Zermelo set theory: Encyclopedia II - Zermelo set theory - Connection with standard set theory

Zermelo set theory - Connection with standard set theory

The accepted standard for set theory is Zermelo-Fraenkel set theory. 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 "Axiom of pairs". If a exists, a and a exist, thus {a,a} exists. By extensionality {a,a} = {a}.) The empty set axiom is already assumed by axiom of infinity, and is now included as part of it.

The axioms do not include the Axiom of regularity and Axiom of replacement. These were added as the result of work by Thoralf Skolem in 1922, based on earlier work by Adolf Fraenkel in the same year.

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 formula with parameters". The notion of "first order formula" was not known in 1904 when Zermelo published his axiom system, and he later rejected this interpretation as being too restrictive.

In the usual cumulative hierarchy V_α of ZFC set theory (for ordinals α), any one of the sets V_α for α a limit ordinal larger than the first infinite ordinal ω forms a model of Zermelo set theory. So the consistency of Zermelo set theory is a theorem of ZFC set theory. Zermelo's axioms do not imply the existence of many infinite cardinals; for example, in the model V_ω+ω of Zermelo set theory the only infinite cardinals are ℵ_α for α a finite ordinal.

The axiom of infinity is usually now modified to assert the existence of the first infinite von Neumann ordinal ω; it is interesting to observe that the original Zermelo axioms cannot prove the existence of this set, nor can the modified Zermelo axioms prove Zermelo's axiom of infinity. Zermelo's axioms (original or modified) cannot prove the existence of V_ω as a set nor of any rank of the cumulative hierarchy of sets with infinite index.

Adapted from the Wikipedia article "Connection with standard set theory", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki