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https://math.cs.kitami-it.ac.jp/~fuchino/xbooks/The-Higher-Infinite-optimized.pdf
The Higher Infinite (Springer)
Akihiro Kanamori The Higher Infinite Large Cardinals in Set Theory from Their Beginnings Second Edition
–kŒ©�H‹Æ‘åŠw 2004
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PXIV
but shorn of these features it became the standard Zermelo-
Fraenkel (ZFC) one when recast in the soon to emerge terms of first-order logic.
The Foundation Axiom had been prefigured as a restricting possibility by Dmitry
Mirimanov [17], Skolem [23], and von Neumann [25]. Zermelo offered a synthetic
view of a succession of natural models for set theory, each a member of
a next, essentially realizing that Foundation ranks the sets in these models into a
cumulative hierarchy. In current terms the axiom stratifies the formal universe V
of sets as �¾ƒ¿Vƒ¿, where V0 is ∅,Vƒ¿+1 is the power set of Vƒ¿, and VƒÂ for limit
ordinals ƒÂ is the union of the Vƒ¿�fs for ƒ¿ < ƒÂ.
In a notable inversion this iterative
conception came to be accepted after Godel�fs later advocacy as a heuristic for
motivating the axioms of set theory generally, its open-endedness moreover promoting
a principle of tolerance for motivating new hypotheses mediating toward
Cantor�fs Absolute. Foundation is the one axiom unnecessary for the recasting
of mathematics in set-theoretic terms, but it came to be the salient feature that
distinguishes structural investigations specific to set theory. Indeed, it can be
fairly said that modern set theory is the study of well-foundedness, the Cantorian
well-ordering doctrines adapted to the Zermelian combinatorial conception of sets.

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