User blog comment:Rgetar/Higher weakly inaccessible and weakly Mahlo cardinals/@comment-28606698-20200106214759/@comment-32213734-20200121021320

> I did not know such a terminology of "on α". Is it correct?

I modified definition of normal function on α form [Frank R. Drake. Set Theory. An Introduction to Large Cardinals] in order to extend it to α and beyond. The definition was "For ordinal α, a function f: δ → α is a normal function on α if f is increasing, continuous, and its range is unbounded in α, i.e., ∪Range(f) = α". Also I added "for any δ < α there is x such as δ < f(x) < α" condition in order not to write a definition of "unbounded".

> It is just a normal function such that α is not a successor ordinal and belongs to the image of f.

What is "it"? Normal function on α according another definition? Because let f(x) = ωx, α = ω2, then f(x) is a normal function, α is not a successor ordinal, α = f(2), so α belongs to the image of f, but if δ = ω, then δ < α, and there is no x such as δ < f(x) < α, that is ω < ωx < ω2.

> You need to assume that the given set of fixed points is not empty.

Can empty set be considered as a "set of fixed points"? I thought that "set of fixed points" must include at least one fixed point, so it cannot be empty.

> It has counterexamples. For example, cosider f(x) = ω+x. Then f is a normal function on ω+ω by the definition of your terminology of "on α", but {n|f(n)<ω+ω} = {n|n<ω} is not a fundamental sequence of ω+ω.

17 states that {f(n)|f(n) < α} forms a fundamental sequence of α, not {n|f(n) < α}; {ω + n|ω + n < ω + ω} = {ω + n|n < ω} is a fundamental sequence of ω + ω