Is $\circledcirc\equiv * \omega$? Or is $* \omega \in \circledcirc$?

Question

In Winning Ways Volume 2 (pg. 398) :
$\circledcirc$ (sunny) is used "instead of $0\star\rightarrow$" as "the collection $0, \star1, \star2, \star3, \star4,...$" eg.

$$\circledcirc=\{0,\star1,\star2,\star3,...\}$$

This is how I would think $\star \omega$ would be defined.

Is $\circledcirc\equiv\star\omega$? Or is $\star\omega \in \circledcirc$? If $\circledcirc\equiv\star\omega$, is there a symbol or notation for the set of all transfinite nimbers?

Answer 1

I am using https://fse.studenttheses.ub.rug.nl/27904/1/bMATH_2022_TielmanBB.pdf.pdf to get some Idea about the Question :

Page 9 :

Page 24 :

I think , it is safe to say that the collection does not contain that element.