User blog comment:Fejfo/Uncountable indexed veblen function/@comment-32213734-20180513045616/@comment-32213734-20180513201259

Does strictly increasing and continuous guarantees a fixed point? I read in Wikipedia, article Continuous function, that it is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output. So, β + 1 is continuous function? It is also strictly increasing, but does not have fixed points.

Now i'm sure that φβ(β) > β for all β, since if β > 0, then φα(β) > φα(0) for all α, and

φβ(0) > β, if β is not Γ number

φβ(0) = β, if β is Γ number

But I guess φΩ(β) = φβ(0) may work.