Yes. It is a well-ordering.</span>
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<span style="font-family: 'book antiqua', palatino;">No. It is not a well-ordering.</span>
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Problem 3
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<span style="font-family: 'book antiqua', palatino;">Is the following ordering a well-ordering?</span>
</p>
<blockquote>
<p>
<span style="font-family: 'book antiqua', palatino;">The natural numbers, under an unusual ordering, <span class="math inline">\(&lt;_0\)</span>, in which 0 is counted as bigger than every positive number but the remaining numbers are ordered in the standard way: <span class="math display">\[1&lt;_02&lt;_03&lt;_04 &lt;_0 \dots &lt;_0 0\]</span></span>
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<span style="font-family: 'book antiqua', palatino;">Yes. It is a well-ordering.</span>
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<span style="font-family: 'book antiqua', palatino;">No. It is not a well-ordering.</span>
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<span style="font-family: 'book antiqua', palatino;">Is the following ordering a well-ordering?</span>
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<span style="font-family: 'book antiqua', palatino;">A finite set of real numbers, under the standard ordering <span class="math inline">\(&lt;_\mathbb{R}\)</span>.</span>
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<h2 class="hd hd-2 unit-title">The Shapes of Well-Orderings</h2>
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<p><span style="font-family: 'book antiqua', palatino;">There are many different ways of well-ordering the natural numbers. There is, of course, the standard ordering, <span class="math inline">\(<_\mathbb{N}\)</span>. But one can also well-order the natural numbers using an ordering, <span class="math inline">\(<_\backsim\)</span>, which is like the standard ordering except that it reverses the position of each even number and its successor: <span class="math display">\[1<_\backsim 0<_\backsim 3 <_\backsim 2 <_\backsim 5 <_\backsim 4 <_\backsim \dots\]</span> Although <span class="math inline">\(<_\mathbb{N}\)</span> and <span class="math inline">\(<_\backsim\)</span> correspond to different ways of ordering the natural numbers, they have exactly the same structure. The way to see this is to think of each of the two orderings as consisting of a sequence of "positions", each of which is occupied by a particular number:</span></p>
<center><span style="font-family: 'book antiqua', palatino;"><img src="/assets/courseware/v1/e9f03f4953fb1d5ec1a4b2b15bf32d81/asset-v1:MITx+24.118x+2T2020+type@asset+block/OrderType1.png" type="saveimage" target="[object Object]" height="30%" width="30%" /></span></center>
<p><span style="font-family: 'book antiqua', palatino;">When we abstract away from the numbers that occupy the positions and consider only the positions themselves, we get the exact same structure in both cases:</span></p>
<center><span style="font-family: 'book antiqua', palatino;"><img src="/assets/courseware/v1/e9522d6e673985fdd7f767295f14833c/asset-v1:MITx+24.118x+2T2020+type@asset+block/OrderType2.png" alt="" type="saveimage" target="[object Object]" height="30%" width="30%" /></span></center>
<p><span style="font-family: 'book antiqua', palatino;">Accordingly, we can say that <span class="math inline">\(<_\mathbb{N}\)</span> and <span class="math inline">\(<_\backsim\)</span> are well-orderings with the same "shape". I will sometimes represent this shape schematically, by shrinking the width of squares, until they look like lines: <span class="math display">\[| | | | | | \dots\]</span></span></p>
<p><span style="font-family: 'book antiqua', palatino;">Formally, we say that the well-orderings <span class="math inline">\(<_1\)</span> and <span class="math inline">\(<_2\)</span> have the same structure—or the same "shape"—if they are <strong>isomorphic</strong> to one another, in the following sense:</span></p>
<blockquote>
<p><span style="font-family: 'book antiqua', palatino;">there is a bijection <span class="math inline">\(f\)</span> from the domain of <span class="math inline">\(<_1\)</span> to the domain of <span class="math inline">\(<_2\)</span> such that, for every <span class="math inline">\(x\)</span> and <span class="math inline">\(y\)</span> in the domain of <span class="math inline">\(<_1\)</span>, <span class="math inline">\(x<_1y\)</span> if and only if <span class="math inline">\(f(x)<_2f(y)\)</span></span></p>
</blockquote>
<p><span style="font-family: 'book antiqua', palatino;">Not all well-orderings of <span class="math inline">\(\mathbb{N}\)</span> are isomorphic to one another. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">To see this, consider the well-ordering <span class="math inline">\(<_0\)</span>, which figures as an exercise below. It is just like the standard ordering, except that it takes every positive number to precede zero: <span class="math display">\[1<_02<_03<_04<_0\ldots<_00\]</span> Note that <span class="math inline">\(<_0\)</span> is not isomorphic to <span class="math inline">\(<_\mathbb{N}\)</span>, since its shape is "<span class="math inline">\(| | | | | \ldots | \)"</span>, which is different from the shape of of <span class="math inline">\(<_\mathbb{N}\)</span>, "<span class="math inline">\(| | | | | \dots\)"</span>.</span></p>
<p><span style="font-family: 'book antiqua', palatino;">In this lecture we won’t be interested in individual well-orderings, like <span class="math inline">\(<_\mathbb{N}\)</span> or <span class="math inline">\(<_0\)</span>. Instead, we will be focusing on <em>types</em> of well-orderings, or <strong>well-order types</strong>. Intuitively, each well-order type corresponds to a particular shape of well-ordering. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">Formally, a well-order type is an isomorphism-class of well-orderings: a class that consists of a well-ordering, and every well-ordering it is isomorphic to. For instance, <span class="math inline">\(<_\mathbb{N}\)</span> and <span class="math inline">\(<_\backsim\)</span> fall under the same well-order type, because they are isomorphic to one another. But <span class="math inline">\(<_\mathbb{N}\)</span> and <span class="math inline">\(<_0\)</span> do not fall under the same well-order type, because they are not isomorphic to one another.</span></p>
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<span style="font-family: 'book antiqua', palatino;">Let <span class="math inline">\(&lt;_\mathbb{N}\)</span> be the standard ordering of the natural numbers.</span>
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<span style="font-family: 'book antiqua', palatino;">Of each of the following pairs of sets, determine whether they correspond to the same well-order type when ordered by <span class="math inline">\(&lt;_\mathbb{N}\)</span>:</span>
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<span class="math inline">\(\{7, 2, 13, 12\}\)</span>
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<span class="math inline">\(\{412, 708, 20081\}\)</span>
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<td>the set of natural numbers</td>
<td>the set of natural numbers greater than 17</td>
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<h2 class="hd hd-2 unit-title">Well-Ordering Finite and Infinite Sets</h2>
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<p><span style="font-family: 'book antiqua', palatino;">As long as a set has more than one element, it can be well-ordered in multiple ways. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">Notice, however, that the <em>well-orderings</em> of a <em>finite</em> set are always isomorphic to one another. For example, the finite set <span class="math inline">\(\{a,b,c\}\)</span> has six different well-orderings: <span class="math display">\[\begin{array}{ccc} a<b<c \ &\ b<a<c \ & \ c<a<b \\ a<c<b \ & b<c<a \ & \ c<b<a \\ \end{array}\]</span> But they all fall under a single well-order type, the type corresponding to the shape "<span class="math inline">\(| | |\)"</span>. This is true in general: every well-ordering of an <span class="math inline">\(n\)</span>-element finite set is isomorphic to any other.</span></p>
<p><span style="font-family: 'book antiqua', palatino;">In contrast, there are always non-isomorphic ways of well-ordering an infinite set. We saw an example of this above, when we noted that there are well-orderings of the set of natural numbers with each of following two shapes: <span class="math display">\[\begin{array}{c} {| | | | | \dots}\\ {| | | | | \dots |} \end{array}\]</span> There are also well-orderings of the set of natural numbers with each of the following shapes: <span class="math display">\[\begin{array}{c} {| | | | | \dots | |} \\ {| | | | | \dots | | |} \\ {| | | | | \dots | | | | } \\ \vdots \\ {| | | | | \dots | | | | | \dots} \\ {| | | | | \dots | | | | | \dots |} \\ {| | | | | \dots | | | | | \dots | |} \\ {| | | | | \dots | | | | | \dots | | |} \\ {\vdots}\\ {| | | | | \dots | | | | | \dots | | | | | \dots} \\ {| | | | | \dots | | | | | \dots | | | | | \dots |} \\ {\vdots} \end{array}\]</span> Note that even though each of these shapes corresponds to a different well-order type, the sets that are ordered by well-orderings of these types <em>all have the same cardinality</em>: they all have the cardinality of the natural numbers.</span></p>
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<span style="font-family: 'book antiqua', palatino;">Which of the following orderings of the natural numbers, if any, has the following well-ordering shape:</span>
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<span class="math inline">\(\begin{array}{c} {| | | | | \dots | | |} \end{array}\)</span>
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<span style="font-family: 'book antiqua', palatino;">Which of the following orderings of the natural numbers, if any, has the following well-ordering shape:</span>
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<h2 class="hd hd-2 unit-title">Ordinals</h2>
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<p><span style="font-family: book antiqua, palatino;">We've been doing our best to keep track of well-order types using diagrams of lines and dots. But this notation soon becomes unmanageable. A better way of keeping track of big well-order types is to use <strong>ordinals</strong>. </span><br /><br /><span style="font-family: book antiqua, palatino;">Ordinals are <em>sets</em> of a particular kind. It is useful to think of them as introduced in stages, in accordance with the following principles:</span></p>
<ul>
<li><span style="font-family: book antiqua, palatino;"><strong>Open-Endedness Principle</strong> However many stages have occurred, there is always a "next'' stage: a first stage after every stage considered so far.</span></li>
<li><span style="font-family: book antiqua, palatino;"><strong>Construction Principle</strong> At each stage, we introduce a new ordinal, namely: the set of all ordinals that have been introduced at previous stages.</span></li>
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<p><span style="font-family: book antiqua, palatino;">(We will interpret the Open-Endedness Principle as entailing that there is no such thing as "all'' stages, with the result that there is no such thing as "all'' ordinals. This is an important point, to which we will return later.)</span></p>
<p><span style="font-family: 'book antiqua', palatino;">Let me describe the first few steps of the process. (They are also summarized in below.) </span></p>
<p><span style="font-family: 'book antiqua', palatino;">The Open-Endedness Principle entails that there must be a first stage to the process (i.e. the "next" stage after nothing has happened). At this first stage, no ordinals have been introduced. So the Construction Principle tells us to introduce the set with no members: <span class="math inline">\(\{\}\)</span>. We refer to this set as the ordinal <span class="math inline">\(0\)</span>.</span></p>
<p><span style="font-family: 'book antiqua', palatino;">The Open-Endedness Principle then tells us that there must be a next stage to the process. At this next stage, <span class="math inline">\(0\)</span> has been introduced. So the Construction Principle tells us to introduce the set <span class="math inline">\(\{0\}\)</span>. We refer to this set as the ordinal <span class="math inline">\(0'\)</span> (read: "successor of <span class="math inline">\(0\)"</span>). Again, Open-Endedness Principle tells us that there must be a next stage to the process. At this next stage, <span class="math inline">\(0\)</span> and <span class="math inline">\(0'\)</span> have been introduced. So the Construction Principle tells us to introduce the set <span class="math inline">\(\{0,0'\}\)</span>. We refer to this set as the ordinal <span class="math inline">\(0''\)</span> (read "successor of successor of <span class="math inline">\(0\)"</span>).</span></p>
<p><span style="font-family: 'book antiqua', palatino;">\[\begin{array}{ccc} \text{Stage} & \text {Ordinal introduced} &\text{Name of ordinal}\\ \hline 0 & \{\} & 0\\ 1 & \{0\} & 0'\\ 2 & \{0,0'\} & 0''\\ 3 & \{0,0',0''\} & 0'''\\ \vdots & \vdots & \vdots\\ \infty & \{0,0',0'',0''',\dots\} & \omega\\ \infty + 1 & \{0,0',0'',0''',\dots,\omega\} & \omega'\\ \infty + 2 & \{0,0',0'',\dots,\omega, \omega'\} & \omega''\\ \vdots & \vdots & \vdots\\ \end{array}\]</span></p>
<p><span style="font-family: 'book antiqua', palatino;">We carry out this process once for each natural number, yielding the sets <span class="math inline">\(0,0',0'',\ldots\)</span>. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">But the Open-Endedness Principle tells us that there is a next stage after that. And the Construction Principle tells us to introduce the set of everything that’s been introduced so far at that next stage. So we introduce the set <span class="math inline">\(\{0,0',0'',\ldots\}\)</span>. We refer to this set as the ordinal <span class="math inline">\(\omega\)</span> (read: "omega"). </span></p>
<p><span style="font-family: 'book antiqua', palatino;">And the Open-Endedness Principle tells us to keep going. At the next stage of the process, we introduce the set <span class="math inline">\(\{0,0',0'',\ldots,\omega\}\)</span>. We refer to this set as the ordinal <span class="math inline">\(\omega'\)</span> (read "successor of <span class="math inline">\(\omega\)"</span>). </span></p>
<p><span style="font-family: 'book antiqua', palatino;">And so on.</span></p>
<p><span style="font-family: 'book antiqua', palatino;">As I described the first few levels of the hierarchy of ordinals, I introduced an important piece of notation: the successor operation symbol "<span class="math inline">\(\,'\,\)"</span>. This operation can be defined rigorously, as follows: <span class="math inline">\(\alpha' = \alpha \cup \{\alpha\}\)</span>. Notice, moreover, that the Construction Principle delivers ordinals of two different kinds, <em>successor ordinals</em> and <em>limit ordinals</em>:</span></p>
<dl><dd>
<p><span style="font-family: 'book antiqua', palatino;">A <strong>successor ordinal</strong> is an ordinal <span class="math inline">\(\alpha\)</span> such that <span class="math inline">\(\alpha = \beta'\)</span> for some <span class="math inline">\(\beta\)</span>.</span></p>
</dd><dd>
<p><span style="font-family: 'book antiqua', palatino;">A <strong>limit ordinal</strong> is an ordinal that is not a successor ordinal.</span></p>
</dd></dl>
<p><span style="font-family: 'book antiqua', palatino;"><span class="math inline">\(0\)</span> and <span class="math inline">\(\omega\)</span> are both limit ordinals, since neither of them can be reached by applying the successor operation to an ordinal; <span class="math inline">\(0''\)</span> and <span class="math inline">\(\omega'\)</span> are both successor ordinals. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">The distinction between successor ordinals and limit ordinals will be important in Lecture 2.4, when we return to the project of using ordinals to build sets of greater and greater cardinality.</span></p>
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<h2 class="hd hd-2 unit-title">Well-Ordering Ordinals</h2>
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<p><span style="font-family: 'book antiqua', palatino;">We have been considering a picture on which the ordinals are introduced in stages, and the stages never run out. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">What is the shape of the resulting hierarchy of stages? As you’ll be asked to verify below, the answer is that <em>the stages of the hierarchy are well-ordered</em>. More precisely:</span></p>
<p style="padding-left: 30px;"><span style="font-family: 'book antiqua', palatino;"><strong>Stage Well-Ordering</strong></span><br /><span style="font-family: 'book antiqua', palatino;">Every set of stages is well-ordered by the relation of occurring-earlier-than.</span></p>
<p><span style="font-family: 'book antiqua', palatino;">An immediate consequence of this result is that <em>every set of ordinals is well-ordered</em>. To see this, let us start by introducing a relation <span class="math inline">\(<_o\)</span> for ordering ordinals: <span class="math display">\[\alpha <_o \beta \leftrightarrow \alpha \in \beta\]</span> Since <span class="math inline">\(\alpha \in \beta\)</span> if and only if <span class="math inline">\(\alpha\)</span> was introduced at an earlier stage than <span class="math inline">\(\beta\)</span>, <span class="math inline">\(<_o\)</span> is, in fact, the relation of being-introduced-at-an-earlier-stage-than. Since every set of stages is well-ordered by the relation of occurring-earlier-than, this means that <em>every set of ordinals is well-ordered by <span class="math inline">\(<_o\)</span></em>. And since every ordinal is a set of ordinals, this yields:</span></p>
<p style="padding-left: 30px;"><span style="font-family: 'book antiqua', palatino;"><strong>Ordinal Well-Ordering</strong></span><br /><span style="font-family: 'book antiqua', palatino;">Every ordinal is well-ordered by <span class="math inline">\(<_o\)</span>.</span></p>
<p><span style="font-family: 'book antiqua', palatino;">Ordinal Well-Ordering is one of the key properties of orderings. One reason it is so important is that it allows us to use ordinals to represent well-order types. More specifically: each ordinal can be used as a representative for the well-order type that it itself instantiates under <span class="math inline">\(<_o\)</span>. For example: <span class="math display">\[\begin{array}{ccc} \text{Ordinal} & \text{Ordering under $<_o$} & \text{Well-order type represented}\\ \hline 0 & & \ \\ 0' &0 & | \\ 0'' & 0 <_o 0' & | | \\ 0''' & 0 <_o 0' <_o 0'' & | | | \\ \vdots & \vdots\\ \omega & 0 <_o 0' <_o 0'' \ldots & | | | \dots \\ \omega' & 0 <_o 0' <_o 0'' \ldots <_o \omega & | | | | | \dots | \\ \omega'' & 0 <_o 0' <_o 0'' \ldots <_o \omega <_o \omega' & | | | | | \dots | | \\ \vdots & \vdots & \vdots \\ \end{array}\]</span> Ordinals are such natural representatives for well-order types that I will sometimes blur the distinction between an ordinal and the well-order type it represents, by speaking of ordinals as if they were themselves well-order types.</span></p>
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<h2 class="hd hd-2 unit-title">The Official Definition of Ordinal</h2>
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<p><span style="font-family: 'book antiqua', palatino;">Every mathematically significant feature of the ordinals follows Ordinal Well-Ordering, together with the following principle:</span></p>
<p style="padding-left: 30px;"><span style="font-family: 'book antiqua', palatino;"><strong>Set-transitivity</strong></span><br /><span style="font-family: 'book antiqua', palatino;">Every element of an element of an ordinal <span class="math inline">\(\alpha\)</span> is an element of <span class="math inline">\(\alpha\)</span>.</span></p>
<p><span style="font-family: 'book antiqua', palatino;">As a result, set-theorists often define ordinals on the basis of Ordinal Well-<span style="color: #313131;">Ordering</span> and Set-Transitivity:</span></p>
<p style="padding-left: 30px;"><span style="font-family: 'book antiqua', palatino;"><strong>Official Definition</strong></span><br /><span style="font-family: 'book antiqua', palatino;">An ordinal is a set that is set-transitive and well-ordered by <span class="math inline">\(<_o\)</span>.</span></p>
<p><span style="font-family: 'book antiqua', palatino;">(This assumes that we restrict our attention to pure sets: sets such that all their members are sets, all the members of their members are sets, all the members of their members of their members are sets, and so forth.) </span></p>
<p><span style="font-family: 'book antiqua', palatino;">Unless you want to go deep into the material, though, you won’t need to worry about this definition. Just think of the ordinals as generated by the construction in Lecture 2.2.6.</span></p>
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<span style="font-family: 'book antiqua', palatino;">Justify Set-Transitivity on the basis of the intuitive picture of ordinals we developed in Lecture 2.2.6.</span>
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