Zermelo Set Theory - Connection With Standard 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 Abraham 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", so the separation axiom is replaced by an axiom scheme. 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. 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.

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 (such as V_ω·2) 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 ℵ_ω or larger infinite cardinals, as the model V_ω·2 does not contain such cardinals. (Cardinals have to be defined differently in Zermelo set theory, as the usual definition of cardinals and ordinals does not work very well: with the usual definition it is not even possible to prove the existence of the ordinal ω2.)

The axiom of infinity is usually now modified to assert the existence of the first infinite von Neumann ordinal ; 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 as a set nor of any rank of the cumulative hierarchy of sets with infinite index.

Zermelo set theory is similar in strength to topos theory with a natural number object, or to the system in Principia mathematica. It is strong enough to carry out almost all ordinary mathematics not directly connected with set theory or logic.