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<h2 class="hd hd-2 unit-title">Homework</h2>
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<p>The exercises below will count towards your grade. <strong>You have only one chance to answer these questions. </strong>Take your time, and think carefully before answering.</p>
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Problem 1
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><p>What do the following sentences of \(L\) say?</p>
<p>\(\neg\)\(\forall\)\(x\)(\(x=3\))</p>
<div class="choicegroup capa_inputtype" id="inputtype_3e1c24d124dc4f1bb408eaffcac5a339_2_1">
<fieldset aria-describedby="status_3e1c24d124dc4f1bb408eaffcac5a339_2_1">
<div class="field">
<input type="radio" name="input_3e1c24d124dc4f1bb408eaffcac5a339_2_1" id="input_3e1c24d124dc4f1bb408eaffcac5a339_2_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="3e1c24d124dc4f1bb408eaffcac5a339_2_1-choice_0-label" for="input_3e1c24d124dc4f1bb408eaffcac5a339_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_3e1c24d124dc4f1bb408eaffcac5a339_2_1"> It is not the case that there is something that is identical to 3.
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<input type="radio" name="input_3e1c24d124dc4f1bb408eaffcac5a339_2_1" id="input_3e1c24d124dc4f1bb408eaffcac5a339_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="3e1c24d124dc4f1bb408eaffcac5a339_2_1-choice_1-label" for="input_3e1c24d124dc4f1bb408eaffcac5a339_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_3e1c24d124dc4f1bb408eaffcac5a339_2_1"> There is something that is not identical to 3.
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<input type="radio" name="input_3e1c24d124dc4f1bb408eaffcac5a339_2_1" id="input_3e1c24d124dc4f1bb408eaffcac5a339_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="3e1c24d124dc4f1bb408eaffcac5a339_2_1-choice_2-label" for="input_3e1c24d124dc4f1bb408eaffcac5a339_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_3e1c24d124dc4f1bb408eaffcac5a339_2_1"> There is something that is identical to 3.
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<input type="radio" name="input_3e1c24d124dc4f1bb408eaffcac5a339_2_1" id="input_3e1c24d124dc4f1bb408eaffcac5a339_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="3e1c24d124dc4f1bb408eaffcac5a339_2_1-choice_3-label" for="input_3e1c24d124dc4f1bb408eaffcac5a339_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_3e1c24d124dc4f1bb408eaffcac5a339_2_1"> Nothing is identical to 3.
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<span id="answer_3e1c24d124dc4f1bb408eaffcac5a339_2_1"/>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><p><br/>\(\forall\)\(x\) \(\exists y\) (\(y+1=x\))</p>
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<input type="radio" name="input_3e1c24d124dc4f1bb408eaffcac5a339_3_1" id="input_3e1c24d124dc4f1bb408eaffcac5a339_3_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="3e1c24d124dc4f1bb408eaffcac5a339_3_1-choice_0-label" for="input_3e1c24d124dc4f1bb408eaffcac5a339_3_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_3e1c24d124dc4f1bb408eaffcac5a339_3_1"> Every number plus 1 equals another number. (That is, every number has a successor.)
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<input type="radio" name="input_3e1c24d124dc4f1bb408eaffcac5a339_3_1" id="input_3e1c24d124dc4f1bb408eaffcac5a339_3_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="3e1c24d124dc4f1bb408eaffcac5a339_3_1-choice_1-label" for="input_3e1c24d124dc4f1bb408eaffcac5a339_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_3e1c24d124dc4f1bb408eaffcac5a339_3_1"> Every number is such that some number plus 1 equals it. (That is, every number has a predecessor.)
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<input type="radio" name="input_3e1c24d124dc4f1bb408eaffcac5a339_3_1" id="input_3e1c24d124dc4f1bb408eaffcac5a339_3_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="3e1c24d124dc4f1bb408eaffcac5a339_3_1-choice_2-label" for="input_3e1c24d124dc4f1bb408eaffcac5a339_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_3e1c24d124dc4f1bb408eaffcac5a339_3_1"> Every number is unique.
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<input type="radio" name="input_3e1c24d124dc4f1bb408eaffcac5a339_3_1" id="input_3e1c24d124dc4f1bb408eaffcac5a339_3_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="3e1c24d124dc4f1bb408eaffcac5a339_3_1-choice_3-label" for="input_3e1c24d124dc4f1bb408eaffcac5a339_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_3e1c24d124dc4f1bb408eaffcac5a339_3_1"> Every number is such that 1 can be added to it.
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Problem 2
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<p>Is it possible to construct a Turing Machine \(M_1\) which runs forever outputting sentences of \(L\) in such a way that every arithmetical truth is eventually outputted by \(M_1\)?</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_370cd21bec4048918212b7a2a41891c4_2_1">
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<input type="radio" name="input_370cd21bec4048918212b7a2a41891c4_2_1" id="input_370cd21bec4048918212b7a2a41891c4_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="370cd21bec4048918212b7a2a41891c4_2_1-choice_1-label" for="input_370cd21bec4048918212b7a2a41891c4_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_370cd21bec4048918212b7a2a41891c4_2_1"> No
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</div><p><br/>Is it possible to construct a Turing Machine \(M_2\) which runs forever outputting sentences of \(L\) in such a way that no arithmetical falsehood is ever output by \(M_2\)?</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_370cd21bec4048918212b7a2a41891c4_3_1">
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<input type="radio" name="input_370cd21bec4048918212b7a2a41891c4_3_1" id="input_370cd21bec4048918212b7a2a41891c4_3_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="370cd21bec4048918212b7a2a41891c4_3_1-choice_0-label" for="input_370cd21bec4048918212b7a2a41891c4_3_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_370cd21bec4048918212b7a2a41891c4_3_1"> Yes
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<input type="radio" name="input_370cd21bec4048918212b7a2a41891c4_3_1" id="input_370cd21bec4048918212b7a2a41891c4_3_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="370cd21bec4048918212b7a2a41891c4_3_1-choice_1-label" for="input_370cd21bec4048918212b7a2a41891c4_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_370cd21bec4048918212b7a2a41891c4_3_1"> No
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</div><p><br/>Is it possible to construct a Turing Machine which satisfies both the conditions of \(M_1\) above and the conditions of \(M_2\) above?</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><div class="choicegroup capa_inputtype" id="inputtype_370cd21bec4048918212b7a2a41891c4_4_1">
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<h3 class="hd hd-3 problem-header" id="5cf77975b60142d6bfe620c6879097c8-problem-title" aria-describedby="block-v1:MITx+24.118x+2T2020+type@problem+block@5cf77975b60142d6bfe620c6879097c8-problem-progress" tabindex="-1">
Problem 3
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><p>Say that a sentence \(s\) of \(L\) is <i>provable</i> on the basis of a given axiomatization if there is a finite sequence of sentences of \(L\) with the following two properties:</p>
<p style="padding-left:60px;">\((1)\) the last member of the sequence is \(s\), and</p>
<p style="padding-left:60px;">\((2)\) every member of the sequence is either an axiom or something that results from previous members of the sequence by applying a rule of inference.</p>
<p>Now consider an axiomatization \(\mathfrak{A}\) that consists of the following axioms:</p>
<p style="padding-left:60px;">\( (A0) \; 0 = 0 \)</p>
<p style="padding-left:60px;">\( (A1) \; 1 = 1 \)</p>
<p style="padding-left:60px;">\( (A2) \; 2 = 2 \)</p>
<p style="padding-left:60px;">\( ... \)</p>
<p>and the following rules of inference:</p>
<p style="padding-left:60px;">\( (R1) \;\) You may infer "\( \phi\) &amp; \(\psi \)" if you have \(\phi\) and \(\psi\).</p>
<p style="padding-left:60px;">\( (R2) \;\) You may infer "\(\forall x (x=x))\)" if you have all of the following:</p>
<p style="padding-left:60px;">
\[0=0\]
\[1=1\]
\[2=2\]
\[\vdots\]</p>
<p>Now, is "\(0=0\)" provable on the basis of \(\mathfrak{A}\)?</p>
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<input type="radio" name="input_5cf77975b60142d6bfe620c6879097c8_2_1" id="input_5cf77975b60142d6bfe620c6879097c8_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="5cf77975b60142d6bfe620c6879097c8_2_1-choice_1-label" for="input_5cf77975b60142d6bfe620c6879097c8_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_5cf77975b60142d6bfe620c6879097c8_2_1"> No
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<span id="answer_5cf77975b60142d6bfe620c6879097c8_2_1"/>
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<span id="solution_5cf77975b60142d6bfe620c6879097c8_solution_1"/>
</div><div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><p><br/>Is "\( 1=1 \ \) &amp; \(\ 2=2 \ \) &amp; \( \ 7=7 \)" provable on the basis of \(\mathfrak{A}\)?</p>
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<input type="radio" name="input_5cf77975b60142d6bfe620c6879097c8_3_1" id="input_5cf77975b60142d6bfe620c6879097c8_3_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="5cf77975b60142d6bfe620c6879097c8_3_1-choice_0-label" for="input_5cf77975b60142d6bfe620c6879097c8_3_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_5cf77975b60142d6bfe620c6879097c8_3_1"> Yes
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<span id="solution_5cf77975b60142d6bfe620c6879097c8_solution_2"/>
</div><div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><p><br/>Is the following provable on the basis of \(\mathfrak{A}\)?</p>
<p style="padding-left:60px;"> \(0=0 \ \) &amp; \(\ 1=1 \ \) &amp; \(\ 2=2 \ \) &amp; \(\ 3=3\ \) &amp; \(\ ...\)</p>
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<input type="radio" name="input_5cf77975b60142d6bfe620c6879097c8_4_1" id="input_5cf77975b60142d6bfe620c6879097c8_4_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="5cf77975b60142d6bfe620c6879097c8_4_1-choice_1-label" for="input_5cf77975b60142d6bfe620c6879097c8_4_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_5cf77975b60142d6bfe620c6879097c8_4_1"> No
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<span id="solution_5cf77975b60142d6bfe620c6879097c8_solution_3"/>
</div><div class="wrapper-problem-response" tabindex="-1" aria-label="Question 4" role="group"><p><br/>Does rule \((R2)\) license inferring "\(\forall x (x = x)\)'' from the axioms of \(\mathfrak{A}\)?</p>
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<input type="radio" name="input_5cf77975b60142d6bfe620c6879097c8_5_1" id="input_5cf77975b60142d6bfe620c6879097c8_5_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="5cf77975b60142d6bfe620c6879097c8_5_1-choice_1-label" for="input_5cf77975b60142d6bfe620c6879097c8_5_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_5cf77975b60142d6bfe620c6879097c8_5_1"> No
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<span id="solution_5cf77975b60142d6bfe620c6879097c8_solution_4"/>
</div><div class="wrapper-problem-response" tabindex="-1" aria-label="Question 5" role="group"><p><br/>Is "\( \forall x (x=x) \)" provable on the basis of \(\mathfrak{A}\)?</p>
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<input type="radio" name="input_5cf77975b60142d6bfe620c6879097c8_6_1" id="input_5cf77975b60142d6bfe620c6879097c8_6_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="5cf77975b60142d6bfe620c6879097c8_6_1-choice_1-label" for="input_5cf77975b60142d6bfe620c6879097c8_6_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_5cf77975b60142d6bfe620c6879097c8_6_1"> No
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<h3 class="hd hd-3 problem-header" id="d276f9440fb244e1a93d08ffd28940db-problem-title" aria-describedby="block-v1:MITx+24.118x+2T2020+type@problem+block@d276f9440fb244e1a93d08ffd28940db-problem-progress" tabindex="-1">
Problem 4
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><p>Choose the sentence of \(L\) (plus abbreviations) that expresses <i>Goldbach's Conjecture</i>: the statement that every even number greater than \(2\) is the sum of two primes.</p>
<div class="choicegroup capa_inputtype" id="inputtype_d276f9440fb244e1a93d08ffd28940db_2_1">
<fieldset aria-describedby="status_d276f9440fb244e1a93d08ffd28940db_2_1">
<div class="field">
<input class="field-input input-radio" value="choice_0" id="input_d276f9440fb244e1a93d08ffd28940db_2_1_choice_0" type="radio" name="input_d276f9440fb244e1a93d08ffd28940db_2_1"/><label class="response-label field-label label-inline" id="d276f9440fb244e1a93d08ffd28940db_2_1-choice_0-label" for="input_d276f9440fb244e1a93d08ffd28940db_2_1_choice_0" aria-describedby="status_d276f9440fb244e1a93d08ffd28940db_2_1"> \( \forall x_0 ( ( \textrm{Even}(x_0) \) &amp; \( 2 &lt; x_0 ) \supset \forall x_1 \forall x_2 ( \textrm{Prime}(x_1) \) &amp; \(\textrm{Prime}(x_2)\) &amp; \( x_0 = x_1 + x_2 ) ) \)
</label>
</div>
<div class="field">
<input class="field-input input-radio" value="choice_1" id="input_d276f9440fb244e1a93d08ffd28940db_2_1_choice_1" type="radio" name="input_d276f9440fb244e1a93d08ffd28940db_2_1"/><label class="response-label field-label label-inline" id="d276f9440fb244e1a93d08ffd28940db_2_1-choice_1-label" for="input_d276f9440fb244e1a93d08ffd28940db_2_1_choice_1" aria-describedby="status_d276f9440fb244e1a93d08ffd28940db_2_1"> \( \forall x_0 ( ( \textrm{Even}(x_0) \) &amp; \( 2 &lt; x_0 ) \supset \exists x_1 \exists x_2 ( \textrm{Prime}(x_1) \) &amp; \( \textrm{Prime}(x_2) \) &amp; \( x_0 = x_1 + x_2 ) ) \)
</label>
</div>
<div class="field">
<input class="field-input input-radio" value="choice_2" id="input_d276f9440fb244e1a93d08ffd28940db_2_1_choice_2" type="radio" name="input_d276f9440fb244e1a93d08ffd28940db_2_1"/><label class="response-label field-label label-inline" id="d276f9440fb244e1a93d08ffd28940db_2_1-choice_2-label" for="input_d276f9440fb244e1a93d08ffd28940db_2_1_choice_2" aria-describedby="status_d276f9440fb244e1a93d08ffd28940db_2_1"> \( \forall x_0 ( ( \textrm{Even}(x_0) \) &amp; \( 2 &lt; x_0 ) \vee \exists x_1 \exists x_2 ( \textrm{Prime}(x_1) \) &amp; \(\textrm{Prime}(x_2)\) &amp; \( x_0 = x_1 + x_2 ) ) \)
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<span id="answer_d276f9440fb244e1a93d08ffd28940db_2_1"/>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><p>Choose the sentence of \(L\) (plus abbreviations) that expresses the (incorrect) claim that the natural numbers are dense: between any two natural numbers there is a third.</p>
<div class="choicegroup capa_inputtype" id="inputtype_d276f9440fb244e1a93d08ffd28940db_3_1">
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<input class="field-input input-radio" value="choice_0" id="input_d276f9440fb244e1a93d08ffd28940db_3_1_choice_0" type="radio" name="input_d276f9440fb244e1a93d08ffd28940db_3_1"/><label class="response-label field-label label-inline" id="d276f9440fb244e1a93d08ffd28940db_3_1-choice_0-label" for="input_d276f9440fb244e1a93d08ffd28940db_3_1_choice_0" aria-describedby="status_d276f9440fb244e1a93d08ffd28940db_3_1"> \( \exists x_0 \exists x_1 \forall x_2 ( ( x_0 &lt; x_2 \) &amp; \( x_2 &lt; x_1) \vee (x_1 &lt; x_2 \) &amp; \( x_2 &lt; x_0)) \)
</label>
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<input class="field-input input-radio" value="choice_1" id="input_d276f9440fb244e1a93d08ffd28940db_3_1_choice_1" type="radio" name="input_d276f9440fb244e1a93d08ffd28940db_3_1"/><label class="response-label field-label label-inline" id="d276f9440fb244e1a93d08ffd28940db_3_1-choice_1-label" for="input_d276f9440fb244e1a93d08ffd28940db_3_1_choice_1" aria-describedby="status_d276f9440fb244e1a93d08ffd28940db_3_1"> \( \forall x_0 \forall x_1 \exists x_2 ( ( x_0 &lt; x_2 \) &amp; \( x_2 &lt; x_1) \wedge (x_1 &lt; x_2 \) &amp; \( x_2 &lt; x_0)) \)
</label>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><p>Choose the sentence of \(L\) (plus abbreviations) that expresses Euclid's Theorem: there are infinitely many primes.</p>
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<input class="field-input input-radio" value="choice_0" id="input_d276f9440fb244e1a93d08ffd28940db_4_1_choice_0" type="radio" name="input_d276f9440fb244e1a93d08ffd28940db_4_1"/><label class="response-label field-label label-inline" id="d276f9440fb244e1a93d08ffd28940db_4_1-choice_0-label" for="input_d276f9440fb244e1a93d08ffd28940db_4_1_choice_0" aria-describedby="status_d276f9440fb244e1a93d08ffd28940db_4_1"> \( \exists x_0 ( \textrm{Prime}(x_0)\) &amp; \(\exists x_1 (\textrm{Prime}(x_1) \) &amp; \( x_0 &lt; x_1)) \)
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<input class="field-input input-radio" value="choice_2" id="input_d276f9440fb244e1a93d08ffd28940db_4_1_choice_2" type="radio" name="input_d276f9440fb244e1a93d08ffd28940db_4_1"/><label class="response-label field-label label-inline" id="d276f9440fb244e1a93d08ffd28940db_4_1-choice_2-label" for="input_d276f9440fb244e1a93d08ffd28940db_4_1_choice_2" aria-describedby="status_d276f9440fb244e1a93d08ffd28940db_4_1"> \( \forall x_0 ( \textrm{Prime}(x_0) \supset \forall x_1 (\textrm{Prime}(x_1) \) &amp; \( x_0 &lt; x_1)) \)
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