@Frpzzd oops...

@Secret lol

Has been improved quite a bit

o/

$$C_0(\alpha, \xi,\pi)=\{0\} \cup\xi\\C_{n+ 1}(\alpha,\xi, \pi)=\{\gamma+\delta, \Omega_\mu,\vartheta_\gamma(\eta)~|~ \gamma,\delta,\mu,\eta \in C_n(\alpha,\xi,\pi)\land \pi\in\zeta\land\eta \in\alpha\}\\C(\alpha ,\xi,\pi)=\bigcup_{n\in \omega}C_n(\alpha,\xi, \pi)\\\rm closure(A) =A\cup\{\sup B ~|~B\subset A\} \\\vartheta_\pi(\alpha )=C(\alpha,\min\{\xi~|~ \alpha\in{\rm closure}(C(\alpha,\xi, \pi))\},\pi)\cap \Omega_{\pi+1}$$

@Secret @user21820 This creates a function which behaves like the Veblen fubction and an ordinal collapsing function =)

It starts off with $$\vartheta_\pi(\alpha<\Omega_{\pi+1})=\omega^\alpha$$

And indeed I believe it has the relationship of $$\vartheta_\pi(\Omega_{\pi+1}^{ \gamma_n}\cdot\delta_n +\dots+\Omega_{\pi+1}^{ \gamma_0}\cdot\delta_0 )=\varphi( \delta_n\text @\gamma_n ,\dots,\delta_0\text @ \gamma_0)\\\gamma,\delta<\Omega_{\pi+1} \\\gamma_n>\dots>\gamma_0$$

Where $\varphi$ is the Veblen function with infinitely many arguments and we have things like $$\begin{align} \varphi(1\text @\omega)&=\sup\{\varphi(1), \varphi(1,0), \varphi(1,0,0), \dots\}\\ &=\sup\{ \varphi(1\text @n)~| ~n\in\mathbb N\}$$

Hopefully I didn't make any mistakes with writing that

o/ lol

Hi

;-; MathJax mistake

DANG IT

Oh hi @YuriyS

Hi!, @SimplyBeautifulArt

$$\begin{align} \varphi(1\text @\omega)&=\sup\{\varphi(1), \varphi(1,0), \varphi(1,0,0), \dots\}\\ &=\sup\{ \varphi(1\text @n)~| ~n\in\mathbb N\} \end{align}$$

o.o lots of random-ish people suddenly enter the chat

I almost never use chat, and that's a pity, thank you for your welcome

:P May I ask what brings you here? @YuriyS

Is a k-linear map a linear map over vector space with scalar from the field k? Or is it a linear map from a vector space to the field k?

.-. idk, you could try asking in the main chat room @WilliamOliver unless someone in here happens to know that

I was in a discussion with an OP just now, and then I looked at the chatrooms and was reminded of this place, I think I've been here before

Oh sorry I meant to post it there, the mobile site is weird

How helpful, thanks:)

np

@YuriyS D: You should learn about ordinal collapsing functions. I think they are cool (though very possibly not your type of thing)

@Simply, most likely they are not, but I'll look them up. If you've seen my post you probably noticed, I'm not even close to being a matematician :p

:P

They are just some cool recursive functions relating to ordinals

@SimplyBeautifulArt: Thanks for telling me about your new ordinal notation, but honestly I've too much stuff on my hands now haha..

haha np

Anyway, have you popped into your realm recently? Didn't see you around.

:P

Was busy editing my codegolf.SE answers

@SimplyBeautifulArt $\zeta$ does not appear anywhere inside the $C_{n+1}$?

Unrelated:

$A(0)=0$

$A(1)=1$

$A(2)=\min (\{\alpha | \beta +\beta < \alpha, \forall \beta < \omega_1^{CK}\})$

Also what is:

$$\sup \{0,2,\omega 2, \omega^2 2,... ,\epsilon_0 2, \Gamma_0 2, ...\}$$

@Secret $A(2)=\omega_1^{\rm CK}$

@Secret Not clear what the terms are

@Secret You mean $\zeta_\alpha$?

Depends on the arguments of $C$

googology.wikia.com/wiki/User_blog:Simply_Beautiful_Art/…

^ Read

$$\mu=\min(\{\alpha~|~\beta+\beta<\alpha\forall\beta<\mu\})\forall\mu\in\rm Lim$$