Editing Universe (mathematics) (section)

==In set theory==
It is possible to give a precise meaning to the claim that '''SN''' is the universe of ordinary mathematics; it is a [[model theory|model]] of [[Zermelo set theory]], the [[axiomatic set theory]] originally developed by [[Ernst Zermelo]] in 1908. Zermelo set theory was successful precisely because it was capable of axiomatising "ordinary" mathematics, fulfilling the programme begun by Cantor over 30 years earlier. But Zermelo set theory proved insufficient for the further development of axiomatic set theory and other work in the [[foundations of mathematics]], especially [[model theory]].

For a dramatic example, the description of the superstructure process above cannot itself be carried out in Zermelo set theory. The final step, forming '''S''' as an infinitary union, requires the [[axiom of replacement]], which was added to Zermelo set theory in 1922 to form [[Zermelo–Fraenkel set theory]], the set of axioms most widely accepted today. So while ordinary mathematics may be done ''in'' '''SN''', discussion ''of'' '''SN''' goes beyond the "ordinary", into [[metamathematics]].

But if high-powered set theory is brought in, the superstructure process above reveals itself to be merely the beginning of a [[transfinite recursion]].
Going back to ''X'' = {}, the empty set, and introducing the (standard) notation ''V''<sub>''i''</sub> for '''S'''<sub>''i''</sub>{}, ''V''<sub>0</sub> = {}, ''V''<sub>1</sub> = '''P'''{}, and so on as before. But what used to be called "superstructure" is now just the next item on the list: ''V''<sub>ω</sub>, where ω is the first [[Infinity|infinite]] [[ordinal number]]. This can be extended to arbitrary [[ordinal number]]s:
: <math> V_{i} := \bigcup_{j<i} \mathbf{P}V_j \! </math>
defines ''V''<sub>''i''</sub> for ''any'' ordinal number ''i''.
The union of all of the ''V''<sub>''i''</sub> is the [[von Neumann universe]] ''V'':
: <math> V := \bigcup_{i} V_{i} \! </math>.
Every individual ''V''<sub>''i''</sub> is a set, but their union ''V'' is a [[proper class]]. The [[axiom of foundation]], which was added to [[Zermelo–Fraenkel set theory|ZF]] set theory at around the same time as the axiom of replacement, says that ''every'' set belongs to ''V''.

: ''[[Kurt Gödel]]'s [[constructible universe]] ''L'' and the [[axiom of constructibility]]''
: ''[[Inaccessible cardinal]]s yield models of ZF and sometimes additional axioms, and are equivalent to the existence of the [[Grothendieck universe]] set''