Why does the set {1,3,5,7… ; 2,4,6,8…} qualify as well-ordered? How to explain this notation?












2












$begingroup$


The set {odd natural numbers greater than 0 }U {even natural numbers}



that is, the set



Union { {1,3,5,7...}, {2,4,6,8...} }



also strangely written



{1,3,5,7... ; 2,4,6,8...} .



is often given as example of a well ordered set ( a set such that for all subset there is a first element).



I have some problems with this example.



(1) First, which relation orders this set? could this relation be defined explicitely? is this relation somewhat analogous to lexigraphic order? So I don't understand how this set can be an ordered set.



(2) Second, I don't understand how it is well-ordered. In order to be well ordered, every subset should have a first element. But apparently the set {7,2} is a subset of my set. What is the first element of {7,2}?










share|cite|improve this question











$endgroup$

















    2












    $begingroup$


    The set {odd natural numbers greater than 0 }U {even natural numbers}



    that is, the set



    Union { {1,3,5,7...}, {2,4,6,8...} }



    also strangely written



    {1,3,5,7... ; 2,4,6,8...} .



    is often given as example of a well ordered set ( a set such that for all subset there is a first element).



    I have some problems with this example.



    (1) First, which relation orders this set? could this relation be defined explicitely? is this relation somewhat analogous to lexigraphic order? So I don't understand how this set can be an ordered set.



    (2) Second, I don't understand how it is well-ordered. In order to be well ordered, every subset should have a first element. But apparently the set {7,2} is a subset of my set. What is the first element of {7,2}?










    share|cite|improve this question











    $endgroup$















      2












      2








      2





      $begingroup$


      The set {odd natural numbers greater than 0 }U {even natural numbers}



      that is, the set



      Union { {1,3,5,7...}, {2,4,6,8...} }



      also strangely written



      {1,3,5,7... ; 2,4,6,8...} .



      is often given as example of a well ordered set ( a set such that for all subset there is a first element).



      I have some problems with this example.



      (1) First, which relation orders this set? could this relation be defined explicitely? is this relation somewhat analogous to lexigraphic order? So I don't understand how this set can be an ordered set.



      (2) Second, I don't understand how it is well-ordered. In order to be well ordered, every subset should have a first element. But apparently the set {7,2} is a subset of my set. What is the first element of {7,2}?










      share|cite|improve this question











      $endgroup$




      The set {odd natural numbers greater than 0 }U {even natural numbers}



      that is, the set



      Union { {1,3,5,7...}, {2,4,6,8...} }



      also strangely written



      {1,3,5,7... ; 2,4,6,8...} .



      is often given as example of a well ordered set ( a set such that for all subset there is a first element).



      I have some problems with this example.



      (1) First, which relation orders this set? could this relation be defined explicitely? is this relation somewhat analogous to lexigraphic order? So I don't understand how this set can be an ordered set.



      (2) Second, I don't understand how it is well-ordered. In order to be well ordered, every subset should have a first element. But apparently the set {7,2} is a subset of my set. What is the first element of {7,2}?







      elementary-number-theory elementary-set-theory order-theory






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited 1 hour ago







      Ray LittleRock

















      asked 1 hour ago









      Ray LittleRockRay LittleRock

      12110




      12110






















          3 Answers
          3






          active

          oldest

          votes


















          3












          $begingroup$

          As suggested, a well-ordered set is one in which each subset has a first element. The natural numbers $mathbb{N}$ with their usual relation $le$ form a prototypical example:



          $$0 < 1 < 2 < 3 < 4 < 5 < cdots$$



          however what I believe your example, written as



          $${ 1, 3, 5, 7, cdots ; 2, 4, 6, 8, cdots }$$



          is supposed to indicate (what book or other reference is this from?) is a well-ordered set in which the order looks like this:



          $$1 < 3 < 5 < 7 < cdots < 2 < 4 < 6 < 8 < cdots$$



          in other words, it's all the odd numbers ordered in the usual way, and then after ALL of them, all the even numbers, again, ordered in the usual way. The trick is that every odd number is "less" than every even number in this new ordering, and thus shows a well-order which is different from the usual ordering of the natural numbers. It's meant to show that there are more forms a well-order can take than one might at first expect. In particular, it is a well-order which has an "internal infinity" in that there are an infinite number of elements between any odd number and any even number.



          With regard to the question of the initial element of the subset ${7, 2}$, the answer is simple: looking at the sequence as shown above and where 7 and 2 fall therein, it is seen at a glance to be 7 (not 2). In terms of the mathematical definition, since 7 is odd, and 2 is even, 2 must be "greater" than every odd number, hence also greater than 7 and 7 is the initial element.






          share|cite|improve this answer











          $endgroup$





















            3












            $begingroup$

            Sets have no "inherent order". They can have a natural order (such as $Bbb N$ or $Bbb R$ having a natural ordering that we consider somehow "part of the set"), but they do not have an inherent ordering.



            Writing ${1,3,5,dots;0,2,4,dots}$ is a terrible abuse of notation.



            But it is indeed a well-ordering of $Bbb N$ which is not its natural ordering (which is also a well-ordering). Here is an explicit definition:



            $$mprec niff begin{cases}mtext{ is odd and }ntext{ is even}, &text{ or}\mequiv npmod 2text{ and }m<n.end{cases}$$



            To see it is a well-ordering, note that each of the subsets, evens and odds, is ordered in the natural way which makes that part a well-order, and so given a non-empty set, it either has a smallest odd number or it is a subset of the even numbers.






            share|cite|improve this answer











            $endgroup$









            • 1




              $begingroup$
              @AsafKalaglia. Thanks a lot! So in the context of my question, putting first odd numbers and second even ones, the first element of { 7,2} is 7, am I right?
              $endgroup$
              – Ray LittleRock
              52 mins ago






            • 1




              $begingroup$
              Yes. I misremembered between reading the question and writing my answer (mainly because whenever I gave that example, I would put the even numbers first. I'll edit to fix this.)
              $endgroup$
              – Asaf Karagila
              52 mins ago










            • $begingroup$
              @AsafKalaglia. The set ( with its abusive notation) is given in my book as the union of {odds} and {even} to show, as example, that the union of a family of WOS is also a WOS. Does it really qualify as such a union? As noted JoséCarlosSantos in a previous answer, normally the union of these two sets is simply N with its natural ordering.
              $endgroup$
              – Ray LittleRock
              49 mins ago








            • 1




              $begingroup$
              I don't know who wrote the book. There is a notion of summation of orders, and the sum of well-orders over a well-ordered index, is again a well-order. My guess is that this was their meaning.
              $endgroup$
              – Asaf Karagila
              48 mins ago










            • $begingroup$
              @AsafKalaglia.The reference is Schaum's Outline of Set Theory.
              $endgroup$
              – Ray LittleRock
              45 mins ago



















            1












            $begingroup$

            The set that you described is simply $mathbb N$, the set of all natural numbers. The order relation that we use here is the usual one. And the first element of ${7,2}$ is $2$.






            share|cite|improve this answer









            $endgroup$













            • $begingroup$
              @JoséCarlosSantos.Thanks. Could you also explain the following notation for this set ( a notation I have encountered) : {1,3,5... ; 2,4,6...}
              $endgroup$
              – Ray LittleRock
              1 hour ago








            • 2




              $begingroup$
              No, I cannot; I have never seen this notation.
              $endgroup$
              – José Carlos Santos
              1 hour ago










            • $begingroup$
              @JoséCarlosSantos.Thanks for your help.
              $endgroup$
              – Ray LittleRock
              1 hour ago










            • $begingroup$
              I let the question live its life for some hours, and after that, I promise I will mark an answer as accepted.
              $endgroup$
              – Ray LittleRock
              1 hour ago












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            3 Answers
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            active

            oldest

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            3 Answers
            3






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes









            3












            $begingroup$

            As suggested, a well-ordered set is one in which each subset has a first element. The natural numbers $mathbb{N}$ with their usual relation $le$ form a prototypical example:



            $$0 < 1 < 2 < 3 < 4 < 5 < cdots$$



            however what I believe your example, written as



            $${ 1, 3, 5, 7, cdots ; 2, 4, 6, 8, cdots }$$



            is supposed to indicate (what book or other reference is this from?) is a well-ordered set in which the order looks like this:



            $$1 < 3 < 5 < 7 < cdots < 2 < 4 < 6 < 8 < cdots$$



            in other words, it's all the odd numbers ordered in the usual way, and then after ALL of them, all the even numbers, again, ordered in the usual way. The trick is that every odd number is "less" than every even number in this new ordering, and thus shows a well-order which is different from the usual ordering of the natural numbers. It's meant to show that there are more forms a well-order can take than one might at first expect. In particular, it is a well-order which has an "internal infinity" in that there are an infinite number of elements between any odd number and any even number.



            With regard to the question of the initial element of the subset ${7, 2}$, the answer is simple: looking at the sequence as shown above and where 7 and 2 fall therein, it is seen at a glance to be 7 (not 2). In terms of the mathematical definition, since 7 is odd, and 2 is even, 2 must be "greater" than every odd number, hence also greater than 7 and 7 is the initial element.






            share|cite|improve this answer











            $endgroup$


















              3












              $begingroup$

              As suggested, a well-ordered set is one in which each subset has a first element. The natural numbers $mathbb{N}$ with their usual relation $le$ form a prototypical example:



              $$0 < 1 < 2 < 3 < 4 < 5 < cdots$$



              however what I believe your example, written as



              $${ 1, 3, 5, 7, cdots ; 2, 4, 6, 8, cdots }$$



              is supposed to indicate (what book or other reference is this from?) is a well-ordered set in which the order looks like this:



              $$1 < 3 < 5 < 7 < cdots < 2 < 4 < 6 < 8 < cdots$$



              in other words, it's all the odd numbers ordered in the usual way, and then after ALL of them, all the even numbers, again, ordered in the usual way. The trick is that every odd number is "less" than every even number in this new ordering, and thus shows a well-order which is different from the usual ordering of the natural numbers. It's meant to show that there are more forms a well-order can take than one might at first expect. In particular, it is a well-order which has an "internal infinity" in that there are an infinite number of elements between any odd number and any even number.



              With regard to the question of the initial element of the subset ${7, 2}$, the answer is simple: looking at the sequence as shown above and where 7 and 2 fall therein, it is seen at a glance to be 7 (not 2). In terms of the mathematical definition, since 7 is odd, and 2 is even, 2 must be "greater" than every odd number, hence also greater than 7 and 7 is the initial element.






              share|cite|improve this answer











              $endgroup$
















                3












                3








                3





                $begingroup$

                As suggested, a well-ordered set is one in which each subset has a first element. The natural numbers $mathbb{N}$ with their usual relation $le$ form a prototypical example:



                $$0 < 1 < 2 < 3 < 4 < 5 < cdots$$



                however what I believe your example, written as



                $${ 1, 3, 5, 7, cdots ; 2, 4, 6, 8, cdots }$$



                is supposed to indicate (what book or other reference is this from?) is a well-ordered set in which the order looks like this:



                $$1 < 3 < 5 < 7 < cdots < 2 < 4 < 6 < 8 < cdots$$



                in other words, it's all the odd numbers ordered in the usual way, and then after ALL of them, all the even numbers, again, ordered in the usual way. The trick is that every odd number is "less" than every even number in this new ordering, and thus shows a well-order which is different from the usual ordering of the natural numbers. It's meant to show that there are more forms a well-order can take than one might at first expect. In particular, it is a well-order which has an "internal infinity" in that there are an infinite number of elements between any odd number and any even number.



                With regard to the question of the initial element of the subset ${7, 2}$, the answer is simple: looking at the sequence as shown above and where 7 and 2 fall therein, it is seen at a glance to be 7 (not 2). In terms of the mathematical definition, since 7 is odd, and 2 is even, 2 must be "greater" than every odd number, hence also greater than 7 and 7 is the initial element.






                share|cite|improve this answer











                $endgroup$



                As suggested, a well-ordered set is one in which each subset has a first element. The natural numbers $mathbb{N}$ with their usual relation $le$ form a prototypical example:



                $$0 < 1 < 2 < 3 < 4 < 5 < cdots$$



                however what I believe your example, written as



                $${ 1, 3, 5, 7, cdots ; 2, 4, 6, 8, cdots }$$



                is supposed to indicate (what book or other reference is this from?) is a well-ordered set in which the order looks like this:



                $$1 < 3 < 5 < 7 < cdots < 2 < 4 < 6 < 8 < cdots$$



                in other words, it's all the odd numbers ordered in the usual way, and then after ALL of them, all the even numbers, again, ordered in the usual way. The trick is that every odd number is "less" than every even number in this new ordering, and thus shows a well-order which is different from the usual ordering of the natural numbers. It's meant to show that there are more forms a well-order can take than one might at first expect. In particular, it is a well-order which has an "internal infinity" in that there are an infinite number of elements between any odd number and any even number.



                With regard to the question of the initial element of the subset ${7, 2}$, the answer is simple: looking at the sequence as shown above and where 7 and 2 fall therein, it is seen at a glance to be 7 (not 2). In terms of the mathematical definition, since 7 is odd, and 2 is even, 2 must be "greater" than every odd number, hence also greater than 7 and 7 is the initial element.







                share|cite|improve this answer














                share|cite|improve this answer



                share|cite|improve this answer








                edited 53 mins ago

























                answered 58 mins ago









                The_SympathizerThe_Sympathizer

                8,0152246




                8,0152246























                    3












                    $begingroup$

                    Sets have no "inherent order". They can have a natural order (such as $Bbb N$ or $Bbb R$ having a natural ordering that we consider somehow "part of the set"), but they do not have an inherent ordering.



                    Writing ${1,3,5,dots;0,2,4,dots}$ is a terrible abuse of notation.



                    But it is indeed a well-ordering of $Bbb N$ which is not its natural ordering (which is also a well-ordering). Here is an explicit definition:



                    $$mprec niff begin{cases}mtext{ is odd and }ntext{ is even}, &text{ or}\mequiv npmod 2text{ and }m<n.end{cases}$$



                    To see it is a well-ordering, note that each of the subsets, evens and odds, is ordered in the natural way which makes that part a well-order, and so given a non-empty set, it either has a smallest odd number or it is a subset of the even numbers.






                    share|cite|improve this answer











                    $endgroup$









                    • 1




                      $begingroup$
                      @AsafKalaglia. Thanks a lot! So in the context of my question, putting first odd numbers and second even ones, the first element of { 7,2} is 7, am I right?
                      $endgroup$
                      – Ray LittleRock
                      52 mins ago






                    • 1




                      $begingroup$
                      Yes. I misremembered between reading the question and writing my answer (mainly because whenever I gave that example, I would put the even numbers first. I'll edit to fix this.)
                      $endgroup$
                      – Asaf Karagila
                      52 mins ago










                    • $begingroup$
                      @AsafKalaglia. The set ( with its abusive notation) is given in my book as the union of {odds} and {even} to show, as example, that the union of a family of WOS is also a WOS. Does it really qualify as such a union? As noted JoséCarlosSantos in a previous answer, normally the union of these two sets is simply N with its natural ordering.
                      $endgroup$
                      – Ray LittleRock
                      49 mins ago








                    • 1




                      $begingroup$
                      I don't know who wrote the book. There is a notion of summation of orders, and the sum of well-orders over a well-ordered index, is again a well-order. My guess is that this was their meaning.
                      $endgroup$
                      – Asaf Karagila
                      48 mins ago










                    • $begingroup$
                      @AsafKalaglia.The reference is Schaum's Outline of Set Theory.
                      $endgroup$
                      – Ray LittleRock
                      45 mins ago
















                    3












                    $begingroup$

                    Sets have no "inherent order". They can have a natural order (such as $Bbb N$ or $Bbb R$ having a natural ordering that we consider somehow "part of the set"), but they do not have an inherent ordering.



                    Writing ${1,3,5,dots;0,2,4,dots}$ is a terrible abuse of notation.



                    But it is indeed a well-ordering of $Bbb N$ which is not its natural ordering (which is also a well-ordering). Here is an explicit definition:



                    $$mprec niff begin{cases}mtext{ is odd and }ntext{ is even}, &text{ or}\mequiv npmod 2text{ and }m<n.end{cases}$$



                    To see it is a well-ordering, note that each of the subsets, evens and odds, is ordered in the natural way which makes that part a well-order, and so given a non-empty set, it either has a smallest odd number or it is a subset of the even numbers.






                    share|cite|improve this answer











                    $endgroup$









                    • 1




                      $begingroup$
                      @AsafKalaglia. Thanks a lot! So in the context of my question, putting first odd numbers and second even ones, the first element of { 7,2} is 7, am I right?
                      $endgroup$
                      – Ray LittleRock
                      52 mins ago






                    • 1




                      $begingroup$
                      Yes. I misremembered between reading the question and writing my answer (mainly because whenever I gave that example, I would put the even numbers first. I'll edit to fix this.)
                      $endgroup$
                      – Asaf Karagila
                      52 mins ago










                    • $begingroup$
                      @AsafKalaglia. The set ( with its abusive notation) is given in my book as the union of {odds} and {even} to show, as example, that the union of a family of WOS is also a WOS. Does it really qualify as such a union? As noted JoséCarlosSantos in a previous answer, normally the union of these two sets is simply N with its natural ordering.
                      $endgroup$
                      – Ray LittleRock
                      49 mins ago








                    • 1




                      $begingroup$
                      I don't know who wrote the book. There is a notion of summation of orders, and the sum of well-orders over a well-ordered index, is again a well-order. My guess is that this was their meaning.
                      $endgroup$
                      – Asaf Karagila
                      48 mins ago










                    • $begingroup$
                      @AsafKalaglia.The reference is Schaum's Outline of Set Theory.
                      $endgroup$
                      – Ray LittleRock
                      45 mins ago














                    3












                    3








                    3





                    $begingroup$

                    Sets have no "inherent order". They can have a natural order (such as $Bbb N$ or $Bbb R$ having a natural ordering that we consider somehow "part of the set"), but they do not have an inherent ordering.



                    Writing ${1,3,5,dots;0,2,4,dots}$ is a terrible abuse of notation.



                    But it is indeed a well-ordering of $Bbb N$ which is not its natural ordering (which is also a well-ordering). Here is an explicit definition:



                    $$mprec niff begin{cases}mtext{ is odd and }ntext{ is even}, &text{ or}\mequiv npmod 2text{ and }m<n.end{cases}$$



                    To see it is a well-ordering, note that each of the subsets, evens and odds, is ordered in the natural way which makes that part a well-order, and so given a non-empty set, it either has a smallest odd number or it is a subset of the even numbers.






                    share|cite|improve this answer











                    $endgroup$



                    Sets have no "inherent order". They can have a natural order (such as $Bbb N$ or $Bbb R$ having a natural ordering that we consider somehow "part of the set"), but they do not have an inherent ordering.



                    Writing ${1,3,5,dots;0,2,4,dots}$ is a terrible abuse of notation.



                    But it is indeed a well-ordering of $Bbb N$ which is not its natural ordering (which is also a well-ordering). Here is an explicit definition:



                    $$mprec niff begin{cases}mtext{ is odd and }ntext{ is even}, &text{ or}\mequiv npmod 2text{ and }m<n.end{cases}$$



                    To see it is a well-ordering, note that each of the subsets, evens and odds, is ordered in the natural way which makes that part a well-order, and so given a non-empty set, it either has a smallest odd number or it is a subset of the even numbers.







                    share|cite|improve this answer














                    share|cite|improve this answer



                    share|cite|improve this answer








                    edited 50 mins ago

























                    answered 1 hour ago









                    Asaf KaragilaAsaf Karagila

                    309k33442776




                    309k33442776








                    • 1




                      $begingroup$
                      @AsafKalaglia. Thanks a lot! So in the context of my question, putting first odd numbers and second even ones, the first element of { 7,2} is 7, am I right?
                      $endgroup$
                      – Ray LittleRock
                      52 mins ago






                    • 1




                      $begingroup$
                      Yes. I misremembered between reading the question and writing my answer (mainly because whenever I gave that example, I would put the even numbers first. I'll edit to fix this.)
                      $endgroup$
                      – Asaf Karagila
                      52 mins ago










                    • $begingroup$
                      @AsafKalaglia. The set ( with its abusive notation) is given in my book as the union of {odds} and {even} to show, as example, that the union of a family of WOS is also a WOS. Does it really qualify as such a union? As noted JoséCarlosSantos in a previous answer, normally the union of these two sets is simply N with its natural ordering.
                      $endgroup$
                      – Ray LittleRock
                      49 mins ago








                    • 1




                      $begingroup$
                      I don't know who wrote the book. There is a notion of summation of orders, and the sum of well-orders over a well-ordered index, is again a well-order. My guess is that this was their meaning.
                      $endgroup$
                      – Asaf Karagila
                      48 mins ago










                    • $begingroup$
                      @AsafKalaglia.The reference is Schaum's Outline of Set Theory.
                      $endgroup$
                      – Ray LittleRock
                      45 mins ago














                    • 1




                      $begingroup$
                      @AsafKalaglia. Thanks a lot! So in the context of my question, putting first odd numbers and second even ones, the first element of { 7,2} is 7, am I right?
                      $endgroup$
                      – Ray LittleRock
                      52 mins ago






                    • 1




                      $begingroup$
                      Yes. I misremembered between reading the question and writing my answer (mainly because whenever I gave that example, I would put the even numbers first. I'll edit to fix this.)
                      $endgroup$
                      – Asaf Karagila
                      52 mins ago










                    • $begingroup$
                      @AsafKalaglia. The set ( with its abusive notation) is given in my book as the union of {odds} and {even} to show, as example, that the union of a family of WOS is also a WOS. Does it really qualify as such a union? As noted JoséCarlosSantos in a previous answer, normally the union of these two sets is simply N with its natural ordering.
                      $endgroup$
                      – Ray LittleRock
                      49 mins ago








                    • 1




                      $begingroup$
                      I don't know who wrote the book. There is a notion of summation of orders, and the sum of well-orders over a well-ordered index, is again a well-order. My guess is that this was their meaning.
                      $endgroup$
                      – Asaf Karagila
                      48 mins ago










                    • $begingroup$
                      @AsafKalaglia.The reference is Schaum's Outline of Set Theory.
                      $endgroup$
                      – Ray LittleRock
                      45 mins ago








                    1




                    1




                    $begingroup$
                    @AsafKalaglia. Thanks a lot! So in the context of my question, putting first odd numbers and second even ones, the first element of { 7,2} is 7, am I right?
                    $endgroup$
                    – Ray LittleRock
                    52 mins ago




                    $begingroup$
                    @AsafKalaglia. Thanks a lot! So in the context of my question, putting first odd numbers and second even ones, the first element of { 7,2} is 7, am I right?
                    $endgroup$
                    – Ray LittleRock
                    52 mins ago




                    1




                    1




                    $begingroup$
                    Yes. I misremembered between reading the question and writing my answer (mainly because whenever I gave that example, I would put the even numbers first. I'll edit to fix this.)
                    $endgroup$
                    – Asaf Karagila
                    52 mins ago




                    $begingroup$
                    Yes. I misremembered between reading the question and writing my answer (mainly because whenever I gave that example, I would put the even numbers first. I'll edit to fix this.)
                    $endgroup$
                    – Asaf Karagila
                    52 mins ago












                    $begingroup$
                    @AsafKalaglia. The set ( with its abusive notation) is given in my book as the union of {odds} and {even} to show, as example, that the union of a family of WOS is also a WOS. Does it really qualify as such a union? As noted JoséCarlosSantos in a previous answer, normally the union of these two sets is simply N with its natural ordering.
                    $endgroup$
                    – Ray LittleRock
                    49 mins ago






                    $begingroup$
                    @AsafKalaglia. The set ( with its abusive notation) is given in my book as the union of {odds} and {even} to show, as example, that the union of a family of WOS is also a WOS. Does it really qualify as such a union? As noted JoséCarlosSantos in a previous answer, normally the union of these two sets is simply N with its natural ordering.
                    $endgroup$
                    – Ray LittleRock
                    49 mins ago






                    1




                    1




                    $begingroup$
                    I don't know who wrote the book. There is a notion of summation of orders, and the sum of well-orders over a well-ordered index, is again a well-order. My guess is that this was their meaning.
                    $endgroup$
                    – Asaf Karagila
                    48 mins ago




                    $begingroup$
                    I don't know who wrote the book. There is a notion of summation of orders, and the sum of well-orders over a well-ordered index, is again a well-order. My guess is that this was their meaning.
                    $endgroup$
                    – Asaf Karagila
                    48 mins ago












                    $begingroup$
                    @AsafKalaglia.The reference is Schaum's Outline of Set Theory.
                    $endgroup$
                    – Ray LittleRock
                    45 mins ago




                    $begingroup$
                    @AsafKalaglia.The reference is Schaum's Outline of Set Theory.
                    $endgroup$
                    – Ray LittleRock
                    45 mins ago











                    1












                    $begingroup$

                    The set that you described is simply $mathbb N$, the set of all natural numbers. The order relation that we use here is the usual one. And the first element of ${7,2}$ is $2$.






                    share|cite|improve this answer









                    $endgroup$













                    • $begingroup$
                      @JoséCarlosSantos.Thanks. Could you also explain the following notation for this set ( a notation I have encountered) : {1,3,5... ; 2,4,6...}
                      $endgroup$
                      – Ray LittleRock
                      1 hour ago








                    • 2




                      $begingroup$
                      No, I cannot; I have never seen this notation.
                      $endgroup$
                      – José Carlos Santos
                      1 hour ago










                    • $begingroup$
                      @JoséCarlosSantos.Thanks for your help.
                      $endgroup$
                      – Ray LittleRock
                      1 hour ago










                    • $begingroup$
                      I let the question live its life for some hours, and after that, I promise I will mark an answer as accepted.
                      $endgroup$
                      – Ray LittleRock
                      1 hour ago
















                    1












                    $begingroup$

                    The set that you described is simply $mathbb N$, the set of all natural numbers. The order relation that we use here is the usual one. And the first element of ${7,2}$ is $2$.






                    share|cite|improve this answer









                    $endgroup$













                    • $begingroup$
                      @JoséCarlosSantos.Thanks. Could you also explain the following notation for this set ( a notation I have encountered) : {1,3,5... ; 2,4,6...}
                      $endgroup$
                      – Ray LittleRock
                      1 hour ago








                    • 2




                      $begingroup$
                      No, I cannot; I have never seen this notation.
                      $endgroup$
                      – José Carlos Santos
                      1 hour ago










                    • $begingroup$
                      @JoséCarlosSantos.Thanks for your help.
                      $endgroup$
                      – Ray LittleRock
                      1 hour ago










                    • $begingroup$
                      I let the question live its life for some hours, and after that, I promise I will mark an answer as accepted.
                      $endgroup$
                      – Ray LittleRock
                      1 hour ago














                    1












                    1








                    1





                    $begingroup$

                    The set that you described is simply $mathbb N$, the set of all natural numbers. The order relation that we use here is the usual one. And the first element of ${7,2}$ is $2$.






                    share|cite|improve this answer









                    $endgroup$



                    The set that you described is simply $mathbb N$, the set of all natural numbers. The order relation that we use here is the usual one. And the first element of ${7,2}$ is $2$.







                    share|cite|improve this answer












                    share|cite|improve this answer



                    share|cite|improve this answer










                    answered 1 hour ago









                    José Carlos SantosJosé Carlos Santos

                    179k24139254




                    179k24139254












                    • $begingroup$
                      @JoséCarlosSantos.Thanks. Could you also explain the following notation for this set ( a notation I have encountered) : {1,3,5... ; 2,4,6...}
                      $endgroup$
                      – Ray LittleRock
                      1 hour ago








                    • 2




                      $begingroup$
                      No, I cannot; I have never seen this notation.
                      $endgroup$
                      – José Carlos Santos
                      1 hour ago










                    • $begingroup$
                      @JoséCarlosSantos.Thanks for your help.
                      $endgroup$
                      – Ray LittleRock
                      1 hour ago










                    • $begingroup$
                      I let the question live its life for some hours, and after that, I promise I will mark an answer as accepted.
                      $endgroup$
                      – Ray LittleRock
                      1 hour ago


















                    • $begingroup$
                      @JoséCarlosSantos.Thanks. Could you also explain the following notation for this set ( a notation I have encountered) : {1,3,5... ; 2,4,6...}
                      $endgroup$
                      – Ray LittleRock
                      1 hour ago








                    • 2




                      $begingroup$
                      No, I cannot; I have never seen this notation.
                      $endgroup$
                      – José Carlos Santos
                      1 hour ago










                    • $begingroup$
                      @JoséCarlosSantos.Thanks for your help.
                      $endgroup$
                      – Ray LittleRock
                      1 hour ago










                    • $begingroup$
                      I let the question live its life for some hours, and after that, I promise I will mark an answer as accepted.
                      $endgroup$
                      – Ray LittleRock
                      1 hour ago
















                    $begingroup$
                    @JoséCarlosSantos.Thanks. Could you also explain the following notation for this set ( a notation I have encountered) : {1,3,5... ; 2,4,6...}
                    $endgroup$
                    – Ray LittleRock
                    1 hour ago






                    $begingroup$
                    @JoséCarlosSantos.Thanks. Could you also explain the following notation for this set ( a notation I have encountered) : {1,3,5... ; 2,4,6...}
                    $endgroup$
                    – Ray LittleRock
                    1 hour ago






                    2




                    2




                    $begingroup$
                    No, I cannot; I have never seen this notation.
                    $endgroup$
                    – José Carlos Santos
                    1 hour ago




                    $begingroup$
                    No, I cannot; I have never seen this notation.
                    $endgroup$
                    – José Carlos Santos
                    1 hour ago












                    $begingroup$
                    @JoséCarlosSantos.Thanks for your help.
                    $endgroup$
                    – Ray LittleRock
                    1 hour ago




                    $begingroup$
                    @JoséCarlosSantos.Thanks for your help.
                    $endgroup$
                    – Ray LittleRock
                    1 hour ago












                    $begingroup$
                    I let the question live its life for some hours, and after that, I promise I will mark an answer as accepted.
                    $endgroup$
                    – Ray LittleRock
                    1 hour ago




                    $begingroup$
                    I let the question live its life for some hours, and after that, I promise I will mark an answer as accepted.
                    $endgroup$
                    – Ray LittleRock
                    1 hour ago


















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