Zermelo set theory

The Zermelo set theory is the first axiomatic set theory; it is by Ernst Zermelo and is dated July 30, 1907. She was born on February 13, 1908 published studies on the foundations of set theory under the title in Volume 65 (2nd issue) of the Mathematische Annalen and is the basis of the Zermelo - Fraenkel set theory, which today serves as the basis of mathematics.

To set theory to give a solid formal basis, Bertrand Russell had in 1903 published his theory of types which, however, was due to their syntactically complicated shape difficult to access. Zermelo therefore chose the more elegant way of the axiomatic structure of the theory. His seven quantity axioms that ensure above all the existence of sets, proved to be stable and allow the expanded form of the Zermelo -Fraenkel set theory, the total derivative of Cantor's set theory. Zermelo formulated his axioms or verbally; Today, however, they are usually specified in the form of predicate logic.

Zermelo's axioms 1907

Zermelo formulated his seven axioms for a range of things, including the quantities as part of the field. He defined namely quantities as element-containing things or the null set ( empty set). The axiom system allows but as elements other element loose things he later called Elements. It is with him that is, elements and things equate, but not quantities and items. In the original naming and numbering and in the original verbal text which omits the following only commenting bays with synonymous formulations loud his axioms:

I. axiom of determinacy:

Second Axiom of elementary sets:

III. Axiom of separation:

IV axiom of the power set:

V. Axiom of union:

VI. Axiom of choice:

VII axiom of infinity:

The axiom of infinity calls for an inductive set ( closed with respect to the count a '= {a }). Following this, Zermelo gave the first precise explicit definition of the natural numbers as the smallest set Z that satisfies the axiom of infinity. With this definition, all Peano axioms are provable and the proof principle of mathematical induction.

The axiom system is slightly redundant, since the elementary quantity 0 can be obtained and define the elementary set { a} by the pair set {a, a} by separation from the infinite set Z with the class statement x ≠ x. So you need only the third elementary set { a, b}.

Original ZF system in 1930

Zermelo expanded its axioms amount of 1907 in an essay of 1930 and supplemented his foundation axiom and the substitution axiom that Abraham Fraenkel in 1921 introduced the complete derivation of Cantor's set theory; Zermelo took advantage of it but in a generalization for arbitrary things of the area, to expressly include Elements. The extended axiom system he called " Zermelo -Fraenkel system " or " IF system ". The two supplementary axioms of 1930 have the following original text:

Axiom of Replacement:

Axiom of foundation ( second formulation Zermelo ):

The axiom of replacement means that images of sets are also quantities. The foundation axiom excludes compass -like amounts, including amounts which contain themselves as elements. Zermelo himself pointed out that the extended axiom system is redundant: the axiom of separation is provable with the axiom of replacement, and the elementary quantity can be derived with the replacement axiom of the power set and the null set ( because {a, b} is the image of twice the power set of the null set ). So Zermelo already knew an optimized ZF axiom system which manages with the Zermelo axioms I, VI, V, VII, VIII, and the replacement of the foundation. Zermelo modified there, however, his early axioms: He let the axiom of choice away and replaced it with the ( metalogical ) well-ordering capability for quantities eliminated the two expendable elementary quantities and let the axiom of infinity away as suggested for the general set theory. He called this modified general set theory ZF' system.

Modified IF systems

Later formalized ZF systems differ in several respects from the original:

• You eliminate Zermelo frame with things and basic elements and are pure quantity teachings, in which all objects are sets, which by a stronger axiom of determinacy ( extensionality ) is reached.
• Do not include the axiom of choice to ZF and call the entire system with selection ZFC (C = Choice ( English) = selection).
• Use since the proposal of Skolem 1922 a predicate logic formal language that differs greatly from Zermelo text. He even used a class logic by Giuseppe Peano and Ernst Schroeder.
• Its stuck in the axiom of infinity counting with is usually replaced by his later counting from the set theory of 1930.