Difference between Kleene's O and the system $S_1$












1












$begingroup$


On pages 207-208, Rogers (Theory of Recursive Functions and Effective Computability) introduces two system of notations for ordinals: the first is what he calls $S_1$ and the second one is Kleene's $mathcal{O}$.



My question is: what is the difference between the two systems?



In the definition of Kleene's $mathcal{O}$, Rogers defines also a partial ordering $<_O$. It seems to me that the only difference is that, when defining the names for a limit ordinal, in the definition of $mathcal{O}$ we have the additional requirement that



$$ forall i, forall jquad i<j Rightarrow (varphi_y(i), varphi_y(j)) text{ is already in} <_O$$



But I don't see how this requirement actually reduces the amount of names for a limit ordinal. I mean, if $gamma$ is a limit ordinal and $(varphi_y(n))_{ninmathbb{N}}$ is an increasing sequence of (names for) ordinals with limit $gamma$, how can it be that we did not already put the pair $(varphi_y(i), varphi_y(j))$ in $<_O$?






computability ordinals transfinite-recursion







share|cite|improve this question









$endgroup$

















    1












    $begingroup$


    On pages 207-208, Rogers (Theory of Recursive Functions and Effective Computability) introduces two system of notations for ordinals: the first is what he calls $S_1$ and the second one is Kleene's $mathcal{O}$.



    My question is: what is the difference between the two systems?



    In the definition of Kleene's $mathcal{O}$, Rogers defines also a partial ordering $<_O$. It seems to me that the only difference is that, when defining the names for a limit ordinal, in the definition of $mathcal{O}$ we have the additional requirement that



    $$ forall i, forall jquad i<j Rightarrow (varphi_y(i), varphi_y(j)) text{ is already in} <_O$$



    But I don't see how this requirement actually reduces the amount of names for a limit ordinal. I mean, if $gamma$ is a limit ordinal and $(varphi_y(n))_{ninmathbb{N}}$ is an increasing sequence of (names for) ordinals with limit $gamma$, how can it be that we did not already put the pair $(varphi_y(i), varphi_y(j))$ in $<_O$?






    computability ordinals transfinite-recursion







    share|cite|improve this question









    $endgroup$















      1












      1








      1





      $begingroup$


      On pages 207-208, Rogers (Theory of Recursive Functions and Effective Computability) introduces two system of notations for ordinals: the first is what he calls $S_1$ and the second one is Kleene's $mathcal{O}$.



      My question is: what is the difference between the two systems?



      In the definition of Kleene's $mathcal{O}$, Rogers defines also a partial ordering $<_O$. It seems to me that the only difference is that, when defining the names for a limit ordinal, in the definition of $mathcal{O}$ we have the additional requirement that



      $$ forall i, forall jquad i<j Rightarrow (varphi_y(i), varphi_y(j)) text{ is already in} <_O$$



      But I don't see how this requirement actually reduces the amount of names for a limit ordinal. I mean, if $gamma$ is a limit ordinal and $(varphi_y(n))_{ninmathbb{N}}$ is an increasing sequence of (names for) ordinals with limit $gamma$, how can it be that we did not already put the pair $(varphi_y(i), varphi_y(j))$ in $<_O$?






      computability ordinals transfinite-recursion







      share|cite|improve this question









      $endgroup$




      On pages 207-208, Rogers (Theory of Recursive Functions and Effective Computability) introduces two system of notations for ordinals: the first is what he calls $S_1$ and the second one is Kleene's $mathcal{O}$.



      My question is: what is the difference between the two systems?



      In the definition of Kleene's $mathcal{O}$, Rogers defines also a partial ordering $<_O$. It seems to me that the only difference is that, when defining the names for a limit ordinal, in the definition of $mathcal{O}$ we have the additional requirement that



      $$ forall i, forall jquad i<j Rightarrow (varphi_y(i), varphi_y(j)) text{ is already in} <_O$$



      But I don't see how this requirement actually reduces the amount of names for a limit ordinal. I mean, if $gamma$ is a limit ordinal and $(varphi_y(n))_{ninmathbb{N}}$ is an increasing sequence of (names for) ordinals with limit $gamma$, how can it be that we did not already put the pair $(varphi_y(i), varphi_y(j))$ in $<_O$?






      computability ordinals transfinite-recursion




      computability ordinals transfinite-recursion






      share|cite|improve this question













      share|cite|improve this question











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      asked Dec 14 '18 at 8:47









      ManlioManlio

      924719




      924719






















          1 Answer
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          $begingroup$

          Short version: the relation "$<_mathcal{O}$" is a proper subrelation of the relation "names a smaller ordinal than."




          The point is the distinction between names and ordinals in determining increasing-ness. Specifically, in general if $a,b$ are names for $alpha,beta$ in the sense of Kleene's $mathcal{O}$ and $alpha<beta$, we need not have $a<_mathcal{O}b$. In particular, we can find a sequence $(varphi_y(n))_{ninmathbb{N}}$ such that




          • each $varphi_y(n)$ is in $mathcal{O}$, and


          • $vertvarphi_y(n)vert<vertvarphi_y(n+1)vert$ for each $ninmathbb{N}$



          (where "$vert avert$" is the ordinal corresponding to the name $a$), but $3cdot 5^y$ is not in $mathcal{O}$ since (say) $varphi_y(0)not<_mathcal{O}varphi_y(1)$.



          In the system $S_1$, we only required that the values of the names be increasing; in $mathcal{O}$, we require that this increasing-ness be witnessed in a very concrete way, namely by the relation $<_mathcal{O}$. This is all in an attempt to make a more "constructive" system, and at this point Kleene's original paper "On notation for ordinal numbers" has a useful paragraph:




          Following Church, a modification is now made in the system $S_1$, which is
          regarded from the finitary viewpoint as a correction, in that it eliminates the
          presupposition of the classical (non-constructive) second number class. The
          modification consists in replacing the ordering relation $<$ between ordinals by a partially ordering relation $<_mathcal{O}$ between the notations.




          (Note that he refers to the system we know and love as $mathcal{O}$ by "$S_2$.")






          share|cite|improve this answer









          $endgroup$













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            1 Answer
            1






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes









            2












            $begingroup$

            Short version: the relation "$<_mathcal{O}$" is a proper subrelation of the relation "names a smaller ordinal than."




            The point is the distinction between names and ordinals in determining increasing-ness. Specifically, in general if $a,b$ are names for $alpha,beta$ in the sense of Kleene's $mathcal{O}$ and $alpha<beta$, we need not have $a<_mathcal{O}b$. In particular, we can find a sequence $(varphi_y(n))_{ninmathbb{N}}$ such that




            • each $varphi_y(n)$ is in $mathcal{O}$, and


            • $vertvarphi_y(n)vert<vertvarphi_y(n+1)vert$ for each $ninmathbb{N}$



            (where "$vert avert$" is the ordinal corresponding to the name $a$), but $3cdot 5^y$ is not in $mathcal{O}$ since (say) $varphi_y(0)not<_mathcal{O}varphi_y(1)$.



            In the system $S_1$, we only required that the values of the names be increasing; in $mathcal{O}$, we require that this increasing-ness be witnessed in a very concrete way, namely by the relation $<_mathcal{O}$. This is all in an attempt to make a more "constructive" system, and at this point Kleene's original paper "On notation for ordinal numbers" has a useful paragraph:




            Following Church, a modification is now made in the system $S_1$, which is
            regarded from the finitary viewpoint as a correction, in that it eliminates the
            presupposition of the classical (non-constructive) second number class. The
            modification consists in replacing the ordering relation $<$ between ordinals by a partially ordering relation $<_mathcal{O}$ between the notations.




            (Note that he refers to the system we know and love as $mathcal{O}$ by "$S_2$.")






            share|cite|improve this answer









            $endgroup$


















              2












              $begingroup$

              Short version: the relation "$<_mathcal{O}$" is a proper subrelation of the relation "names a smaller ordinal than."




              The point is the distinction between names and ordinals in determining increasing-ness. Specifically, in general if $a,b$ are names for $alpha,beta$ in the sense of Kleene's $mathcal{O}$ and $alpha<beta$, we need not have $a<_mathcal{O}b$. In particular, we can find a sequence $(varphi_y(n))_{ninmathbb{N}}$ such that




              • each $varphi_y(n)$ is in $mathcal{O}$, and


              • $vertvarphi_y(n)vert<vertvarphi_y(n+1)vert$ for each $ninmathbb{N}$



              (where "$vert avert$" is the ordinal corresponding to the name $a$), but $3cdot 5^y$ is not in $mathcal{O}$ since (say) $varphi_y(0)not<_mathcal{O}varphi_y(1)$.



              In the system $S_1$, we only required that the values of the names be increasing; in $mathcal{O}$, we require that this increasing-ness be witnessed in a very concrete way, namely by the relation $<_mathcal{O}$. This is all in an attempt to make a more "constructive" system, and at this point Kleene's original paper "On notation for ordinal numbers" has a useful paragraph:




              Following Church, a modification is now made in the system $S_1$, which is
              regarded from the finitary viewpoint as a correction, in that it eliminates the
              presupposition of the classical (non-constructive) second number class. The
              modification consists in replacing the ordering relation $<$ between ordinals by a partially ordering relation $<_mathcal{O}$ between the notations.




              (Note that he refers to the system we know and love as $mathcal{O}$ by "$S_2$.")






              share|cite|improve this answer









              $endgroup$
















                2












                2








                2





                $begingroup$

                Short version: the relation "$<_mathcal{O}$" is a proper subrelation of the relation "names a smaller ordinal than."




                The point is the distinction between names and ordinals in determining increasing-ness. Specifically, in general if $a,b$ are names for $alpha,beta$ in the sense of Kleene's $mathcal{O}$ and $alpha<beta$, we need not have $a<_mathcal{O}b$. In particular, we can find a sequence $(varphi_y(n))_{ninmathbb{N}}$ such that




                • each $varphi_y(n)$ is in $mathcal{O}$, and


                • $vertvarphi_y(n)vert<vertvarphi_y(n+1)vert$ for each $ninmathbb{N}$



                (where "$vert avert$" is the ordinal corresponding to the name $a$), but $3cdot 5^y$ is not in $mathcal{O}$ since (say) $varphi_y(0)not<_mathcal{O}varphi_y(1)$.



                In the system $S_1$, we only required that the values of the names be increasing; in $mathcal{O}$, we require that this increasing-ness be witnessed in a very concrete way, namely by the relation $<_mathcal{O}$. This is all in an attempt to make a more "constructive" system, and at this point Kleene's original paper "On notation for ordinal numbers" has a useful paragraph:




                Following Church, a modification is now made in the system $S_1$, which is
                regarded from the finitary viewpoint as a correction, in that it eliminates the
                presupposition of the classical (non-constructive) second number class. The
                modification consists in replacing the ordering relation $<$ between ordinals by a partially ordering relation $<_mathcal{O}$ between the notations.




                (Note that he refers to the system we know and love as $mathcal{O}$ by "$S_2$.")






                share|cite|improve this answer









                $endgroup$



                Short version: the relation "$<_mathcal{O}$" is a proper subrelation of the relation "names a smaller ordinal than."




                The point is the distinction between names and ordinals in determining increasing-ness. Specifically, in general if $a,b$ are names for $alpha,beta$ in the sense of Kleene's $mathcal{O}$ and $alpha<beta$, we need not have $a<_mathcal{O}b$. In particular, we can find a sequence $(varphi_y(n))_{ninmathbb{N}}$ such that




                • each $varphi_y(n)$ is in $mathcal{O}$, and


                • $vertvarphi_y(n)vert<vertvarphi_y(n+1)vert$ for each $ninmathbb{N}$



                (where "$vert avert$" is the ordinal corresponding to the name $a$), but $3cdot 5^y$ is not in $mathcal{O}$ since (say) $varphi_y(0)not<_mathcal{O}varphi_y(1)$.



                In the system $S_1$, we only required that the values of the names be increasing; in $mathcal{O}$, we require that this increasing-ness be witnessed in a very concrete way, namely by the relation $<_mathcal{O}$. This is all in an attempt to make a more "constructive" system, and at this point Kleene's original paper "On notation for ordinal numbers" has a useful paragraph:




                Following Church, a modification is now made in the system $S_1$, which is
                regarded from the finitary viewpoint as a correction, in that it eliminates the
                presupposition of the classical (non-constructive) second number class. The
                modification consists in replacing the ordering relation $<$ between ordinals by a partially ordering relation $<_mathcal{O}$ between the notations.




                (Note that he refers to the system we know and love as $mathcal{O}$ by "$S_2$.")







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Dec 22 '18 at 20:25









                Noah SchweberNoah Schweber

                126k10151290




                126k10151290






























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