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<h2 class="hd hd-2 unit-title">Paradox in Paradise</h2>
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<p><span style="font-family: book antiqua, palatino;">We have been considering the wonderful word of bigger and bigger ordinals -- "the paradise that Cantor has created for us'', as mathematician David Hilbert famously put it.</span></p>
<p><span style="font-family: book antiqua, palatino;">But Cantor's paradise has a terrible instability looming at its core: one can prove that there can be no such thing as the set of all ordinals.</span></p>
<p><span style="font-family: book antiqua, palatino;">More specifically: one can prove that if the set of all ordinals existed, it would have inconsistent properties: it would have to both be an ordinal and not be an ordinal. This result is known as the <strong>Burali-Forti Paradox</strong>, because the Italian mathematician Cesare Burali-Forti was one of the first people to realize that the order-type of all ordinals is problematic. </span><br /><br /><span style="font-family: book antiqua, palatino;">When I introduced the ordinals informally, I cautioned that the <a href="/courses/course-v1:MITx+24.118x+2T2020/jump_to_id/2af5205362d04157b36f5ae90f4aedbb" target="[object Object]">Open-Endedness Principle</a> was to be understood as entailing that there is no such thing as "all'' stages of the construction process. I did so in order to avoid commitment to the existence of a definite totality of "all'' ordinals, and thereby avoid the Burali-Forti Paradox.</span><br /><br /><span style="font-family: book antiqua, palatino;">Here is an informal version of the basic result. Suppose, for <em>reductio</em>, that \(\Omega\) is the set of all ordinals. Then:</span></p>
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<li><span style="font-family: book antiqua, palatino;">Since \(\Omega\) consists of every ordinal, it consists of every ordinal that's been introduced so far. But by the <a href="/courses/course-v1:MITx+24.118x+2T2020/jump_to_id/2af5205362d04157b36f5ae90f4aedbb" target="[object Object]">Construction Principle</a>, one introduces a new ordinal by forming the set consisting of every ordinal that's been introduced so far. So:<strong> \(\Omega\) is an ordinal</strong>.</span></li>
<li><span style="font-family: book antiqua, palatino;">If \(\Omega\) was itself an ordinal, it would be a member of itself (and therefore have itself as a predecessor). But no ordinal can be its own predecessor. So: <strong>\(\Omega\) is not an ordinal</strong>.</span></li>
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<p><span style="font-family: book antiqua, palatino;">One can carry out a rigorous version of this argument using the official definition of ordinal, which I introduced <a href="/courses/course-v1:MITx+24.118x+2T2020/jump_to_id/50852705c81344c0bc56e314d95dcb83" target="[object Object]">above</a>. Specifically, one shows that \(\Omega\) is an ordinal by verifying that it is well-ordered and set-transitive, and one shows that \(\Omega\) is not an ordinal by appealing to a set-theoretic axiom that entails that no set is a member of itself.</span><br /><br /><span style="font-family: book antiqua, palatino;">One reason the Burali-Forti Paradox is important is that it constrains our understanding of sets. It is a natural thought that whenever it makes sense to speak of the \(F\)s, one can introduce the set of \(F\)s. But the Burali-Forti Paradox shows that there is at least one instance in which this cannot be the case: when the \(F\)s are the ordinals, there is no such thing as the set of \(F\).</span><span style="font-family: book antiqua, palatino;"></span></p>
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<h3 class="hd hd-2">Burali-Forti Paradox</h3>
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<h2 class="hd hd-2 unit-title">Set-Theoretic Paradoxes</h2>
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<p><span style="font-family: book antiqua, palatino;">The Burali-Forti Paradox is part of a broader family of set-theoretic paradoxes, leading to similar conclusions. The most famous of these paradoxes is <strong>Russell's Paradox</strong>, which shows that there is no such thing as the set of sets that are not members of themselves. </span></p>
<p style="padding-left: 30px;"><span style="font-family: book antiqua, palatino;">(<em>Proof:</em> suppose there was such as set \(R\). Is \(R\) a member of itself? If it is, it shouldn't be, since only sets that are not members of themselves should be members of \(R\). If it isn't, it should be, since every set that is not a member of itself should be a member of \(R\).)</span></p>
<p><span style="font-family: book antiqua, palatino;">Over the years, set-theorists have identified ways of working with sets while steering clear of the paradoxes. </span></p>
<p><span style="font-family: book antiqua, palatino;">Importantly, they have identified families of set-theoretic axioms that are strong enough to deliver interesting mathematics, but are designed to retain consistency by ensuring that one can never introduce the set of ordinals, or the set of sets that are not members of themselves. </span></p>
<p><span style="font-family: book antiqua, palatino;">One such family of axioms is known as ZFC, after Ernst Zermelo and Abraham Fraenkel. ("C'' is for the Axiom of Choice.) ZFC is what I referred to earlier when I spoke of the "standard'' set-theoretic axioms. Most set-theorists believe that ZFC is a consistent axiom system and that it successfully steers clear of the paradoxes. But, as we will see in Lecture 10, establishing the consistency of an axiom system is not as easy as one might have hoped. </span><br /><br /><span style="font-family: book antiqua, palatino;">Even if axiom systems like ZFC allow us to do set-theory without lapsing into paradox, they do not immediately deliver a diagnosis of the paradoxes. They do not, for example, tell us what it is about the ordinals that make it impossible to collect them into a set. A lot of interesting work has been done on the subject, but a proper discussion is beyond the scope of this text. (I recommend some readings at the end of this lecture.)</span><br /><br /><span style="font-family: book antiqua, palatino;">My own view, in a nutshell, is that when we speak of "the ordinals'', we do not succeed in singling out a definite totality of individuals. In other words: our concept of ordinal fails to settle the question of just how far the hierarchy of ordinals goes. And there are reasons of principle for such indeterminacy: any definite hypothesis about the length of the ordinal hierarchy must fail, since it could be used to characterize a further ordinal, and therefore an ordinal hierarchy of even greater length.</span></p>
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