User blog comment:Simplicityaboveall/The Construction of Extremely Large Numbers/@comment-24920136-20160725013617/@comment-24920136-20160725154843

ω^ω is the limit ordinal of set K, in other words, ω^ω is the smallest ordinal not in set K

Just because the set K is infinite, it doesn't mean it will ever surpass ω^ω,

let me show you why in another way:

Lets say i want to access the 10000 th member, 1 is at the 5-th place and the rest is zeroes, so it is ω^4. Lets call 10000 10^4 i will now make a table of comparison between index and members of set k

10^10 th member = ω^10\

10^99 th member = ω^99

10^25,000,000 th member = ω^25,000,000

10^(10^10,000) = ω^(10^10,000)

10^Grahams number = ω^Grahams number

Surely you must accept natural numbers are infinite, and so my table can be infinitely long, but still every ordinal on the right will be of the form ω^X where X is some finite number. and since ω > X whenever X is a finite number, the following relationship can never be broken:

Members of set K < ω^ω