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<h2 class="hd hd-2 unit-title">Introduction</h2>
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<h3 class="hd hd-2">Video: Introduction to Lecture 10</h3>
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<h4 class="hd hd-5">Transcripts</h4>
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Exercise
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<span style="font-family: 'book antiqua', palatino;">Show that the following formulations of G&#246;del's Theorem are equivalent:</span>
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<strong> G&#246;del's Incompleteness Theorem</strong>
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<span style="font-family: 'book antiqua', palatino;">Let \(L\) be a (rich enough) arithmetical language. Then no Turing Machine \(M\) is such that when run on an empty input: \(M\) runs forever, outputting sentences of \(L\), every true sentence of \(L\) is eventually output by \(M\), and no false sentence of \(L\) is ever output by \(M\).</span>
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<span style="font-family: 'book antiqua', palatino;">
<strong> G&#246;del's Incompleteness Theorem</strong>
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<br/>
<span style="font-family: 'book antiqua', palatino;">Let \(L\) be a (rich enough) arithmetical language. Then no Turing Machine \(M'\) is such that whenever it is given a sentence of \(L\) as input it outputs "1" if the sentence is true and "0'' if the sentence is false.</span>
<span style="font-family: 'book antiqua', palatino;">(In other words: <i>no Turing machine can decide the truth-value of the sentences of \(L\)</i>.)</span>
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<span style="font-family: 'book antiqua', palatino;">(Hint: you may assume that there is a Turing Machine that outputs every sentence of \(L\) and nothing else, and that every sentence in \(L\) has a negation that is also in \(L\).)</span>
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