Mixing
Conway Notation for Large Countable Ordinals
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
asked Nov 20 '18 at 19:37
meowzz
93212
add a comment |
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
asked Nov 20 '18 at 19:37
meowzz
93212
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 '18 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 '18 at 21:43
@JDH I am also curious about your thoughts on the matter!
– meowzz
Nov 21 '18 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 '18 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 '18 at 9:08
add a comment |
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
asked Nov 20 '18 at 19:37
meowzz
93212
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
ordinals online-resources surreal-numbers ordinal-analysis
asked Nov 20 '18 at 19:37
meowzz
93212
asked Nov 20 '18 at 19:37
meowzz
93212
asked Nov 20 '18 at 19:37
meowzz
93212
asked Nov 20 '18 at 19:37
meowzz
93212
asked Nov 20 '18 at 19:37
meowzz
93212
93212
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 '18 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 '18 at 21:43
@JDH I am also curious about your thoughts on the matter!
– meowzz
Nov 21 '18 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 '18 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 '18 at 9:08
add a comment |
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 '18 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 '18 at 21:43
@JDH I am also curious about your thoughts on the matter!
– meowzz
Nov 21 '18 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 '18 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 '18 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 '18 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 '18 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 '18 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 '18 at 21:43
@JDH I am also curious about your thoughts on the matter!
– meowzz
Nov 21 '18 at 21:47
@JDH I am also curious about your thoughts on the matter!
– meowzz
Nov 21 '18 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 '18 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 '18 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 '18 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 '18 at 9:08
add a comment |
1 Answer
1
active
oldest
votes
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.
answered Nov 22 '18 at 9:18
nombre
2,634913
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3006787%2fconway-notation-for-large-countable-ordinals%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
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.
answered Nov 22 '18 at 9:18
nombre
2,634913
add a comment |
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.
answered Nov 22 '18 at 9:18
nombre
2,634913
add a comment |
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.
answered Nov 22 '18 at 9:18
nombre
2,634913
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.
answered Nov 22 '18 at 9:18
nombre
2,634913
answered Nov 22 '18 at 9:18
nombre
2,634913
answered Nov 22 '18 at 9:18
nombre
2,634913
answered Nov 22 '18 at 9:18
nombre
2,634913
2,634913
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3006787%2fconway-notation-for-large-countable-ordinals%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
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 '18 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 '18 at 21:43
@JDH I am also curious about your thoughts on the matter!
– meowzz
Nov 21 '18 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 '18 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 '18 at 9:08