model theory - completeness of a theory












0












$begingroup$



For an $L$-structure $mathcal{A}$, the $L$-theory of $mathcal{A}$ is the set of $L$-sentences:
$$
mathrm{Th}_L(mathcal A) = {sigma : mathcal A models sigma}
$$



Prove that $mathrm{Th}_L(mathcal A)$ is complete.




Whys is this true? Why can't there be an $L$-sentence such that it is neither true or false?






logic model-theory







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$endgroup$

















    0












    $begingroup$



    For an $L$-structure $mathcal{A}$, the $L$-theory of $mathcal{A}$ is the set of $L$-sentences:
    $$
    mathrm{Th}_L(mathcal A) = {sigma : mathcal A models sigma}
    $$



    Prove that $mathrm{Th}_L(mathcal A)$ is complete.




    Whys is this true? Why can't there be an $L$-sentence such that it is neither true or false?






    logic model-theory







    share|cite|improve this question









    $endgroup$















      0












      0








      0





      $begingroup$



      For an $L$-structure $mathcal{A}$, the $L$-theory of $mathcal{A}$ is the set of $L$-sentences:
      $$
      mathrm{Th}_L(mathcal A) = {sigma : mathcal A models sigma}
      $$



      Prove that $mathrm{Th}_L(mathcal A)$ is complete.




      Whys is this true? Why can't there be an $L$-sentence such that it is neither true or false?






      logic model-theory







      share|cite|improve this question









      $endgroup$





      For an $L$-structure $mathcal{A}$, the $L$-theory of $mathcal{A}$ is the set of $L$-sentences:
      $$
      mathrm{Th}_L(mathcal A) = {sigma : mathcal A models sigma}
      $$



      Prove that $mathrm{Th}_L(mathcal A)$ is complete.




      Whys is this true? Why can't there be an $L$-sentence such that it is neither true or false?






      logic model-theory




      logic model-theory






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Dec 8 '18 at 16:20









      bofbof

      152




      152






















          1 Answer
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          $begingroup$

          Assume there's an $L$-sentence $sigma$ that's neither true nor false. Then we have that $mathcal{A}nvDashsigma$ and $mathcal{A}nvDashnegsigma$. But this is a contradiction since $mathcal{A}vDashsigma$ if and only if $mathcal{A}nvDashnegsigma$, and vice versa, by definition of what a structure is.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            I'm sorry if this is a really basic question but why is it that you can construct such sentence that is neither true nor false under Godel's incompleteness theorem but not here? Thanks!
            $endgroup$
            – bof
            Dec 8 '18 at 16:26










          • $begingroup$
            For a detailed explanation, see here. But basically, Godel's incompleteness theorem is not a negation of Godel's completeness theorem. Also, here we are dealing with a structure, not a set of sentences (which is what Godel's incompleteness theorem talks about).
            $endgroup$
            – quanticbolt
            Dec 8 '18 at 16:39








          • 1




            $begingroup$
            @bof That's an incorrect statement of Godel's incompleteness theorem. There is a distinction between truth in a model/structure (fixing a structure $M$, everything is either true in $M$ or false in $M$) and provability from a theory (= truth in every model of that theory; since a given theory might have many disagreeing models, there are in general independent sentences, and this is what Godel's incompleteness theorem is about).
            $endgroup$
            – Noah Schweber
            Dec 8 '18 at 16:40










          • $begingroup$
            @bof For any sentence $varphi$ and any structure $A$ the definition of $Amodels lnot varphi$ is that $Anotmodels varphi$. It immediately follows that either $Amodels varphi$ or $Amodels lnot varphi$.
            $endgroup$
            – Alex Kruckman
            Dec 8 '18 at 18:34










          • $begingroup$
            thank you everyone
            $endgroup$
            – bof
            Dec 8 '18 at 19:27











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          $begingroup$

          Assume there's an $L$-sentence $sigma$ that's neither true nor false. Then we have that $mathcal{A}nvDashsigma$ and $mathcal{A}nvDashnegsigma$. But this is a contradiction since $mathcal{A}vDashsigma$ if and only if $mathcal{A}nvDashnegsigma$, and vice versa, by definition of what a structure is.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            I'm sorry if this is a really basic question but why is it that you can construct such sentence that is neither true nor false under Godel's incompleteness theorem but not here? Thanks!
            $endgroup$
            – bof
            Dec 8 '18 at 16:26










          • $begingroup$
            For a detailed explanation, see here. But basically, Godel's incompleteness theorem is not a negation of Godel's completeness theorem. Also, here we are dealing with a structure, not a set of sentences (which is what Godel's incompleteness theorem talks about).
            $endgroup$
            – quanticbolt
            Dec 8 '18 at 16:39








          • 1




            $begingroup$
            @bof That's an incorrect statement of Godel's incompleteness theorem. There is a distinction between truth in a model/structure (fixing a structure $M$, everything is either true in $M$ or false in $M$) and provability from a theory (= truth in every model of that theory; since a given theory might have many disagreeing models, there are in general independent sentences, and this is what Godel's incompleteness theorem is about).
            $endgroup$
            – Noah Schweber
            Dec 8 '18 at 16:40










          • $begingroup$
            @bof For any sentence $varphi$ and any structure $A$ the definition of $Amodels lnot varphi$ is that $Anotmodels varphi$. It immediately follows that either $Amodels varphi$ or $Amodels lnot varphi$.
            $endgroup$
            – Alex Kruckman
            Dec 8 '18 at 18:34










          • $begingroup$
            thank you everyone
            $endgroup$
            – bof
            Dec 8 '18 at 19:27
















          2












          $begingroup$

          Assume there's an $L$-sentence $sigma$ that's neither true nor false. Then we have that $mathcal{A}nvDashsigma$ and $mathcal{A}nvDashnegsigma$. But this is a contradiction since $mathcal{A}vDashsigma$ if and only if $mathcal{A}nvDashnegsigma$, and vice versa, by definition of what a structure is.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            I'm sorry if this is a really basic question but why is it that you can construct such sentence that is neither true nor false under Godel's incompleteness theorem but not here? Thanks!
            $endgroup$
            – bof
            Dec 8 '18 at 16:26










          • $begingroup$
            For a detailed explanation, see here. But basically, Godel's incompleteness theorem is not a negation of Godel's completeness theorem. Also, here we are dealing with a structure, not a set of sentences (which is what Godel's incompleteness theorem talks about).
            $endgroup$
            – quanticbolt
            Dec 8 '18 at 16:39








          • 1




            $begingroup$
            @bof That's an incorrect statement of Godel's incompleteness theorem. There is a distinction between truth in a model/structure (fixing a structure $M$, everything is either true in $M$ or false in $M$) and provability from a theory (= truth in every model of that theory; since a given theory might have many disagreeing models, there are in general independent sentences, and this is what Godel's incompleteness theorem is about).
            $endgroup$
            – Noah Schweber
            Dec 8 '18 at 16:40










          • $begingroup$
            @bof For any sentence $varphi$ and any structure $A$ the definition of $Amodels lnot varphi$ is that $Anotmodels varphi$. It immediately follows that either $Amodels varphi$ or $Amodels lnot varphi$.
            $endgroup$
            – Alex Kruckman
            Dec 8 '18 at 18:34










          • $begingroup$
            thank you everyone
            $endgroup$
            – bof
            Dec 8 '18 at 19:27














          2












          2








          2





          $begingroup$

          Assume there's an $L$-sentence $sigma$ that's neither true nor false. Then we have that $mathcal{A}nvDashsigma$ and $mathcal{A}nvDashnegsigma$. But this is a contradiction since $mathcal{A}vDashsigma$ if and only if $mathcal{A}nvDashnegsigma$, and vice versa, by definition of what a structure is.






          share|cite|improve this answer









          $endgroup$



          Assume there's an $L$-sentence $sigma$ that's neither true nor false. Then we have that $mathcal{A}nvDashsigma$ and $mathcal{A}nvDashnegsigma$. But this is a contradiction since $mathcal{A}vDashsigma$ if and only if $mathcal{A}nvDashnegsigma$, and vice versa, by definition of what a structure is.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 8 '18 at 16:24









          quanticboltquanticbolt

          769514




          769514












          • $begingroup$
            I'm sorry if this is a really basic question but why is it that you can construct such sentence that is neither true nor false under Godel's incompleteness theorem but not here? Thanks!
            $endgroup$
            – bof
            Dec 8 '18 at 16:26










          • $begingroup$
            For a detailed explanation, see here. But basically, Godel's incompleteness theorem is not a negation of Godel's completeness theorem. Also, here we are dealing with a structure, not a set of sentences (which is what Godel's incompleteness theorem talks about).
            $endgroup$
            – quanticbolt
            Dec 8 '18 at 16:39








          • 1




            $begingroup$
            @bof That's an incorrect statement of Godel's incompleteness theorem. There is a distinction between truth in a model/structure (fixing a structure $M$, everything is either true in $M$ or false in $M$) and provability from a theory (= truth in every model of that theory; since a given theory might have many disagreeing models, there are in general independent sentences, and this is what Godel's incompleteness theorem is about).
            $endgroup$
            – Noah Schweber
            Dec 8 '18 at 16:40










          • $begingroup$
            @bof For any sentence $varphi$ and any structure $A$ the definition of $Amodels lnot varphi$ is that $Anotmodels varphi$. It immediately follows that either $Amodels varphi$ or $Amodels lnot varphi$.
            $endgroup$
            – Alex Kruckman
            Dec 8 '18 at 18:34










          • $begingroup$
            thank you everyone
            $endgroup$
            – bof
            Dec 8 '18 at 19:27


















          • $begingroup$
            I'm sorry if this is a really basic question but why is it that you can construct such sentence that is neither true nor false under Godel's incompleteness theorem but not here? Thanks!
            $endgroup$
            – bof
            Dec 8 '18 at 16:26










          • $begingroup$
            For a detailed explanation, see here. But basically, Godel's incompleteness theorem is not a negation of Godel's completeness theorem. Also, here we are dealing with a structure, not a set of sentences (which is what Godel's incompleteness theorem talks about).
            $endgroup$
            – quanticbolt
            Dec 8 '18 at 16:39








          • 1




            $begingroup$
            @bof That's an incorrect statement of Godel's incompleteness theorem. There is a distinction between truth in a model/structure (fixing a structure $M$, everything is either true in $M$ or false in $M$) and provability from a theory (= truth in every model of that theory; since a given theory might have many disagreeing models, there are in general independent sentences, and this is what Godel's incompleteness theorem is about).
            $endgroup$
            – Noah Schweber
            Dec 8 '18 at 16:40










          • $begingroup$
            @bof For any sentence $varphi$ and any structure $A$ the definition of $Amodels lnot varphi$ is that $Anotmodels varphi$. It immediately follows that either $Amodels varphi$ or $Amodels lnot varphi$.
            $endgroup$
            – Alex Kruckman
            Dec 8 '18 at 18:34










          • $begingroup$
            thank you everyone
            $endgroup$
            – bof
            Dec 8 '18 at 19:27
















          $begingroup$
          I'm sorry if this is a really basic question but why is it that you can construct such sentence that is neither true nor false under Godel's incompleteness theorem but not here? Thanks!
          $endgroup$
          – bof
          Dec 8 '18 at 16:26




          $begingroup$
          I'm sorry if this is a really basic question but why is it that you can construct such sentence that is neither true nor false under Godel's incompleteness theorem but not here? Thanks!
          $endgroup$
          – bof
          Dec 8 '18 at 16:26












          $begingroup$
          For a detailed explanation, see here. But basically, Godel's incompleteness theorem is not a negation of Godel's completeness theorem. Also, here we are dealing with a structure, not a set of sentences (which is what Godel's incompleteness theorem talks about).
          $endgroup$
          – quanticbolt
          Dec 8 '18 at 16:39






          $begingroup$
          For a detailed explanation, see here. But basically, Godel's incompleteness theorem is not a negation of Godel's completeness theorem. Also, here we are dealing with a structure, not a set of sentences (which is what Godel's incompleteness theorem talks about).
          $endgroup$
          – quanticbolt
          Dec 8 '18 at 16:39






          1




          1




          $begingroup$
          @bof That's an incorrect statement of Godel's incompleteness theorem. There is a distinction between truth in a model/structure (fixing a structure $M$, everything is either true in $M$ or false in $M$) and provability from a theory (= truth in every model of that theory; since a given theory might have many disagreeing models, there are in general independent sentences, and this is what Godel's incompleteness theorem is about).
          $endgroup$
          – Noah Schweber
          Dec 8 '18 at 16:40




          $begingroup$
          @bof That's an incorrect statement of Godel's incompleteness theorem. There is a distinction between truth in a model/structure (fixing a structure $M$, everything is either true in $M$ or false in $M$) and provability from a theory (= truth in every model of that theory; since a given theory might have many disagreeing models, there are in general independent sentences, and this is what Godel's incompleteness theorem is about).
          $endgroup$
          – Noah Schweber
          Dec 8 '18 at 16:40












          $begingroup$
          @bof For any sentence $varphi$ and any structure $A$ the definition of $Amodels lnot varphi$ is that $Anotmodels varphi$. It immediately follows that either $Amodels varphi$ or $Amodels lnot varphi$.
          $endgroup$
          – Alex Kruckman
          Dec 8 '18 at 18:34




          $begingroup$
          @bof For any sentence $varphi$ and any structure $A$ the definition of $Amodels lnot varphi$ is that $Anotmodels varphi$. It immediately follows that either $Amodels varphi$ or $Amodels lnot varphi$.
          $endgroup$
          – Alex Kruckman
          Dec 8 '18 at 18:34












          $begingroup$
          thank you everyone
          $endgroup$
          – bof
          Dec 8 '18 at 19:27




          $begingroup$
          thank you everyone
          $endgroup$
          – bof
          Dec 8 '18 at 19:27


















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