Just below first inaccessible cardinal

[tzimie]

First inaccessible is "above" all $$\aleph_0, \aleph_{\aleph_0}, \aleph_{\aleph_{\aleph_0}}$$ etc.

My question is what is the "height" of that tower, can it be not only infinite, but also as huge as $$\aleph_{\aleph_{\aleph_...}} \big\} \aleph_{\aleph_0}$$ so the height of the tower is defined by another huge cardinal?

Does adding more levels to that structure (cardinal defines the height of a tower of another cardinal, which defines the height a tower on level 2, etc)

$$\aleph_{\aleph_{\aleph_...}} \big\} \aleph_{\aleph_{\aleph_...}} \big\} \aleph_{\aleph_0}$$

bring anything new, or any "multilevel" cardinal can be defined as using a single level notation?

 

[SSequence]

First inaccessible is "above" all $$\aleph_0, \aleph_{\aleph_0}, \aleph_{\aleph_{\aleph_0}}$$ etc.

My question is what is the "height" of that tower, can it be not only infinite, but also as huge as $$\aleph_{\aleph_{\aleph_...}} \big\} \aleph_{\aleph_0}$$ so the height of the tower is defined by another huge cardinal?

I don't know about the notation using the symbol N₀, N₁, N₂ etc. This cardinality topic is somewhat beyond my knowledge. Probably someone more knowledgeable and/or comfortable with this topic can give a better response (hopefully my response doesn't contain some major inaccuracy). But nevertheless, still here is my thought on it.

I think it should make sense. Simply thinking about it in terms of Ω (first uncountable). Let's write Ω as Ω₀. Let's simply define a function f (from ordinals to ordinals I guess?) such as:
f(x)=Ω_x=x-th uncountable ordinal

Now assuming that the fixed points for this function work in the same way as for countable ordinals (f should be a normal function), we have its first fixed point as the supremum of (fⁿ denotes composition of f n times):
A={f(0),f²(0),f³(0),f⁴(0),...}

And this essentially corresponds the "height" of the tower you described being equal to ω. If we then denote g as the function enumerating fixed points of f, then we can also write something like:
g(Ω_Ω0)=g(f²(0))
it essentially corresponds to the term in quote above. Only sort of ... I think(?) because N₀ is equal to ω and N₁ is equal to Ω?. In that case we can just write: g(Ω_ω).