Hierarchy theory as FOUNDATION

Hierarchy theory is the theory obtained by adding a one place function V to the first order language of set theory.

Define ‘‘ordinal"‘‘ordinal" along Von Neumann's, also ‘‘<" is defined in the usual manner over von Neumann ordinals only.

Add axioms of: Extensionality; Separation, stipulated in the usual manner.

Stages: ∀ ordinal α:Vα={x⊆Vβ|β<α} 

Foundation: ∀x∃ ordinal α (x∈Vα) 

Infinity: ∃α[α≠∅∧∀β<α∃γ(β<γ<α)] 

Height: ∀x[x is well ordered →∃ ordinal α:α=ord(x)] 

Where ord(x)  means the order type of x.

/ Theory definition finished.

I think the above axioms are the ones that truly capture the notion of an infinite hierarchy of sets that is built from below, the last axiom [to me] makes full sense and it does belong to the notion of a hierarchy itself, since it reflects the build up from below notion, i.e. we can add up heights to the hierarchy as long as those are formed at earlier stages!

Now of course we can add up further ordinal lengths like: fixed points, inaccessible ones, etc.. and thereby obtain higher hierarchies.

Also we can thin up stages by filtering out special kinds of sets like taking out all non well ordered sets (thereby getting choice), or all non-constructible sets (thereby getting V=L) , etc.. [ I mean by that paragraph adding axioms that all sets are well ordered, or all of them are constructible, thereby filtering out any discourse about non-well ordered sets, or about non-contructible sets]

However those additions of lengths to the hierarchy and those thinnings of its stages, I don't see any of them really stemming from the very basic notion of hierarchy itself, they can be conceived as manipulations on that notion rather than stemming essentially from it. However these manipulations seem to preserve the notion of a hierarchy, and so can be seen as constituting different versions of it.

So for example ZFC    can be seen as a special case of a hierarchy, namely when the hierarchy has inaccessible height and thinned to only well ordered sets.

So the overall notion of a hierarchy with various lengthening and\or thinning of it, is a bigger notion than the axioms of ZFC , and it is the foundational notion about sets and mathematics thereof. And I see the above theory being the basic form of it, not ZFC  which only captures a special manipulation of its height and width.