Zermelo–Fraenkel set theory

The Zermelo -Fraenkel set theory is a common axiomatic set theory, which is named after Ernst Zermelo and Abraham Fraenkel Adolf. Today, it is the basis of almost all branches of mathematics. The Zermelo -Fraenkel set theory without the axiom of choice is abbreviated by ZF with the axiom of choice by ZFC ( the C in the Engl. Word choice, so selection or choice is ).

• 7.1 Primary sources ( in chronological order )
• 7.2 secondary literature
• 7.3 Notes and references

History

The Zermelo -Fraenkel set theory is an extension of Zermelo set theory of 1907, based on axioms and suggestions from Fraenkel of 1921. Fraenkel completed the replacement axiom and pleaded for regular quantities without circular element chains and for a pure set theory whose objects are sets. Zermelo completed in 1930, the axiom system of Zermelo -Fraenkel set theory, which he himself described as an IF system: he took the axiom replacing Fraenkel on and added the foundation axiom added to exclude circular element chains. The original ZF system is verbally and also calculates a primordial elements that are not sets. In such Elements later formalized ZF systems usually do without, thereby setting Fraenkel ideas to complete. The first precise predicate logic formalization of pure ZF set theory Thoralf Skolem created in 1929 (still without the foundation axiom ). This tradition has prevailed, so that today the abbreviation ZF stands for the pure Zermelo -Fraenkel set theory. The IF system closer to the original standing version with original elements but is now also used and referred to the clear distinction as ZFUs.

Importance

It has been shown - this is an empirical finding - that can be as good as any known mathematical statements formulated so as to be provable statements derived from ZFC. The ZFC set theory has therefore become a proven and widely accepted framework for the whole of mathematics. Exceptions can be found anywhere where you have to work with real classes or wants. It then uses some extensions of ZFC that provide classes or additional very large quantities available, such as an extension to the ZFC - class logic or the Neumann - Bernays - Gödel set theory or a Grothendieck universe. In any case ZFC but is now regarded as the fundamental axiom system for mathematics.

Due to the fundamental importance of ZFC set theory in mathematics a consistency proof for the set theory has been sought since 1918 in the framework of the Hilbert program. Godel, who participated with important contributions in this program, but could 1930 show in his Second Incompleteness that such a consistency proof is impossible in the context of a non-contradictory ZFC set theory. Therefore, the assumption of consistency of ZFC is a hardened through experience working hypothesis of mathematicians:

" The fact that ZFC has been studied for decades and used in mathematics, without a contradiction has shown, however, speaks for the consistency of ZFC. "

The axioms of ZF and ZFC

ZF has infinitely many axioms, since two axiom schemes ( 8 and 9 ) are used, which specify for each predicate with specific properties depending on an axiom. The logical basis of the first stage with identity element and the undefined predicate predicate logic used.

1 extensionality or axiom of determinacy: quantities are equal if and only if they contain the same elements.

2 empty or null set axiom axiom: There are a lot of zero elements.

3rd pair axiom: For all A and B, there is a set C that has exactly A and B as elements.

4 union axiom: For any set A, there exists a set B that contains exactly the elements of the elements of A as elements.

Fifth axiom of infinity: There is a set A, the empty set and every element x also includes the amount (see Inductive amount ).

6 power set axiom: For any set A, there is a set P whose elements are exactly the subsets of A.

7 foundation axiom of regularity or: Every non-empty set A contains an element of B such that A and B are disjoint.

8 axiom: Here is an axiom schema with one axiom for each digit predicate P: For every set A, there exists a subset B of A that contains exactly the elements C of A for which P (C ) is true.

9 substitution axiom ( Fraenkel ): If A is a set and each element of A clearly replaced by any amount, so A goes on in a lot. The substitution is specified by two -place predicates with similar properties to a function, as an axiom scheme for each binary predicate:

In mathematics, the axiom of choice is also often used, which expands to IF ZFC:

10 axiom of choice: If A is a set of pairwise disjoint non -empty sets, then there is a set containing exactly one element from every element of A. This axiom has a complicated formula that can be simplified with the Eindeutigkeitsquantor something:

ZF with basic elements

Zermelo formulated the original ZF system on quantities which he defined as element-containing things or the null set, and for Elements as things without any elements. The null set he regarded as an excellent primary element that extends the IF language as a given constant. Quantities and Elements are then determined by the following defined predicates:

From the usual pure ZF set theory set theory is distinguished by primordial elements with appended U. loud The axioms of ZFUs and ZFCU apart from the empty set axiom verbally as the axioms of ZF or ZFC, but formalized differently because of the different conditions; derivable quantity conditions may be omitted.

ZFUs

ZFUs includes the following axioms:

From the ZFUs axioms and the axiom apparently the ZF axioms follow. For out of the substitution axiom is as in ZF (see below), the pair axiom derivable and also the axiom of separation, the latter here in the following form for each place predicate P:

ZFCU

ZFCU includes the axioms of ZFUs and the following axiom of choice:

Simplified IF system (redundancy)

The IF system is redundant, that is, it has dispensable axioms that can be derived from others. ZF or ZFUs is already completely described by the axiom of extensionality, union axiom, power set axiom, axiom of infinity, the foundation axiom and substitution axiom. This is due to the following points:

• This axiom follows from the substitution axiom ( Zermelo ).
• The empty set axiom follows from the axiom and the existence of any amount, which is a provable tautology in the common calculus of predicate logic, but also results from the axiom of infinity in any case.
• The pair axiom follows from the substitution axiom and the power set axiom ( Zermelo ).

Pair axiom, Union axiom and power set axiom can also be gained from the statement that any amount is an element of a step. Axiom of infinity axiom and substitution are within the scope of the remaining axioms equivalent to the reflection principle. By combining these two insights Dana Scott ZF formulated to Scott 's equivalent to axiom system.

IF system without equality

You can ZF and ZFUs on a predicate logic without equality to build and define equality. But is not sufficient to define the extensionality. The derivation of all equality axioms backs up only the one used in the logic of identity definition:

This definition makes the extensionality not superfluous, because it would not be derived from the definition.

A definition of equality by the extensionality as an alternative to IF only possible if one adds the axiom, which ensures the derivability of the formula above. This option differs from naturally at ZFUs.