I am studying set theory, ordinal part.
Set theory is new to me.
I know that commutativity of addition and multiplication
can be false in infinite ordinal world.
$ \omega $ = limit of sequence $\, 1,2,3,...$
$ \omega\cdot 2 $ = limit of sequence $\, \omega+1,\,\omega+2,\,\omega+3,\,...$
$ \omega \,^ 2 $ = limit of sequence $\, \omega \cdot 1,\, \omega \cdot 2, \omega \cdot 3,\,...$
$ \omega \,^ \omega $ = limit of seuqnce $\,\omega \,^ 1,\, \omega \,^ 2,\, \omega \,^ 3,\, ...$
$ \epsilon_0 $ = limit of sequence $\, \omega, \, \omega \,^ \omega, \omega \,^ {\omega \,^ \omega},... $
$ \epsilon_\omega $ = limit of sequence $ \epsilon_0 , \epsilon_1, \epsilon_2, ... $
So it seems that for relatively small countable limit ordinal $O$,
there is a very natural, very intuitive sequence $S_n,$
(sequence means a function whose domain is positive integers)
whose limit is $O$.
I hope this logic can be applied to all countable limit ordinal.
If you think so, please explain it,
if you don't think so, you can suggest an ordinal (of course the smallest ordinal if you are possible) which can not be expressed as a limit of very natural, very intuitive sequence of ordinals.
I also recognize the problem with my question. The term very natural, very intuitive sequence is not a strict term. So I wish you could clearly define this term as well.
