Definition of the Number Two in ZF Set Theory












0












$begingroup$


This page shows how natural numbers may be defined as unions of sets.



What is a straightforward way to appeal to intuitive notions to dispel the misconception that

{Ø} + {Ø} = {{Ø}, {Ø}, Ø} and instead show that

{Ø} + {Ø} = {Ø, {Ø}}
?






elementary-set-theory natural-numbers







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$endgroup$








  • 1




    $begingroup$
    But ${{emptyset},{emptyset},emptyset}$ is ${emptyset,{emptyset}}$ !
    $endgroup$
    – Saucy O'Path
    Dec 14 '18 at 21:04








  • 1




    $begingroup$
    Furthermore, there's no particular reason why $0$ has to be defined as the empty set, $1$ has to be ${emptyset}$, and so on. We could equally well define $0$ to $ {emptyset }$, and $1$ to be ${emptyset, {emptyset} }$, etc. Of course, we'd then have to modify the definition of successor, and of addition, etc. --- but all we're looking for is a collection of distinct things that we can use to represent the natural numbers (in some orderly way to make other steps easier).
    $endgroup$
    – John Hughes
    Dec 14 '18 at 21:07


















0












$begingroup$


This page shows how natural numbers may be defined as unions of sets.



What is a straightforward way to appeal to intuitive notions to dispel the misconception that

{Ø} + {Ø} = {{Ø}, {Ø}, Ø} and instead show that

{Ø} + {Ø} = {Ø, {Ø}}
?






elementary-set-theory natural-numbers







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    But ${{emptyset},{emptyset},emptyset}$ is ${emptyset,{emptyset}}$ !
    $endgroup$
    – Saucy O'Path
    Dec 14 '18 at 21:04








  • 1




    $begingroup$
    Furthermore, there's no particular reason why $0$ has to be defined as the empty set, $1$ has to be ${emptyset}$, and so on. We could equally well define $0$ to $ {emptyset }$, and $1$ to be ${emptyset, {emptyset} }$, etc. Of course, we'd then have to modify the definition of successor, and of addition, etc. --- but all we're looking for is a collection of distinct things that we can use to represent the natural numbers (in some orderly way to make other steps easier).
    $endgroup$
    – John Hughes
    Dec 14 '18 at 21:07
















0












0








0





$begingroup$


This page shows how natural numbers may be defined as unions of sets.



What is a straightforward way to appeal to intuitive notions to dispel the misconception that

{Ø} + {Ø} = {{Ø}, {Ø}, Ø} and instead show that

{Ø} + {Ø} = {Ø, {Ø}}
?






elementary-set-theory natural-numbers







share|cite|improve this question











$endgroup$




This page shows how natural numbers may be defined as unions of sets.



What is a straightforward way to appeal to intuitive notions to dispel the misconception that

{Ø} + {Ø} = {{Ø}, {Ø}, Ø} and instead show that

{Ø} + {Ø} = {Ø, {Ø}}
?






elementary-set-theory natural-numbers




elementary-set-theory natural-numbers






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 15 '18 at 20:01







bblohowiak

















asked Dec 14 '18 at 21:01









bblohowiakbblohowiak

1099




1099








  • 1




    $begingroup$
    But ${{emptyset},{emptyset},emptyset}$ is ${emptyset,{emptyset}}$ !
    $endgroup$
    – Saucy O'Path
    Dec 14 '18 at 21:04








  • 1




    $begingroup$
    Furthermore, there's no particular reason why $0$ has to be defined as the empty set, $1$ has to be ${emptyset}$, and so on. We could equally well define $0$ to $ {emptyset }$, and $1$ to be ${emptyset, {emptyset} }$, etc. Of course, we'd then have to modify the definition of successor, and of addition, etc. --- but all we're looking for is a collection of distinct things that we can use to represent the natural numbers (in some orderly way to make other steps easier).
    $endgroup$
    – John Hughes
    Dec 14 '18 at 21:07
















  • 1




    $begingroup$
    But ${{emptyset},{emptyset},emptyset}$ is ${emptyset,{emptyset}}$ !
    $endgroup$
    – Saucy O'Path
    Dec 14 '18 at 21:04








  • 1




    $begingroup$
    Furthermore, there's no particular reason why $0$ has to be defined as the empty set, $1$ has to be ${emptyset}$, and so on. We could equally well define $0$ to $ {emptyset }$, and $1$ to be ${emptyset, {emptyset} }$, etc. Of course, we'd then have to modify the definition of successor, and of addition, etc. --- but all we're looking for is a collection of distinct things that we can use to represent the natural numbers (in some orderly way to make other steps easier).
    $endgroup$
    – John Hughes
    Dec 14 '18 at 21:07










1




1




$begingroup$
But ${{emptyset},{emptyset},emptyset}$ is ${emptyset,{emptyset}}$ !
$endgroup$
– Saucy O'Path
Dec 14 '18 at 21:04






$begingroup$
But ${{emptyset},{emptyset},emptyset}$ is ${emptyset,{emptyset}}$ !
$endgroup$
– Saucy O'Path
Dec 14 '18 at 21:04






1




1




$begingroup$
Furthermore, there's no particular reason why $0$ has to be defined as the empty set, $1$ has to be ${emptyset}$, and so on. We could equally well define $0$ to $ {emptyset }$, and $1$ to be ${emptyset, {emptyset} }$, etc. Of course, we'd then have to modify the definition of successor, and of addition, etc. --- but all we're looking for is a collection of distinct things that we can use to represent the natural numbers (in some orderly way to make other steps easier).
$endgroup$
– John Hughes
Dec 14 '18 at 21:07






$begingroup$
Furthermore, there's no particular reason why $0$ has to be defined as the empty set, $1$ has to be ${emptyset}$, and so on. We could equally well define $0$ to $ {emptyset }$, and $1$ to be ${emptyset, {emptyset} }$, etc. Of course, we'd then have to modify the definition of successor, and of addition, etc. --- but all we're looking for is a collection of distinct things that we can use to represent the natural numbers (in some orderly way to make other steps easier).
$endgroup$
– John Hughes
Dec 14 '18 at 21:07












1 Answer
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$begingroup$

Note that ${x,x,y}={x,y}$. So the two sets you suggest are in fact the same.






share|cite|improve this answer









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    active

    oldest

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    4












    $begingroup$

    Note that ${x,x,y}={x,y}$. So the two sets you suggest are in fact the same.






    share|cite|improve this answer









    $endgroup$


















      4












      $begingroup$

      Note that ${x,x,y}={x,y}$. So the two sets you suggest are in fact the same.






      share|cite|improve this answer









      $endgroup$
















        4












        4








        4





        $begingroup$

        Note that ${x,x,y}={x,y}$. So the two sets you suggest are in fact the same.






        share|cite|improve this answer









        $endgroup$



        Note that ${x,x,y}={x,y}$. So the two sets you suggest are in fact the same.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Dec 14 '18 at 21:04









        Asaf Karagila♦Asaf Karagila

        306k33437767




        306k33437767






























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