How do I prove the anti-symmetry and there is a minimal element for each subset of $alpha$Prove using transfinite induction that if ordinals $alpha$ and $beta$ are countable, then so is $alpha + beta$.Element of ordinal a subset of the same ordinalQuestion about the proof of this lemma: If $alpha$, $beta$ are ordinals, then either $alpha subset beta$ or $beta subset alpha.$For every ordinal $alpha$, there is a cardinal number greater then $alpha$.Prove divisibility is a partial order relation over natural numbersTrichotomy of Ordinals. Is $K$ a set?Transitivity and anti-symmetry of setHow to prove that a subset of an oridnal is an element of the ordinal by transitivity.Ordinal numbers, the Burali-Forti paradox, and anti-foundation axiomsI'm trying to prove that any finite partially ordered set has a minimal element.

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How do I prove the anti-symmetry and there is a minimal element for each subset of $alpha$


Prove using transfinite induction that if ordinals $alpha$ and $beta$ are countable, then so is $alpha + beta$.Element of ordinal a subset of the same ordinalQuestion about the proof of this lemma: If $alpha$, $beta$ are ordinals, then either $alpha subset beta$ or $beta subset alpha.$For every ordinal $alpha$, there is a cardinal number greater then $alpha$.Prove divisibility is a partial order relation over natural numbersTrichotomy of Ordinals. Is $K$ a set?Transitivity and anti-symmetry of setHow to prove that a subset of an oridnal is an element of the ordinal by transitivity.Ordinal numbers, the Burali-Forti paradox, and anti-foundation axiomsI'm trying to prove that any finite partially ordered set has a minimal element.













1












$begingroup$



An ordinal number is a set $alpha$ with the following properties:



(a) If $x,y in alpha,$ then either $xin y$, $yin x$, or $x=y.$



(b)If $yin alpha$ and $xin y$, then $xin alpha$.




Let $alpha$ be an ordinal number. For any two numbers $x$ and $y$ of $alpha$, define $leq$ on $alpha$ by $xleq y$ iff $x=y$ or $xin y$.



Theorem:-
Let $alpha$ is an ordinal number. Then $(alpha,leq)$ is a well ordered set.



Proof:-



I don't have the doubts in the proof of Reflexivity, Transitivity. I could be able to prove each element is comparable in $alpha$.



Doubt 1. anti-symmetric
$xleq y$ and $yleq x$. Our claim is $x=y.$



Case 1 $xleq y$ and $yleq x$ means $x=y$ and $yin x implies xin x$. Which is a paradox. How do I prove the anti-symmetry?



Case 2 $xleq y$ and $yleq x$ means $x=y$ and $y= x implies x= y$.



Case 3 $xleq y$ and $yleq x$ means $xin y$ and $y= x implies xin x$.



Case 4 $xleq y$ and $yleq x$ means $xin y$ and $yin x implies x= y$.



Also How do I prove there is a minimal elemnt for each subset of $alpha$?










share|cite|improve this question











$endgroup$
















    1












    $begingroup$



    An ordinal number is a set $alpha$ with the following properties:



    (a) If $x,y in alpha,$ then either $xin y$, $yin x$, or $x=y.$



    (b)If $yin alpha$ and $xin y$, then $xin alpha$.




    Let $alpha$ be an ordinal number. For any two numbers $x$ and $y$ of $alpha$, define $leq$ on $alpha$ by $xleq y$ iff $x=y$ or $xin y$.



    Theorem:-
    Let $alpha$ is an ordinal number. Then $(alpha,leq)$ is a well ordered set.



    Proof:-



    I don't have the doubts in the proof of Reflexivity, Transitivity. I could be able to prove each element is comparable in $alpha$.



    Doubt 1. anti-symmetric
    $xleq y$ and $yleq x$. Our claim is $x=y.$



    Case 1 $xleq y$ and $yleq x$ means $x=y$ and $yin x implies xin x$. Which is a paradox. How do I prove the anti-symmetry?



    Case 2 $xleq y$ and $yleq x$ means $x=y$ and $y= x implies x= y$.



    Case 3 $xleq y$ and $yleq x$ means $xin y$ and $y= x implies xin x$.



    Case 4 $xleq y$ and $yleq x$ means $xin y$ and $yin x implies x= y$.



    Also How do I prove there is a minimal elemnt for each subset of $alpha$?










    share|cite|improve this question











    $endgroup$














      1












      1








      1





      $begingroup$



      An ordinal number is a set $alpha$ with the following properties:



      (a) If $x,y in alpha,$ then either $xin y$, $yin x$, or $x=y.$



      (b)If $yin alpha$ and $xin y$, then $xin alpha$.




      Let $alpha$ be an ordinal number. For any two numbers $x$ and $y$ of $alpha$, define $leq$ on $alpha$ by $xleq y$ iff $x=y$ or $xin y$.



      Theorem:-
      Let $alpha$ is an ordinal number. Then $(alpha,leq)$ is a well ordered set.



      Proof:-



      I don't have the doubts in the proof of Reflexivity, Transitivity. I could be able to prove each element is comparable in $alpha$.



      Doubt 1. anti-symmetric
      $xleq y$ and $yleq x$. Our claim is $x=y.$



      Case 1 $xleq y$ and $yleq x$ means $x=y$ and $yin x implies xin x$. Which is a paradox. How do I prove the anti-symmetry?



      Case 2 $xleq y$ and $yleq x$ means $x=y$ and $y= x implies x= y$.



      Case 3 $xleq y$ and $yleq x$ means $xin y$ and $y= x implies xin x$.



      Case 4 $xleq y$ and $yleq x$ means $xin y$ and $yin x implies x= y$.



      Also How do I prove there is a minimal elemnt for each subset of $alpha$?










      share|cite|improve this question











      $endgroup$





      An ordinal number is a set $alpha$ with the following properties:



      (a) If $x,y in alpha,$ then either $xin y$, $yin x$, or $x=y.$



      (b)If $yin alpha$ and $xin y$, then $xin alpha$.




      Let $alpha$ be an ordinal number. For any two numbers $x$ and $y$ of $alpha$, define $leq$ on $alpha$ by $xleq y$ iff $x=y$ or $xin y$.



      Theorem:-
      Let $alpha$ is an ordinal number. Then $(alpha,leq)$ is a well ordered set.



      Proof:-



      I don't have the doubts in the proof of Reflexivity, Transitivity. I could be able to prove each element is comparable in $alpha$.



      Doubt 1. anti-symmetric
      $xleq y$ and $yleq x$. Our claim is $x=y.$



      Case 1 $xleq y$ and $yleq x$ means $x=y$ and $yin x implies xin x$. Which is a paradox. How do I prove the anti-symmetry?



      Case 2 $xleq y$ and $yleq x$ means $x=y$ and $y= x implies x= y$.



      Case 3 $xleq y$ and $yleq x$ means $xin y$ and $y= x implies xin x$.



      Case 4 $xleq y$ and $yleq x$ means $xin y$ and $yin x implies x= y$.



      Also How do I prove there is a minimal elemnt for each subset of $alpha$?







      elementary-set-theory ordinals






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Mar 17 at 12:43









      Andrés E. Caicedo

      65.8k8160251




      65.8k8160251










      asked Mar 17 at 11:59









      Math geekMath geek

      50111




      50111




















          2 Answers
          2






          active

          oldest

          votes


















          1












          $begingroup$

          Anti-symmetry may be proven the following way: Let $x, yin alpha$ such that $xleq y$ and $yleq x$. This means that we have
          $$
          xin y text or x = y\
          textand\
          yin x text or y = x
          $$

          So, one possibility is certainly that $x = y$. If $xneq y$, then that means that we must have $xin y$ and $yin x$, which cannot be true (as the set $x, y$ would violate the axiom of foundation / regularity). This proves anti-symmetry.



          As for well-orderedness, take a non-empty subset $Ssubseteq alpha$. By the axiom of foundation, there is an element $xin S$ such that $xcap S = varnothing$. I claim that this $x$ is minimal in $S$.



          Assume, for contradiction, that $x$ is not minimal. Then there is a $yin S$ with $yleq x, yneq x$. This means that $yin x$. But then we have $yin xcap S$, which is a contradiction.






          share|cite|improve this answer









          $endgroup$












          • $begingroup$
            You mean axiom of foundation as Russel's paradox?
            $endgroup$
            – Math geek
            Mar 17 at 12:28






          • 2




            $begingroup$
            @Mathgeek What do you mean? If you want to argue about sets on a fundamental level, you have to be aware of your axiomatic system. I assumed the ZF axioms, because that's the most common one. The axiom of foundation is one of the ZF axioms, and is specifically needed to avoid sets like $xin x$, or as in my case, $xin y, yin x$. You can do set theory with the ZF axioms except foundation, and in that case $xin x$ is entirely possible and not paradoxical at all. Particularily, there is no Russel's paradox in sight anywhere.
            $endgroup$
            – Arthur
            Mar 17 at 12:36



















          1












          $begingroup$

          I preassume that the axiom of regularity is accepted in this context.



          Then consequently it cannot happen that $xin x$ so the cases 1 and 3 will not show up.



          Also the axiom excludes the situation that $xin yin x$ so also case 4 is excluded.



          Our conclusion is then that we are dealing with case 2 as was to be shown.



          Proved is now that $leq$ is an anti-symmetric relation on $alpha$.




          Let $A$ be a non-empty subset of $alpha$.



          According to the axiom this set contains an element $a$ such that all elements of $a$ are not elements of $A$.



          So if $xin A$ then we do not have: $xin a$.



          But what we do have is: $xin avee x=avee ain x$.



          So what remains is: $x=avee ain x$ or equivalently $aleq x$.



          This proves that $a$ serves as least element of $A$ in the order $(alpha,leq)$.






          share|cite|improve this answer









          $endgroup$












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            2 Answers
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            2 Answers
            2






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            active

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            active

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            1












            $begingroup$

            Anti-symmetry may be proven the following way: Let $x, yin alpha$ such that $xleq y$ and $yleq x$. This means that we have
            $$
            xin y text or x = y\
            textand\
            yin x text or y = x
            $$

            So, one possibility is certainly that $x = y$. If $xneq y$, then that means that we must have $xin y$ and $yin x$, which cannot be true (as the set $x, y$ would violate the axiom of foundation / regularity). This proves anti-symmetry.



            As for well-orderedness, take a non-empty subset $Ssubseteq alpha$. By the axiom of foundation, there is an element $xin S$ such that $xcap S = varnothing$. I claim that this $x$ is minimal in $S$.



            Assume, for contradiction, that $x$ is not minimal. Then there is a $yin S$ with $yleq x, yneq x$. This means that $yin x$. But then we have $yin xcap S$, which is a contradiction.






            share|cite|improve this answer









            $endgroup$












            • $begingroup$
              You mean axiom of foundation as Russel's paradox?
              $endgroup$
              – Math geek
              Mar 17 at 12:28






            • 2




              $begingroup$
              @Mathgeek What do you mean? If you want to argue about sets on a fundamental level, you have to be aware of your axiomatic system. I assumed the ZF axioms, because that's the most common one. The axiom of foundation is one of the ZF axioms, and is specifically needed to avoid sets like $xin x$, or as in my case, $xin y, yin x$. You can do set theory with the ZF axioms except foundation, and in that case $xin x$ is entirely possible and not paradoxical at all. Particularily, there is no Russel's paradox in sight anywhere.
              $endgroup$
              – Arthur
              Mar 17 at 12:36
















            1












            $begingroup$

            Anti-symmetry may be proven the following way: Let $x, yin alpha$ such that $xleq y$ and $yleq x$. This means that we have
            $$
            xin y text or x = y\
            textand\
            yin x text or y = x
            $$

            So, one possibility is certainly that $x = y$. If $xneq y$, then that means that we must have $xin y$ and $yin x$, which cannot be true (as the set $x, y$ would violate the axiom of foundation / regularity). This proves anti-symmetry.



            As for well-orderedness, take a non-empty subset $Ssubseteq alpha$. By the axiom of foundation, there is an element $xin S$ such that $xcap S = varnothing$. I claim that this $x$ is minimal in $S$.



            Assume, for contradiction, that $x$ is not minimal. Then there is a $yin S$ with $yleq x, yneq x$. This means that $yin x$. But then we have $yin xcap S$, which is a contradiction.






            share|cite|improve this answer









            $endgroup$












            • $begingroup$
              You mean axiom of foundation as Russel's paradox?
              $endgroup$
              – Math geek
              Mar 17 at 12:28






            • 2




              $begingroup$
              @Mathgeek What do you mean? If you want to argue about sets on a fundamental level, you have to be aware of your axiomatic system. I assumed the ZF axioms, because that's the most common one. The axiom of foundation is one of the ZF axioms, and is specifically needed to avoid sets like $xin x$, or as in my case, $xin y, yin x$. You can do set theory with the ZF axioms except foundation, and in that case $xin x$ is entirely possible and not paradoxical at all. Particularily, there is no Russel's paradox in sight anywhere.
              $endgroup$
              – Arthur
              Mar 17 at 12:36














            1












            1








            1





            $begingroup$

            Anti-symmetry may be proven the following way: Let $x, yin alpha$ such that $xleq y$ and $yleq x$. This means that we have
            $$
            xin y text or x = y\
            textand\
            yin x text or y = x
            $$

            So, one possibility is certainly that $x = y$. If $xneq y$, then that means that we must have $xin y$ and $yin x$, which cannot be true (as the set $x, y$ would violate the axiom of foundation / regularity). This proves anti-symmetry.



            As for well-orderedness, take a non-empty subset $Ssubseteq alpha$. By the axiom of foundation, there is an element $xin S$ such that $xcap S = varnothing$. I claim that this $x$ is minimal in $S$.



            Assume, for contradiction, that $x$ is not minimal. Then there is a $yin S$ with $yleq x, yneq x$. This means that $yin x$. But then we have $yin xcap S$, which is a contradiction.






            share|cite|improve this answer









            $endgroup$



            Anti-symmetry may be proven the following way: Let $x, yin alpha$ such that $xleq y$ and $yleq x$. This means that we have
            $$
            xin y text or x = y\
            textand\
            yin x text or y = x
            $$

            So, one possibility is certainly that $x = y$. If $xneq y$, then that means that we must have $xin y$ and $yin x$, which cannot be true (as the set $x, y$ would violate the axiom of foundation / regularity). This proves anti-symmetry.



            As for well-orderedness, take a non-empty subset $Ssubseteq alpha$. By the axiom of foundation, there is an element $xin S$ such that $xcap S = varnothing$. I claim that this $x$ is minimal in $S$.



            Assume, for contradiction, that $x$ is not minimal. Then there is a $yin S$ with $yleq x, yneq x$. This means that $yin x$. But then we have $yin xcap S$, which is a contradiction.







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered Mar 17 at 12:16









            ArthurArthur

            120k7120203




            120k7120203











            • $begingroup$
              You mean axiom of foundation as Russel's paradox?
              $endgroup$
              – Math geek
              Mar 17 at 12:28






            • 2




              $begingroup$
              @Mathgeek What do you mean? If you want to argue about sets on a fundamental level, you have to be aware of your axiomatic system. I assumed the ZF axioms, because that's the most common one. The axiom of foundation is one of the ZF axioms, and is specifically needed to avoid sets like $xin x$, or as in my case, $xin y, yin x$. You can do set theory with the ZF axioms except foundation, and in that case $xin x$ is entirely possible and not paradoxical at all. Particularily, there is no Russel's paradox in sight anywhere.
              $endgroup$
              – Arthur
              Mar 17 at 12:36

















            • $begingroup$
              You mean axiom of foundation as Russel's paradox?
              $endgroup$
              – Math geek
              Mar 17 at 12:28






            • 2




              $begingroup$
              @Mathgeek What do you mean? If you want to argue about sets on a fundamental level, you have to be aware of your axiomatic system. I assumed the ZF axioms, because that's the most common one. The axiom of foundation is one of the ZF axioms, and is specifically needed to avoid sets like $xin x$, or as in my case, $xin y, yin x$. You can do set theory with the ZF axioms except foundation, and in that case $xin x$ is entirely possible and not paradoxical at all. Particularily, there is no Russel's paradox in sight anywhere.
              $endgroup$
              – Arthur
              Mar 17 at 12:36
















            $begingroup$
            You mean axiom of foundation as Russel's paradox?
            $endgroup$
            – Math geek
            Mar 17 at 12:28




            $begingroup$
            You mean axiom of foundation as Russel's paradox?
            $endgroup$
            – Math geek
            Mar 17 at 12:28




            2




            2




            $begingroup$
            @Mathgeek What do you mean? If you want to argue about sets on a fundamental level, you have to be aware of your axiomatic system. I assumed the ZF axioms, because that's the most common one. The axiom of foundation is one of the ZF axioms, and is specifically needed to avoid sets like $xin x$, or as in my case, $xin y, yin x$. You can do set theory with the ZF axioms except foundation, and in that case $xin x$ is entirely possible and not paradoxical at all. Particularily, there is no Russel's paradox in sight anywhere.
            $endgroup$
            – Arthur
            Mar 17 at 12:36





            $begingroup$
            @Mathgeek What do you mean? If you want to argue about sets on a fundamental level, you have to be aware of your axiomatic system. I assumed the ZF axioms, because that's the most common one. The axiom of foundation is one of the ZF axioms, and is specifically needed to avoid sets like $xin x$, or as in my case, $xin y, yin x$. You can do set theory with the ZF axioms except foundation, and in that case $xin x$ is entirely possible and not paradoxical at all. Particularily, there is no Russel's paradox in sight anywhere.
            $endgroup$
            – Arthur
            Mar 17 at 12:36












            1












            $begingroup$

            I preassume that the axiom of regularity is accepted in this context.



            Then consequently it cannot happen that $xin x$ so the cases 1 and 3 will not show up.



            Also the axiom excludes the situation that $xin yin x$ so also case 4 is excluded.



            Our conclusion is then that we are dealing with case 2 as was to be shown.



            Proved is now that $leq$ is an anti-symmetric relation on $alpha$.




            Let $A$ be a non-empty subset of $alpha$.



            According to the axiom this set contains an element $a$ such that all elements of $a$ are not elements of $A$.



            So if $xin A$ then we do not have: $xin a$.



            But what we do have is: $xin avee x=avee ain x$.



            So what remains is: $x=avee ain x$ or equivalently $aleq x$.



            This proves that $a$ serves as least element of $A$ in the order $(alpha,leq)$.






            share|cite|improve this answer









            $endgroup$

















              1












              $begingroup$

              I preassume that the axiom of regularity is accepted in this context.



              Then consequently it cannot happen that $xin x$ so the cases 1 and 3 will not show up.



              Also the axiom excludes the situation that $xin yin x$ so also case 4 is excluded.



              Our conclusion is then that we are dealing with case 2 as was to be shown.



              Proved is now that $leq$ is an anti-symmetric relation on $alpha$.




              Let $A$ be a non-empty subset of $alpha$.



              According to the axiom this set contains an element $a$ such that all elements of $a$ are not elements of $A$.



              So if $xin A$ then we do not have: $xin a$.



              But what we do have is: $xin avee x=avee ain x$.



              So what remains is: $x=avee ain x$ or equivalently $aleq x$.



              This proves that $a$ serves as least element of $A$ in the order $(alpha,leq)$.






              share|cite|improve this answer









              $endgroup$















                1












                1








                1





                $begingroup$

                I preassume that the axiom of regularity is accepted in this context.



                Then consequently it cannot happen that $xin x$ so the cases 1 and 3 will not show up.



                Also the axiom excludes the situation that $xin yin x$ so also case 4 is excluded.



                Our conclusion is then that we are dealing with case 2 as was to be shown.



                Proved is now that $leq$ is an anti-symmetric relation on $alpha$.




                Let $A$ be a non-empty subset of $alpha$.



                According to the axiom this set contains an element $a$ such that all elements of $a$ are not elements of $A$.



                So if $xin A$ then we do not have: $xin a$.



                But what we do have is: $xin avee x=avee ain x$.



                So what remains is: $x=avee ain x$ or equivalently $aleq x$.



                This proves that $a$ serves as least element of $A$ in the order $(alpha,leq)$.






                share|cite|improve this answer









                $endgroup$



                I preassume that the axiom of regularity is accepted in this context.



                Then consequently it cannot happen that $xin x$ so the cases 1 and 3 will not show up.



                Also the axiom excludes the situation that $xin yin x$ so also case 4 is excluded.



                Our conclusion is then that we are dealing with case 2 as was to be shown.



                Proved is now that $leq$ is an anti-symmetric relation on $alpha$.




                Let $A$ be a non-empty subset of $alpha$.



                According to the axiom this set contains an element $a$ such that all elements of $a$ are not elements of $A$.



                So if $xin A$ then we do not have: $xin a$.



                But what we do have is: $xin avee x=avee ain x$.



                So what remains is: $x=avee ain x$ or equivalently $aleq x$.



                This proves that $a$ serves as least element of $A$ in the order $(alpha,leq)$.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Mar 17 at 12:29









                drhabdrhab

                103k545136




                103k545136



























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