Why is the “axiom of extension” an axiom? [duplicate]












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  • What's the point of the axiom of extensionality?

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I think this is a definition of = because it explains the meaning of the new symbol =. However, I wonder why the set theory thinks this as an axiom rather than a definition.






logic set-theory axioms







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marked as duplicate by José Carlos Santos, Asaf Karagila♦ logic
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 Jan 21 at 9:41


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.























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    This question already has an answer here:




    • What's the point of the axiom of extensionality?

      2 answers




    I think this is a definition of = because it explains the meaning of the new symbol =. However, I wonder why the set theory thinks this as an axiom rather than a definition.






    logic set-theory axioms







    share|cite|improve this question











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    marked as duplicate by José Carlos Santos, Asaf Karagila♦ logic
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     Jan 21 at 9:41


    This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.





















      1












      1








      1





      $begingroup$



      This question already has an answer here:




      • What's the point of the axiom of extensionality?

        2 answers




      I think this is a definition of = because it explains the meaning of the new symbol =. However, I wonder why the set theory thinks this as an axiom rather than a definition.






      logic set-theory axioms







      share|cite|improve this question











      $endgroup$





      This question already has an answer here:




      • What's the point of the axiom of extensionality?

        2 answers




      I think this is a definition of = because it explains the meaning of the new symbol =. However, I wonder why the set theory thinks this as an axiom rather than a definition.





      This question already has an answer here:




      • What's the point of the axiom of extensionality?

        2 answers







      logic set-theory axioms




      logic set-theory axioms






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Jan 21 at 10:57









      Mauro ALLEGRANZA

      66.4k449115




      66.4k449115










      asked Jan 21 at 8:52









      amoogaeamoogae

      487




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       Jan 21 at 9:41


      This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.









      marked as duplicate by José Carlos Santos, Asaf Karagila♦ logic
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       Jan 21 at 9:41


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          1 Answer
          1






          active

          oldest

          votes


















          5












          $begingroup$

          See Axiom of extensionality.



          There are two possibilities :



          (i) the underlying logic is predicate calculus with equality.



          In this case, the symbol $=$ is already defined by the equality axiom and Extensionality is needed only to legislate the "interaction" with $in$ :




          $∀x∀y∀z [(z ∈ x leftrightarrow z ∈ y) to x = y]$.




          (ii) the underlying logic is predicate calculus without equality.



          In this case you are right : we need a specific definition for equality :




          $a=b =_{def} forall x [x in a leftrightarrow x in b]$




          and a different version of Extensionality :




          $forall z forall x forall y [x = y to (x in z to y in z)]$.




          From them we can derive the usual properties of equality.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            +1 Very enlightening.
            $endgroup$
            – drhab
            Jan 21 at 9:22










          • $begingroup$
            Thank you very much! Can you tell me exactly what the (slightly different) version of Extensionality is?
            $endgroup$
            – amoogae
            Jan 21 at 9:36










          • $begingroup$
            I just read your edited answer. Thank you so much!
            $endgroup$
            – amoogae
            Jan 21 at 11:01


















          1 Answer
          1






          active

          oldest

          votes








          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          5












          $begingroup$

          See Axiom of extensionality.



          There are two possibilities :



          (i) the underlying logic is predicate calculus with equality.



          In this case, the symbol $=$ is already defined by the equality axiom and Extensionality is needed only to legislate the "interaction" with $in$ :




          $∀x∀y∀z [(z ∈ x leftrightarrow z ∈ y) to x = y]$.




          (ii) the underlying logic is predicate calculus without equality.



          In this case you are right : we need a specific definition for equality :




          $a=b =_{def} forall x [x in a leftrightarrow x in b]$




          and a different version of Extensionality :




          $forall z forall x forall y [x = y to (x in z to y in z)]$.




          From them we can derive the usual properties of equality.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            +1 Very enlightening.
            $endgroup$
            – drhab
            Jan 21 at 9:22










          • $begingroup$
            Thank you very much! Can you tell me exactly what the (slightly different) version of Extensionality is?
            $endgroup$
            – amoogae
            Jan 21 at 9:36










          • $begingroup$
            I just read your edited answer. Thank you so much!
            $endgroup$
            – amoogae
            Jan 21 at 11:01
















          5












          $begingroup$

          See Axiom of extensionality.



          There are two possibilities :



          (i) the underlying logic is predicate calculus with equality.



          In this case, the symbol $=$ is already defined by the equality axiom and Extensionality is needed only to legislate the "interaction" with $in$ :




          $∀x∀y∀z [(z ∈ x leftrightarrow z ∈ y) to x = y]$.




          (ii) the underlying logic is predicate calculus without equality.



          In this case you are right : we need a specific definition for equality :




          $a=b =_{def} forall x [x in a leftrightarrow x in b]$




          and a different version of Extensionality :




          $forall z forall x forall y [x = y to (x in z to y in z)]$.




          From them we can derive the usual properties of equality.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            +1 Very enlightening.
            $endgroup$
            – drhab
            Jan 21 at 9:22










          • $begingroup$
            Thank you very much! Can you tell me exactly what the (slightly different) version of Extensionality is?
            $endgroup$
            – amoogae
            Jan 21 at 9:36










          • $begingroup$
            I just read your edited answer. Thank you so much!
            $endgroup$
            – amoogae
            Jan 21 at 11:01














          5












          5








          5





          $begingroup$

          See Axiom of extensionality.



          There are two possibilities :



          (i) the underlying logic is predicate calculus with equality.



          In this case, the symbol $=$ is already defined by the equality axiom and Extensionality is needed only to legislate the "interaction" with $in$ :




          $∀x∀y∀z [(z ∈ x leftrightarrow z ∈ y) to x = y]$.




          (ii) the underlying logic is predicate calculus without equality.



          In this case you are right : we need a specific definition for equality :




          $a=b =_{def} forall x [x in a leftrightarrow x in b]$




          and a different version of Extensionality :




          $forall z forall x forall y [x = y to (x in z to y in z)]$.




          From them we can derive the usual properties of equality.






          share|cite|improve this answer











          $endgroup$



          See Axiom of extensionality.



          There are two possibilities :



          (i) the underlying logic is predicate calculus with equality.



          In this case, the symbol $=$ is already defined by the equality axiom and Extensionality is needed only to legislate the "interaction" with $in$ :




          $∀x∀y∀z [(z ∈ x leftrightarrow z ∈ y) to x = y]$.




          (ii) the underlying logic is predicate calculus without equality.



          In this case you are right : we need a specific definition for equality :




          $a=b =_{def} forall x [x in a leftrightarrow x in b]$




          and a different version of Extensionality :




          $forall z forall x forall y [x = y to (x in z to y in z)]$.




          From them we can derive the usual properties of equality.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Jan 21 at 15:11

























          answered Jan 21 at 9:00









          Mauro ALLEGRANZAMauro ALLEGRANZA

          66.4k449115




          66.4k449115












          • $begingroup$
            +1 Very enlightening.
            $endgroup$
            – drhab
            Jan 21 at 9:22










          • $begingroup$
            Thank you very much! Can you tell me exactly what the (slightly different) version of Extensionality is?
            $endgroup$
            – amoogae
            Jan 21 at 9:36










          • $begingroup$
            I just read your edited answer. Thank you so much!
            $endgroup$
            – amoogae
            Jan 21 at 11:01


















          • $begingroup$
            +1 Very enlightening.
            $endgroup$
            – drhab
            Jan 21 at 9:22










          • $begingroup$
            Thank you very much! Can you tell me exactly what the (slightly different) version of Extensionality is?
            $endgroup$
            – amoogae
            Jan 21 at 9:36










          • $begingroup$
            I just read your edited answer. Thank you so much!
            $endgroup$
            – amoogae
            Jan 21 at 11:01
















          $begingroup$
          +1 Very enlightening.
          $endgroup$
          – drhab
          Jan 21 at 9:22




          $begingroup$
          +1 Very enlightening.
          $endgroup$
          – drhab
          Jan 21 at 9:22












          $begingroup$
          Thank you very much! Can you tell me exactly what the (slightly different) version of Extensionality is?
          $endgroup$
          – amoogae
          Jan 21 at 9:36




          $begingroup$
          Thank you very much! Can you tell me exactly what the (slightly different) version of Extensionality is?
          $endgroup$
          – amoogae
          Jan 21 at 9:36












          $begingroup$
          I just read your edited answer. Thank you so much!
          $endgroup$
          – amoogae
          Jan 21 at 11:01




          $begingroup$
          I just read your edited answer. Thank you so much!
          $endgroup$
          – amoogae
          Jan 21 at 11:01



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