proof that weak axiom of pairing and axiom schema of specification imply axiom of pairing












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as the title says I am trying to give a (nearly, but not fully formal) proof that the weak axiom of pairing (i.e. $forall x forall y exists p: x in p wedge y in p$) together with a suitable instance of the axiom schema of specification does imply the axiom of pairing.
I haven't found a suitable instance yet, so this would be the first step to take.






elementary-set-theory logic proof-writing axioms







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$endgroup$

















    0












    $begingroup$


    as the title says I am trying to give a (nearly, but not fully formal) proof that the weak axiom of pairing (i.e. $forall x forall y exists p: x in p wedge y in p$) together with a suitable instance of the axiom schema of specification does imply the axiom of pairing.
    I haven't found a suitable instance yet, so this would be the first step to take.






    elementary-set-theory logic proof-writing axioms







    share|cite|improve this question











    $endgroup$















      0












      0








      0


      1



      $begingroup$


      as the title says I am trying to give a (nearly, but not fully formal) proof that the weak axiom of pairing (i.e. $forall x forall y exists p: x in p wedge y in p$) together with a suitable instance of the axiom schema of specification does imply the axiom of pairing.
      I haven't found a suitable instance yet, so this would be the first step to take.






      elementary-set-theory logic proof-writing axioms







      share|cite|improve this question











      $endgroup$




      as the title says I am trying to give a (nearly, but not fully formal) proof that the weak axiom of pairing (i.e. $forall x forall y exists p: x in p wedge y in p$) together with a suitable instance of the axiom schema of specification does imply the axiom of pairing.
      I haven't found a suitable instance yet, so this would be the first step to take.






      elementary-set-theory logic proof-writing axioms




      elementary-set-theory logic proof-writing axioms






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Jan 12 at 21:16









      Andrés E. Caicedo

      65.2k8158247




      65.2k8158247










      asked Jan 12 at 17:34









      StudentuStudentu

      1228




      1228






















          3 Answers
          3






          active

          oldest

          votes


















          2












          $begingroup$

          Here is a 'nearly, but not fully, formal proof' (OK, it is fully formal, but it is not fully completed):



          enter image description here






          share|cite|improve this answer











          $endgroup$









          • 1




            $begingroup$
            What software is that?
            $endgroup$
            – J.G.
            Jan 12 at 18:46






          • 1




            $begingroup$
            @J.G. It's called 'Fitch' .. it comes with the book "Language, Proof, and Logic"
            $endgroup$
            – Bram28
            Jan 12 at 19:14










          • $begingroup$
            This software seems awesome! (Though I don't fully understand the notions there.) Thank you for your reply!
            $endgroup$
            – Studentu
            Jan 13 at 18:10






          • 1




            $begingroup$
            @Studentu The 'Fitch' system is actually a fairly well known and well-used system for creating fully formal proofs. Here is a list with all the rules: math.mcgill.ca/rags/JAC/124/Rules-Strategy-b.pdf
            $endgroup$
            – Bram28
            Jan 13 at 18:21








          • 1




            $begingroup$
            @Studentu Cool. You're welcome! :)
            $endgroup$
            – Bram28
            Jan 15 at 1:31



















          1












          $begingroup$

          Given $x,y$, let $p$ be such that $xin pland yin p$. Then ${x,y}={,tin pmid t=xlor t=y,}$.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Thanks for your answer!
            $endgroup$
            – Studentu
            Jan 13 at 18:09



















          1












          $begingroup$

          Fix $x,y$ and take $p$ such that $x, y in p$. Now take the formula $Phi = ( z = x lor z = y)$ and apply the axiom of specification on $p$.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Thank you for answering!
            $endgroup$
            – Studentu
            Jan 13 at 18:09











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          3 Answers
          3






          active

          oldest

          votes








          3 Answers
          3






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          2












          $begingroup$

          Here is a 'nearly, but not fully, formal proof' (OK, it is fully formal, but it is not fully completed):



          enter image description here






          share|cite|improve this answer











          $endgroup$









          • 1




            $begingroup$
            What software is that?
            $endgroup$
            – J.G.
            Jan 12 at 18:46






          • 1




            $begingroup$
            @J.G. It's called 'Fitch' .. it comes with the book "Language, Proof, and Logic"
            $endgroup$
            – Bram28
            Jan 12 at 19:14










          • $begingroup$
            This software seems awesome! (Though I don't fully understand the notions there.) Thank you for your reply!
            $endgroup$
            – Studentu
            Jan 13 at 18:10






          • 1




            $begingroup$
            @Studentu The 'Fitch' system is actually a fairly well known and well-used system for creating fully formal proofs. Here is a list with all the rules: math.mcgill.ca/rags/JAC/124/Rules-Strategy-b.pdf
            $endgroup$
            – Bram28
            Jan 13 at 18:21








          • 1




            $begingroup$
            @Studentu Cool. You're welcome! :)
            $endgroup$
            – Bram28
            Jan 15 at 1:31
















          2












          $begingroup$

          Here is a 'nearly, but not fully, formal proof' (OK, it is fully formal, but it is not fully completed):



          enter image description here






          share|cite|improve this answer











          $endgroup$









          • 1




            $begingroup$
            What software is that?
            $endgroup$
            – J.G.
            Jan 12 at 18:46






          • 1




            $begingroup$
            @J.G. It's called 'Fitch' .. it comes with the book "Language, Proof, and Logic"
            $endgroup$
            – Bram28
            Jan 12 at 19:14










          • $begingroup$
            This software seems awesome! (Though I don't fully understand the notions there.) Thank you for your reply!
            $endgroup$
            – Studentu
            Jan 13 at 18:10






          • 1




            $begingroup$
            @Studentu The 'Fitch' system is actually a fairly well known and well-used system for creating fully formal proofs. Here is a list with all the rules: math.mcgill.ca/rags/JAC/124/Rules-Strategy-b.pdf
            $endgroup$
            – Bram28
            Jan 13 at 18:21








          • 1




            $begingroup$
            @Studentu Cool. You're welcome! :)
            $endgroup$
            – Bram28
            Jan 15 at 1:31














          2












          2








          2





          $begingroup$

          Here is a 'nearly, but not fully, formal proof' (OK, it is fully formal, but it is not fully completed):



          enter image description here






          share|cite|improve this answer











          $endgroup$



          Here is a 'nearly, but not fully, formal proof' (OK, it is fully formal, but it is not fully completed):



          enter image description here







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Jan 12 at 19:18

























          answered Jan 12 at 18:42









          Bram28Bram28

          61.4k44792




          61.4k44792








          • 1




            $begingroup$
            What software is that?
            $endgroup$
            – J.G.
            Jan 12 at 18:46






          • 1




            $begingroup$
            @J.G. It's called 'Fitch' .. it comes with the book "Language, Proof, and Logic"
            $endgroup$
            – Bram28
            Jan 12 at 19:14










          • $begingroup$
            This software seems awesome! (Though I don't fully understand the notions there.) Thank you for your reply!
            $endgroup$
            – Studentu
            Jan 13 at 18:10






          • 1




            $begingroup$
            @Studentu The 'Fitch' system is actually a fairly well known and well-used system for creating fully formal proofs. Here is a list with all the rules: math.mcgill.ca/rags/JAC/124/Rules-Strategy-b.pdf
            $endgroup$
            – Bram28
            Jan 13 at 18:21








          • 1




            $begingroup$
            @Studentu Cool. You're welcome! :)
            $endgroup$
            – Bram28
            Jan 15 at 1:31














          • 1




            $begingroup$
            What software is that?
            $endgroup$
            – J.G.
            Jan 12 at 18:46






          • 1




            $begingroup$
            @J.G. It's called 'Fitch' .. it comes with the book "Language, Proof, and Logic"
            $endgroup$
            – Bram28
            Jan 12 at 19:14










          • $begingroup$
            This software seems awesome! (Though I don't fully understand the notions there.) Thank you for your reply!
            $endgroup$
            – Studentu
            Jan 13 at 18:10






          • 1




            $begingroup$
            @Studentu The 'Fitch' system is actually a fairly well known and well-used system for creating fully formal proofs. Here is a list with all the rules: math.mcgill.ca/rags/JAC/124/Rules-Strategy-b.pdf
            $endgroup$
            – Bram28
            Jan 13 at 18:21








          • 1




            $begingroup$
            @Studentu Cool. You're welcome! :)
            $endgroup$
            – Bram28
            Jan 15 at 1:31








          1




          1




          $begingroup$
          What software is that?
          $endgroup$
          – J.G.
          Jan 12 at 18:46




          $begingroup$
          What software is that?
          $endgroup$
          – J.G.
          Jan 12 at 18:46




          1




          1




          $begingroup$
          @J.G. It's called 'Fitch' .. it comes with the book "Language, Proof, and Logic"
          $endgroup$
          – Bram28
          Jan 12 at 19:14




          $begingroup$
          @J.G. It's called 'Fitch' .. it comes with the book "Language, Proof, and Logic"
          $endgroup$
          – Bram28
          Jan 12 at 19:14












          $begingroup$
          This software seems awesome! (Though I don't fully understand the notions there.) Thank you for your reply!
          $endgroup$
          – Studentu
          Jan 13 at 18:10




          $begingroup$
          This software seems awesome! (Though I don't fully understand the notions there.) Thank you for your reply!
          $endgroup$
          – Studentu
          Jan 13 at 18:10




          1




          1




          $begingroup$
          @Studentu The 'Fitch' system is actually a fairly well known and well-used system for creating fully formal proofs. Here is a list with all the rules: math.mcgill.ca/rags/JAC/124/Rules-Strategy-b.pdf
          $endgroup$
          – Bram28
          Jan 13 at 18:21






          $begingroup$
          @Studentu The 'Fitch' system is actually a fairly well known and well-used system for creating fully formal proofs. Here is a list with all the rules: math.mcgill.ca/rags/JAC/124/Rules-Strategy-b.pdf
          $endgroup$
          – Bram28
          Jan 13 at 18:21






          1




          1




          $begingroup$
          @Studentu Cool. You're welcome! :)
          $endgroup$
          – Bram28
          Jan 15 at 1:31




          $begingroup$
          @Studentu Cool. You're welcome! :)
          $endgroup$
          – Bram28
          Jan 15 at 1:31











          1












          $begingroup$

          Given $x,y$, let $p$ be such that $xin pland yin p$. Then ${x,y}={,tin pmid t=xlor t=y,}$.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Thanks for your answer!
            $endgroup$
            – Studentu
            Jan 13 at 18:09
















          1












          $begingroup$

          Given $x,y$, let $p$ be such that $xin pland yin p$. Then ${x,y}={,tin pmid t=xlor t=y,}$.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Thanks for your answer!
            $endgroup$
            – Studentu
            Jan 13 at 18:09














          1












          1








          1





          $begingroup$

          Given $x,y$, let $p$ be such that $xin pland yin p$. Then ${x,y}={,tin pmid t=xlor t=y,}$.






          share|cite|improve this answer









          $endgroup$



          Given $x,y$, let $p$ be such that $xin pland yin p$. Then ${x,y}={,tin pmid t=xlor t=y,}$.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Jan 12 at 17:37









          Hagen von EitzenHagen von Eitzen

          278k22269498




          278k22269498












          • $begingroup$
            Thanks for your answer!
            $endgroup$
            – Studentu
            Jan 13 at 18:09


















          • $begingroup$
            Thanks for your answer!
            $endgroup$
            – Studentu
            Jan 13 at 18:09
















          $begingroup$
          Thanks for your answer!
          $endgroup$
          – Studentu
          Jan 13 at 18:09




          $begingroup$
          Thanks for your answer!
          $endgroup$
          – Studentu
          Jan 13 at 18:09











          1












          $begingroup$

          Fix $x,y$ and take $p$ such that $x, y in p$. Now take the formula $Phi = ( z = x lor z = y)$ and apply the axiom of specification on $p$.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Thank you for answering!
            $endgroup$
            – Studentu
            Jan 13 at 18:09
















          1












          $begingroup$

          Fix $x,y$ and take $p$ such that $x, y in p$. Now take the formula $Phi = ( z = x lor z = y)$ and apply the axiom of specification on $p$.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Thank you for answering!
            $endgroup$
            – Studentu
            Jan 13 at 18:09














          1












          1








          1





          $begingroup$

          Fix $x,y$ and take $p$ such that $x, y in p$. Now take the formula $Phi = ( z = x lor z = y)$ and apply the axiom of specification on $p$.






          share|cite|improve this answer











          $endgroup$



          Fix $x,y$ and take $p$ such that $x, y in p$. Now take the formula $Phi = ( z = x lor z = y)$ and apply the axiom of specification on $p$.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Jan 12 at 20:46

























          answered Jan 12 at 17:40









          Lucas HenriqueLucas Henrique

          1,032414




          1,032414












          • $begingroup$
            Thank you for answering!
            $endgroup$
            – Studentu
            Jan 13 at 18:09


















          • $begingroup$
            Thank you for answering!
            $endgroup$
            – Studentu
            Jan 13 at 18:09
















          $begingroup$
          Thank you for answering!
          $endgroup$
          – Studentu
          Jan 13 at 18:09




          $begingroup$
          Thank you for answering!
          $endgroup$
          – Studentu
          Jan 13 at 18:09


















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