User blog comment:Mh314159/new YIP notation/@comment-39585023-20190708030201/@comment-35470197-20190708040007

> Am I right in estimating that \(f_x(x)\) is greater than \(x \uparrow^x x\)

Right. But both appriximetely correspond to the same ordinal, i.e. \(\omega\), in FGH.

> [1] reaches at least w+1?

FGH is a function, and an ordinal in FGH is used in ordr to describe the growth rate. Since \([1]\) is a large number, which is not a large function, it is not described as an ordinal. Anyway, \([1]\) is approximately between \(F_{\omega}(10^{100})\) and \(F_{\omega \times 2}(10^{100})\), and hence of level 5 or 6 in my googological ruler.

> Does that also imply that [2] reaches only w+2?

\([2]\) is approximately between \(F_{\omega^2}(10^{100})\) and \(F_{\omega^2+1}(10^{100})\), and hence is of level 8.

> And that this is why [n] is in the region of w2?

No. It is in the region of \(\omega^2 + 1\), which is much greater than \(\omega 2\). Your subsystem with \(\{m\}_y(x)\) is sufficiently strong, i.e. of level \(\omega^2\), and \([n]\), which is given by iterationg \(\{m\}_y(x)\), is of level \(\omega^2 + 1\).

> Is it fixed by {0}0(x) = [x,b-1]?

Good. The correction works well.

> I will consider indicating the variables as you described, although I'm worried that it will make the notation harder to follow for newcomers especially since I'm already using subscripts for another purpose.

Right. It actually can be complicated if you do so. On the other hand, it can be helpful for you and newbies to understand the precise dependence. You can choose either one of the merits. You can keep your original abbreviations if you think that you can avoid a similar mistake. It is just a recommendation, which you do not have to follow. I am just helping you to consider a method to find errors.