Editing Zermelo set theory (section)

== Connection with standard set theory ==
The most widely used and accepted set theory is known as ZFC, which consists of [[Zermelo–Fraenkel set theory]] including the [[axiom of choice]] (AC).  The links show where the axioms of Zermelo's theory correspond.  There is no exact match for "elementary sets".  (It was later shown that the singleton set could be derived from what is now called the "Axiom of pairs".  If ''a'' exists, ''a'' and ''a'' exist, thus {''a'',''a''} exists, and so by extensionality {''a'',''a''} = {''a''}.)  The empty set axiom is already assumed by axiom of infinity, and is now included as part of it.

Zermelo set theory does not include the axioms of [[axiom of replacement|replacement]] and [[axiom of regularity|regularity]]. The axiom of replacement was first published in 1922 by [[Abraham Fraenkel]] and [[Thoralf Skolem]], who had independently discovered that Zermelo's axioms cannot prove the existence of the set {''Z''<sub>0</sub>,&nbsp;''Z''<sub>1</sub>,&nbsp;''Z''<sub>2</sub>,&nbsp;...} where ''Z''<sub>0</sub> is the set of [[natural number]]s and ''Z''<sub>''n''+1</sub> is the [[power set]] of ''Z''<sub>''n''</sub>. They both realized that the axiom of replacement is needed to prove this. The following year, [[John von Neumann]] pointed out that the axiom of regularity is necessary to build [[von Neumann ordinal|his theory of ordinals]]. The axiom of regularity was stated by von Neumann in 1925.{{sfn|Ferreirós|2007|pp=369, 371}}

In the modern ZFC system, the "propositional function" referred to in the axiom of separation is interpreted as "any property definable by a first-order [[Well-formed formula|formula]] with parameters", so the separation axiom is replaced by an [[axiom schema]]. The notion of "first order formula" was not known in 1908 when Zermelo published his axiom system, and he later rejected this interpretation as being too restrictive. Zermelo set theory is usually taken to be a first-order theory with the separation axiom replaced by an axiom scheme with an axiom for each first-order formula. It can also be considered as a theory in [[second-order logic]], where now the separation axiom is just a single axiom. The second-order interpretation of Zermelo set theory is probably closer to Zermelo's own conception of it, and is stronger than the first-order interpretation.

Since <math>(V_\lambda , V_{\lambda + 1})</math>&mdash;where <math>V_\alpha</math> is the rank-<math>\alpha</math> set in the [[Von Neumann universe|cumulative hierarchy]]&mdash;forms a model of second-order Zermelo set theory within ZFC whenever <math>\lambda</math> is a [[limit ordinal]] greater than the smallest infinite ordinal <math>\omega</math>, it follows that the consistency of second-order Zermelo set theory (and therefore also that of first-order Zermelo set theory) is a theorem of ZFC. If we let <math>\lambda = \omega \cdot 2</math>, the existence of an [[uncountable]] [[strong limit cardinal]] is not satisfied in such a model; thus the existence of [[Beth number|''&beth;<sub>&omega;</sub>'']] (the smallest uncountable strong limit cardinal) cannot be proved in second-order Zermelo set theory. Similarly, the set <math>V_{\omega \cdot 2} \cap L</math> (where ''L'' is the [[constructible universe]]) forms a model of first-order Zermelo set theory wherein the existence of an uncountable weak limit cardinal is not satisfied, showing that first-order Zermelo set theory cannot even prove the existence of the smallest [[singular cardinal]], <math>\aleph_\omega</math>. Within such a model, the only infinite cardinals are the [[aleph numbers]] restricted to finite index ordinals.

The [[axiom of infinity]] is usually now modified to assert the existence of the first infinite von Neumann [[ordinal number|ordinal]] <math>\omega</math>;  the original Zermelo axioms cannot prove the existence of this set, nor can the modified Zermelo axioms prove Zermelo's axiom of infinity.<ref>{{cite journal |last1=Drabbe |first1=Jean |title=Les axiomes de l'infini dans la théorie des ensembles sans axiome de substitution |journal=Comptes Rendus de l'Académie des Sciences de Paris |date=20 January 1969 |volume=268 |pages=137–138 |url=https://gallica.bnf.fr/ark:/12148/bpt6k480296q/f140.item |access-date=8 September 2024}}</ref>  Zermelo's axioms (original or modified) cannot prove the existence of <math>V_{\omega}</math> as a set nor of any rank of the cumulative hierarchy of sets with infinite index. In any formulation, Zermelo set theory cannot prove the existence of the von Neumann ordinal <math>\omega \cdot 2</math>, despite proving the existence of such an order type; thus the von Neumann definition of ordinals is not employed for Zermelo set theory.

Zermelo allowed for the existence of [[urelements]] that are not sets and contain no elements; these are now usually omitted from set theories.