By $\sf HT^\psi$ I mean the Hierarchy Theory of $\psi$ height. This is a set theory written in mono-sorted first order logic with equality and membership, with the following axioms:
- Specification: $\forall A \exists! x: \forall y \, (y \in x \leftrightarrow y \in A \land \phi)$, if "$x$" doesn't occur in $\phi$.
- Hierarchy: $\exists \alpha \forall x \forall y: x \in y \to x \in V_\alpha$
- Height: $ \psi(\Omega)$
Where $\Omega$ is the last ordinal.
Definition: $v = V_\alpha \iff \\ \operatorname {ordinal} (\alpha) \land \\ \exists f: {\sf dom}(f)=\alpha \land \\ \forall \beta \in \alpha: f(\beta) = \{y\mid \exists \gamma \in \beta: y \subseteq f(\gamma)\} \land \\ v= \{y \mid \exists \beta \in \alpha: y \subseteq f(\beta)\} $
The expression "$x \in V_\alpha$" stand for the expression "$\exists v: v=V_\alpha \land x \in v$".
Where ordinals are defined as von Neumann ordinals (i.e.; transitive sets well ordered by $\in$). Well-orderings, functions implemented as sets of Kuratowski ordered pairs, and "$\sf dom$" stand for domains, defined as usual.
Now with those hierarchy theories one can directly capture any standard set\class theory either over the element world of these theories or over the whole world of them. For example a theory with the empty set as the sole object is captured when $\psi$ is "is empty". Finite set theory is directly captured over the element world when $\psi$ is "is limit of finite ordinals". While a set theory equivalent to second order arithmetic would follow over the whole world of that theory. Similarly Zermelo set theory would be captured over the element world when $\psi$ is "is a limit ordinal beyond the first". $\sf ZF$ captured with $\psi$ being "is inaccessible". Likewise, $\sf ZFC$ captured with $\psi$ being "is inaccessible square ordinal", where a square ordinal is the index of a stage that is equinumerous with it; and then $\sf MK$ would follow over the whole world. Reinhardt's cardinals can be captured by letting $\psi$ be "is inaccessible and $V_\Omega \prec V_\Omega$"; and Berkeley cardinals when $\psi$ is "is inaccessible and $\exists \kappa \in V_\Omega: \operatorname {Berkeley}(\kappa)$", and so on...
So, the spectrum of these HT theories is what constitutes the standard mainstream set theory. One can stop at any ordinal, and so this is quite adjustable. In my opinion, this is the big picture about set theory. So, actually it's Hierarchy Theory that is the main core of it, the rest is just size adjustments.
Seeing that this is actually the big picture, then why not start studying Set Theory this way?
I mean instead of presenting the axioms of $\sf ZFC$, which does have an ad-hoc look and doesn't seem to reflect a disciplined line of thought, other than being simply understood, why not start this way which is pretty much disciplined and concise, and moreover can be seen as early introduction to the whole standard set theoretic realm.
What I'm seeing is that it is $V_\alpha$ that is the most basic feature in the standard set world. This what needs to be presented early, since grasping this is the real core issue. So, we begin with Specification, prove Extensionality, and then learn how to build sets after predicates, then work with Definitions and learn simple set theoretic constructions of empty, singletons, pairs, and Kuratwoski pair, and subsets, etc.. but those need not be presented as axioms, just as a defined stuff and defined processing of them, and then present implementation of functions, domains, ordinals, limits, and so on without any need for having axioms, then we define $V_\alpha$. Then after that we come to the axiomatic approach and present it as in Hierarchy Theory style, and buildup sets as we increase height gradually, and examine the properties of those buildups, learning cardinality, inaccessibility, ..etc., then deriving the whole axioms of ZFC, in a hierarchy theory with suitable height, then go beyond ZFC.
