Axiom of regularity

The foundation axiom (also: of regularity ) is an axiom of set theory by John von Neumann in 1925, which went up in the Neumann - Bernays - Gödel set theory ( NBG ), and an axiom of popular Zermelo -Fraenkel set theory (ZF ) of 1930. Ernst Zermelo gave him the name and a simple formulation for a range of quantities and basic elements as follows:

Formalized is the foundation axiom for the area in terms of a class of sets and basic elements:

In pure set theory, in which all variables denote quantities, there are shorter formulations of the foundation axiom, which is eliminated from the formula, for example, by the following:

The foundation axiom prevents infinite descending member chains because people would otherwise form a set, which contradicts the axiom: For each, the two sets are not disjoint so. This also cyclical element chains are impossible as a quasi- infinite circles. There is therefore no quantities that contain themselves as elements.

There are also a lot of teachings without foundation axiom. This includes the original Zermelo set theory, Zermelo explicitly in the quantities included in the calculations, or the Ackermann set theory. For both but can be added the foundation axiom, without a ( not previously existing ) to produce a contradiction. Also worth mentioning is the quantity theory of Quine, the individual amounts by defined so that they are zirkel amazing and the foundation axiom definitely does not apply. In such an amount teachings without foundation axiom compass -like quantities are possible, which shows that this does not necessarily produce a contradiction. The formation of certain compass -like quantities as the Allmenge or the set of ordinals that generate contradictions in the naive set theory, is already excluded in the Zermelo set theory without foundation axiom. The foundation axiom is not intended to prevent contradictions, since the addition of an axiom the set of provable sentences only increase, not decrease but may.

Note that it is possible in spite of the foundation axiom from the formula amount not derive a contradiction, assume that ZFC is consistent, because in such a proof by contradiction only finitely many formulas could be used, which would obviously lead to any contradiction. In other words: Because of the compactness theorem there is, if there are models of ZFC, including models that are not well-founded. Nevertheless considering a model of the above-constructed set of formulas, so it meets the foundation axiom, since no amount exists, which consists exactly of the.

Prehistory

The idea to look at sound levels than normal amounts, dates back to Dmitry Mirimanoff which, designated 1916 circular amounts that were allowed in the Zermelo set theory as extra- ordinary. This extra- ordinary amounts wanted Abraham Fraenkel set theory in 1921 eliminated by a boundedness axiom, "that the quantity range imposed the least compatible with the other axioms of scale". But its boundedness axiom is not formulated in the language of set theory. The first correct formula, which reached to exclude extra- vulgar amounts Neumann gave in 1925 in its boundedness axiom which is more complicated but as the common foundation axiom of Zermelo.