Conway Notation for Large Countable Ordinals











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I have not previously seen anything online that dives deeply into On:




In Conway's notation On denotes the ordinal numbers (and No denotes the set of all surreal Numbers). Basically the elements of On are just von Neumann ordinals. -Source




I would appreciate feedback on the following attempt to write large countable ordinals (& the functions that generate them) in Conway notation (my primary source of information in creating these constructions was Large Countable Ordinals):



epsilon-nought
$$varepsilon_{0}={omega,omega^omega,omega^{omega^omega},...|}$$
veblen function
$$phi={omega,varepsilon_0,zeta_{alpha},...|}$$
veblen hierarchy
$$phi_{gamma}(alpha)={phi_{0}(alpha)=omega^{alpha}, phi_{1}(alpha)=varepsilon_{alpha}, phi_{2}(alpha)=zeta_{alpha}...|}$$
feferman-schutte ordinal
$$Gamma_0=phi_{Gamma_0}(0)={phi_0(0),phi_{phi_0(0)}(0),phi_{phi_{phi_0(0)}(0)}(0),...|}$$
small veblen ordinal
$$SVO={phi_1(0), phi_{1,0}(0), phi_{1,0,0}(0),...|}$$
large veblen ordinal
$$LVO=psi(Omega^{Omega^Omega})={[???]...|}$$
bachmann-howard ordinal
$$BHO=psi(varepsilon_{Omega+1})={psi(Omega),psi(Omega^Omega),psi(Omega^{Omega^Omega})|}$$



Additionally, any online resources related to On would be greatly appreciated.










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  • 2




    In all those cases, the Conway's bracket notation could be replaced by the usual notion of supremum so I don't think there's much to gain here from that.
    – nombre
    Nov 21 at 11:38










  • @nombre I was hoping you might make an appearance! I am a) trying to confirm that, as written, everything (LVO excluded) is correct b) seek sources about On (surordinals?). I would like to eventually be able to perform calculations such as ${varepsilon_0 | varepsilon_0}$, ${SVO|LVO}$, ${Gamma_0|Gamma_1}$, etc.
    – meowzz
    Nov 21 at 21:43










  • @JDH I am also curious about your thoughts on the matter!
    – meowzz
    Nov 21 at 21:47










  • @nombre "the relationship between the two is deep enough for me to imagine that it could be wise at times to view ordinals as surreals to gain insight on ordinals" source
    – meowzz
    Nov 29 at 4:23






  • 2




    I think it could be wise at times yes, and for instance I was able to anwser a question of mine about compositions of ordinals using inspiration from surreal numbers; but I wouldn't qualify this as a significant insight on ordinal numbers that naturally comes from surreal numbers.
    – nombre
    Nov 29 at 9:08















up vote
1
down vote

favorite












I have not previously seen anything online that dives deeply into On:




In Conway's notation On denotes the ordinal numbers (and No denotes the set of all surreal Numbers). Basically the elements of On are just von Neumann ordinals. -Source




I would appreciate feedback on the following attempt to write large countable ordinals (& the functions that generate them) in Conway notation (my primary source of information in creating these constructions was Large Countable Ordinals):



epsilon-nought
$$varepsilon_{0}={omega,omega^omega,omega^{omega^omega},...|}$$
veblen function
$$phi={omega,varepsilon_0,zeta_{alpha},...|}$$
veblen hierarchy
$$phi_{gamma}(alpha)={phi_{0}(alpha)=omega^{alpha}, phi_{1}(alpha)=varepsilon_{alpha}, phi_{2}(alpha)=zeta_{alpha}...|}$$
feferman-schutte ordinal
$$Gamma_0=phi_{Gamma_0}(0)={phi_0(0),phi_{phi_0(0)}(0),phi_{phi_{phi_0(0)}(0)}(0),...|}$$
small veblen ordinal
$$SVO={phi_1(0), phi_{1,0}(0), phi_{1,0,0}(0),...|}$$
large veblen ordinal
$$LVO=psi(Omega^{Omega^Omega})={[???]...|}$$
bachmann-howard ordinal
$$BHO=psi(varepsilon_{Omega+1})={psi(Omega),psi(Omega^Omega),psi(Omega^{Omega^Omega})|}$$



Additionally, any online resources related to On would be greatly appreciated.










share|cite|improve this question


















  • 2




    In all those cases, the Conway's bracket notation could be replaced by the usual notion of supremum so I don't think there's much to gain here from that.
    – nombre
    Nov 21 at 11:38










  • @nombre I was hoping you might make an appearance! I am a) trying to confirm that, as written, everything (LVO excluded) is correct b) seek sources about On (surordinals?). I would like to eventually be able to perform calculations such as ${varepsilon_0 | varepsilon_0}$, ${SVO|LVO}$, ${Gamma_0|Gamma_1}$, etc.
    – meowzz
    Nov 21 at 21:43










  • @JDH I am also curious about your thoughts on the matter!
    – meowzz
    Nov 21 at 21:47










  • @nombre "the relationship between the two is deep enough for me to imagine that it could be wise at times to view ordinals as surreals to gain insight on ordinals" source
    – meowzz
    Nov 29 at 4:23






  • 2




    I think it could be wise at times yes, and for instance I was able to anwser a question of mine about compositions of ordinals using inspiration from surreal numbers; but I wouldn't qualify this as a significant insight on ordinal numbers that naturally comes from surreal numbers.
    – nombre
    Nov 29 at 9:08













up vote
1
down vote

favorite









up vote
1
down vote

favorite











I have not previously seen anything online that dives deeply into On:




In Conway's notation On denotes the ordinal numbers (and No denotes the set of all surreal Numbers). Basically the elements of On are just von Neumann ordinals. -Source




I would appreciate feedback on the following attempt to write large countable ordinals (& the functions that generate them) in Conway notation (my primary source of information in creating these constructions was Large Countable Ordinals):



epsilon-nought
$$varepsilon_{0}={omega,omega^omega,omega^{omega^omega},...|}$$
veblen function
$$phi={omega,varepsilon_0,zeta_{alpha},...|}$$
veblen hierarchy
$$phi_{gamma}(alpha)={phi_{0}(alpha)=omega^{alpha}, phi_{1}(alpha)=varepsilon_{alpha}, phi_{2}(alpha)=zeta_{alpha}...|}$$
feferman-schutte ordinal
$$Gamma_0=phi_{Gamma_0}(0)={phi_0(0),phi_{phi_0(0)}(0),phi_{phi_{phi_0(0)}(0)}(0),...|}$$
small veblen ordinal
$$SVO={phi_1(0), phi_{1,0}(0), phi_{1,0,0}(0),...|}$$
large veblen ordinal
$$LVO=psi(Omega^{Omega^Omega})={[???]...|}$$
bachmann-howard ordinal
$$BHO=psi(varepsilon_{Omega+1})={psi(Omega),psi(Omega^Omega),psi(Omega^{Omega^Omega})|}$$



Additionally, any online resources related to On would be greatly appreciated.










share|cite|improve this question













I have not previously seen anything online that dives deeply into On:




In Conway's notation On denotes the ordinal numbers (and No denotes the set of all surreal Numbers). Basically the elements of On are just von Neumann ordinals. -Source




I would appreciate feedback on the following attempt to write large countable ordinals (& the functions that generate them) in Conway notation (my primary source of information in creating these constructions was Large Countable Ordinals):



epsilon-nought
$$varepsilon_{0}={omega,omega^omega,omega^{omega^omega},...|}$$
veblen function
$$phi={omega,varepsilon_0,zeta_{alpha},...|}$$
veblen hierarchy
$$phi_{gamma}(alpha)={phi_{0}(alpha)=omega^{alpha}, phi_{1}(alpha)=varepsilon_{alpha}, phi_{2}(alpha)=zeta_{alpha}...|}$$
feferman-schutte ordinal
$$Gamma_0=phi_{Gamma_0}(0)={phi_0(0),phi_{phi_0(0)}(0),phi_{phi_{phi_0(0)}(0)}(0),...|}$$
small veblen ordinal
$$SVO={phi_1(0), phi_{1,0}(0), phi_{1,0,0}(0),...|}$$
large veblen ordinal
$$LVO=psi(Omega^{Omega^Omega})={[???]...|}$$
bachmann-howard ordinal
$$BHO=psi(varepsilon_{Omega+1})={psi(Omega),psi(Omega^Omega),psi(Omega^{Omega^Omega})|}$$



Additionally, any online resources related to On would be greatly appreciated.







ordinals online-resources surreal-numbers ordinal-analysis






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asked Nov 20 at 19:37









meowzz

82212




82212








  • 2




    In all those cases, the Conway's bracket notation could be replaced by the usual notion of supremum so I don't think there's much to gain here from that.
    – nombre
    Nov 21 at 11:38










  • @nombre I was hoping you might make an appearance! I am a) trying to confirm that, as written, everything (LVO excluded) is correct b) seek sources about On (surordinals?). I would like to eventually be able to perform calculations such as ${varepsilon_0 | varepsilon_0}$, ${SVO|LVO}$, ${Gamma_0|Gamma_1}$, etc.
    – meowzz
    Nov 21 at 21:43










  • @JDH I am also curious about your thoughts on the matter!
    – meowzz
    Nov 21 at 21:47










  • @nombre "the relationship between the two is deep enough for me to imagine that it could be wise at times to view ordinals as surreals to gain insight on ordinals" source
    – meowzz
    Nov 29 at 4:23






  • 2




    I think it could be wise at times yes, and for instance I was able to anwser a question of mine about compositions of ordinals using inspiration from surreal numbers; but I wouldn't qualify this as a significant insight on ordinal numbers that naturally comes from surreal numbers.
    – nombre
    Nov 29 at 9:08














  • 2




    In all those cases, the Conway's bracket notation could be replaced by the usual notion of supremum so I don't think there's much to gain here from that.
    – nombre
    Nov 21 at 11:38










  • @nombre I was hoping you might make an appearance! I am a) trying to confirm that, as written, everything (LVO excluded) is correct b) seek sources about On (surordinals?). I would like to eventually be able to perform calculations such as ${varepsilon_0 | varepsilon_0}$, ${SVO|LVO}$, ${Gamma_0|Gamma_1}$, etc.
    – meowzz
    Nov 21 at 21:43










  • @JDH I am also curious about your thoughts on the matter!
    – meowzz
    Nov 21 at 21:47










  • @nombre "the relationship between the two is deep enough for me to imagine that it could be wise at times to view ordinals as surreals to gain insight on ordinals" source
    – meowzz
    Nov 29 at 4:23






  • 2




    I think it could be wise at times yes, and for instance I was able to anwser a question of mine about compositions of ordinals using inspiration from surreal numbers; but I wouldn't qualify this as a significant insight on ordinal numbers that naturally comes from surreal numbers.
    – nombre
    Nov 29 at 9:08








2




2




In all those cases, the Conway's bracket notation could be replaced by the usual notion of supremum so I don't think there's much to gain here from that.
– nombre
Nov 21 at 11:38




In all those cases, the Conway's bracket notation could be replaced by the usual notion of supremum so I don't think there's much to gain here from that.
– nombre
Nov 21 at 11:38












@nombre I was hoping you might make an appearance! I am a) trying to confirm that, as written, everything (LVO excluded) is correct b) seek sources about On (surordinals?). I would like to eventually be able to perform calculations such as ${varepsilon_0 | varepsilon_0}$, ${SVO|LVO}$, ${Gamma_0|Gamma_1}$, etc.
– meowzz
Nov 21 at 21:43




@nombre I was hoping you might make an appearance! I am a) trying to confirm that, as written, everything (LVO excluded) is correct b) seek sources about On (surordinals?). I would like to eventually be able to perform calculations such as ${varepsilon_0 | varepsilon_0}$, ${SVO|LVO}$, ${Gamma_0|Gamma_1}$, etc.
– meowzz
Nov 21 at 21:43












@JDH I am also curious about your thoughts on the matter!
– meowzz
Nov 21 at 21:47




@JDH I am also curious about your thoughts on the matter!
– meowzz
Nov 21 at 21:47












@nombre "the relationship between the two is deep enough for me to imagine that it could be wise at times to view ordinals as surreals to gain insight on ordinals" source
– meowzz
Nov 29 at 4:23




@nombre "the relationship between the two is deep enough for me to imagine that it could be wise at times to view ordinals as surreals to gain insight on ordinals" source
– meowzz
Nov 29 at 4:23




2




2




I think it could be wise at times yes, and for instance I was able to anwser a question of mine about compositions of ordinals using inspiration from surreal numbers; but I wouldn't qualify this as a significant insight on ordinal numbers that naturally comes from surreal numbers.
– nombre
Nov 29 at 9:08




I think it could be wise at times yes, and for instance I was able to anwser a question of mine about compositions of ordinals using inspiration from surreal numbers; but I wouldn't qualify this as a significant insight on ordinal numbers that naturally comes from surreal numbers.
– nombre
Nov 29 at 9:08










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up vote
3
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I have to say I know as little about large countable ordinals as I do about games. I actually didn't know the ordinal $Gamma_0$ was thought of as the first impredicative ordinal, had a name and so on.



Regarding your definitions, the function $phi_{gamma}(alpha)$ should also be greater than every ordinal $phi_{eta}^{circ n}(phi_{gamma}(beta)+1)$ for $eta < gamma$, $n in mathbb{N}$ and $beta<alpha$. I am not sure about what you mean by Vleben function, and I don't know about SVO, LVO, BHO.



Perhaps something you might find interesting is a phenomenon noticed by Conway and expended upon by Gonshor: the functions $phi_{gamma}$ can be extended to $mathbf{No}$ in a natural way.



For $x={L | R} in mathbf{No}$, you must know about $omega^x=phi_0(x)={0,mathbb{N} phi_0(L) | 2^{-mathbb{N}} phi_0(R)}$.
Then the class of numbers $e$ such that $omega^e=e$ is parametrized by $varepsilon_x=phi_1(x):={phi_0^{circ mathbb{N}}(0),phi_0^{circ mathbb{N}}(phi_1(L)+1) | phi_0^{circ mathbb{N}}(phi_1(R)-1)}$,
and one can keep going on. At every stage $0<gamma$, the function $phi_{gamma}$ parametrizes the class of numbers $e$ with $forall eta < gamma,phi_{eta}(e)=e$.



As for sources on $mathbf{On}$, since this is just the class of ordinals, you can just look into this. I don't know that new insight on ordinal numbers has been gained by seeing them as surreal numbers, at least not in a significant way.






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    down vote













    I have to say I know as little about large countable ordinals as I do about games. I actually didn't know the ordinal $Gamma_0$ was thought of as the first impredicative ordinal, had a name and so on.



    Regarding your definitions, the function $phi_{gamma}(alpha)$ should also be greater than every ordinal $phi_{eta}^{circ n}(phi_{gamma}(beta)+1)$ for $eta < gamma$, $n in mathbb{N}$ and $beta<alpha$. I am not sure about what you mean by Vleben function, and I don't know about SVO, LVO, BHO.



    Perhaps something you might find interesting is a phenomenon noticed by Conway and expended upon by Gonshor: the functions $phi_{gamma}$ can be extended to $mathbf{No}$ in a natural way.



    For $x={L | R} in mathbf{No}$, you must know about $omega^x=phi_0(x)={0,mathbb{N} phi_0(L) | 2^{-mathbb{N}} phi_0(R)}$.
    Then the class of numbers $e$ such that $omega^e=e$ is parametrized by $varepsilon_x=phi_1(x):={phi_0^{circ mathbb{N}}(0),phi_0^{circ mathbb{N}}(phi_1(L)+1) | phi_0^{circ mathbb{N}}(phi_1(R)-1)}$,
    and one can keep going on. At every stage $0<gamma$, the function $phi_{gamma}$ parametrizes the class of numbers $e$ with $forall eta < gamma,phi_{eta}(e)=e$.



    As for sources on $mathbf{On}$, since this is just the class of ordinals, you can just look into this. I don't know that new insight on ordinal numbers has been gained by seeing them as surreal numbers, at least not in a significant way.






    share|cite|improve this answer

























      up vote
      3
      down vote













      I have to say I know as little about large countable ordinals as I do about games. I actually didn't know the ordinal $Gamma_0$ was thought of as the first impredicative ordinal, had a name and so on.



      Regarding your definitions, the function $phi_{gamma}(alpha)$ should also be greater than every ordinal $phi_{eta}^{circ n}(phi_{gamma}(beta)+1)$ for $eta < gamma$, $n in mathbb{N}$ and $beta<alpha$. I am not sure about what you mean by Vleben function, and I don't know about SVO, LVO, BHO.



      Perhaps something you might find interesting is a phenomenon noticed by Conway and expended upon by Gonshor: the functions $phi_{gamma}$ can be extended to $mathbf{No}$ in a natural way.



      For $x={L | R} in mathbf{No}$, you must know about $omega^x=phi_0(x)={0,mathbb{N} phi_0(L) | 2^{-mathbb{N}} phi_0(R)}$.
      Then the class of numbers $e$ such that $omega^e=e$ is parametrized by $varepsilon_x=phi_1(x):={phi_0^{circ mathbb{N}}(0),phi_0^{circ mathbb{N}}(phi_1(L)+1) | phi_0^{circ mathbb{N}}(phi_1(R)-1)}$,
      and one can keep going on. At every stage $0<gamma$, the function $phi_{gamma}$ parametrizes the class of numbers $e$ with $forall eta < gamma,phi_{eta}(e)=e$.



      As for sources on $mathbf{On}$, since this is just the class of ordinals, you can just look into this. I don't know that new insight on ordinal numbers has been gained by seeing them as surreal numbers, at least not in a significant way.






      share|cite|improve this answer























        up vote
        3
        down vote










        up vote
        3
        down vote









        I have to say I know as little about large countable ordinals as I do about games. I actually didn't know the ordinal $Gamma_0$ was thought of as the first impredicative ordinal, had a name and so on.



        Regarding your definitions, the function $phi_{gamma}(alpha)$ should also be greater than every ordinal $phi_{eta}^{circ n}(phi_{gamma}(beta)+1)$ for $eta < gamma$, $n in mathbb{N}$ and $beta<alpha$. I am not sure about what you mean by Vleben function, and I don't know about SVO, LVO, BHO.



        Perhaps something you might find interesting is a phenomenon noticed by Conway and expended upon by Gonshor: the functions $phi_{gamma}$ can be extended to $mathbf{No}$ in a natural way.



        For $x={L | R} in mathbf{No}$, you must know about $omega^x=phi_0(x)={0,mathbb{N} phi_0(L) | 2^{-mathbb{N}} phi_0(R)}$.
        Then the class of numbers $e$ such that $omega^e=e$ is parametrized by $varepsilon_x=phi_1(x):={phi_0^{circ mathbb{N}}(0),phi_0^{circ mathbb{N}}(phi_1(L)+1) | phi_0^{circ mathbb{N}}(phi_1(R)-1)}$,
        and one can keep going on. At every stage $0<gamma$, the function $phi_{gamma}$ parametrizes the class of numbers $e$ with $forall eta < gamma,phi_{eta}(e)=e$.



        As for sources on $mathbf{On}$, since this is just the class of ordinals, you can just look into this. I don't know that new insight on ordinal numbers has been gained by seeing them as surreal numbers, at least not in a significant way.






        share|cite|improve this answer












        I have to say I know as little about large countable ordinals as I do about games. I actually didn't know the ordinal $Gamma_0$ was thought of as the first impredicative ordinal, had a name and so on.



        Regarding your definitions, the function $phi_{gamma}(alpha)$ should also be greater than every ordinal $phi_{eta}^{circ n}(phi_{gamma}(beta)+1)$ for $eta < gamma$, $n in mathbb{N}$ and $beta<alpha$. I am not sure about what you mean by Vleben function, and I don't know about SVO, LVO, BHO.



        Perhaps something you might find interesting is a phenomenon noticed by Conway and expended upon by Gonshor: the functions $phi_{gamma}$ can be extended to $mathbf{No}$ in a natural way.



        For $x={L | R} in mathbf{No}$, you must know about $omega^x=phi_0(x)={0,mathbb{N} phi_0(L) | 2^{-mathbb{N}} phi_0(R)}$.
        Then the class of numbers $e$ such that $omega^e=e$ is parametrized by $varepsilon_x=phi_1(x):={phi_0^{circ mathbb{N}}(0),phi_0^{circ mathbb{N}}(phi_1(L)+1) | phi_0^{circ mathbb{N}}(phi_1(R)-1)}$,
        and one can keep going on. At every stage $0<gamma$, the function $phi_{gamma}$ parametrizes the class of numbers $e$ with $forall eta < gamma,phi_{eta}(e)=e$.



        As for sources on $mathbf{On}$, since this is just the class of ordinals, you can just look into this. I don't know that new insight on ordinal numbers has been gained by seeing them as surreal numbers, at least not in a significant way.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Nov 22 at 9:18









        nombre

        2,559913




        2,559913






























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