User blog comment:Nedherman1/Is it possible something like omega^...(n)...^omega?/@comment-39541634-20191214192519/@comment-39541634-20191214233652

Yeah, but we can't give an actual value for ω↑↑↑ω, without defining ω↑↑(ω+1) first.

By the way, the reason it is generally accepted that ω↑↑↑ω is the limit you stated, is that we usually want the functions f(x)=ω↑↑x to be normal. This means two things:

1. The function is increasing: If a>b then f(a)>(b).

2. The function is continuous: If x=lim(x1,x2,...) then f(x)=lim(f(x1),f(x2),...).

We usually want our functions to be normal, because such functions have many intuitive properties that make them easier to work with. For example, there's a theorem that states that all normal functions have an infinity of fixed-points, which is the basis of how the Veblen functions works.