User blog comment:Chronolegends/Numbers in CSBN/@comment-24676271-20161026190923/@comment-24920136-20161027044328

You seem to have a few core concepts about ordinals wrong. w is not a function. w is a limit ordinal, it is a limit of the sequence(*not* function) 1,2,3,4....

and not just that one, any sequence that converges to the same limit, for example 3,4,5,6... or even 10,100,1000... or 20,30,50,70,110... every sequence that converges to infinity has omega as its limit ordinal. (The concept is defined as the smallest member not in the sequence)

You need to learn the concept of fixed point and what it means, its beyond my skill to teach you clearly, but basically a fixed point remains unaffected by applying a function to it.

e is an ordinal enumerating function, but *it doesnt have a limit*. It simply labels ordinals that are fixed points of the w^x function. For example w^(e_x)=w for all x, starting with e_0.

e_0 is also the limit of any sequence that converges to w^ ^ w, for example  w,  w ^ w ^ w,  w ^ w ^ w ^ w ^ w... or 1,  w ^ 250,   w ^ w ^2500, etc... for a second definition, e_0 is the smallest ordinal not constructible using finite amounts of addition, multiplication, exponentiation, using 0, 1, and w.  e_1 could have the same definition, but you add "and e_0" after "and w"

How to escape the fixed point traps?, well, thats easy, ol' reliable +1 has your back. +1 is your friend. The first ordinal not in the sequence of e_0, is (e_0)+1.

Also, replicating the steps that get you to e_x, (ie. a power tower) only gets you to  e_(x+1), not e_e_0!. after e_(x+1) you need another such recursion for  e_(x+2), then  e_(x+x) after a long while e_(x^x).. e_(x^x+1).. e_(x^x*2).. e_(x^(x+1))  e_(x^(x*8)). e_(x^(x ^ 20))  e_(x^x ^ x ^ x ^ x ^ x ))   etc.. only after enough such recursions you get to  e_(e_(x+1)), another family of them lets you into e_(e_(e_(x+1))) the limit being z_0. (disclaimer: only true for  x < z_0)

Z_0 into Z_1 and beyond is a *much* harder climb than i described above, btw.