Naive set theory
The term "naive set theory " originated in the early 20th century in response to the set theory of the 19th century, in an unregulated or unrestricted quantity of education was practiced. Because contradictions that arise in her, she was later replaced by the axiomatic set theory, in which the amount of education is governed by axioms. The term "naive set theory " is therefore essentially refers to this early form of unregulated set theory and is meant as a contrast to the concept of axiomatic set theory. Not infrequently, but also referred to the math literature after 1960 a descriptive set theory as naive; therefore can also hide a unformalisierte axiomatic set theory under this name or an axiomatic set theory without metalogical considerations.
For the intention of unlimited naive set formation often the quantity definition of Georg Cantor is quoted: By a "set " we understand any collection M of specific well-distinguished objects m of our intuition or our thought (which are called the " elements " of M) to form a whole. On closer examination, however, this does not hold ( see below). A set theory with an unlimited amount of education but are found in other mathematicians of the late 19th century: Richard Dedekind and Giuseppe Peano Gottlob Frege. It is therefore quite typical of the early set theory. From the perspective of mathematicians of the 20th century it was called naive set theory, since it leads in certain extreme amount of training to contradictions. Known antinomies, which are also known as logical paradoxes are, for example, the following in the naive set theory:
- The set of all ordinals leads to Burali - Forti paradox of 1897 (first published antinomy ).
- The set of all cardinal numbers produced the first Cantor's antinomy of 1897.
- The set of all things or quantities produced the second Cantor's antinomy of 1899.
- The set of all sets that do not contain themselves as elements, gives Russell's antinomy of 1902.
Such genuine logical contradictions are only provable if naively assumed axioms stipulate the existence of all sets to any properties. This applies for example to the oldest amounts or classes calculus of Giuseppe Peano arithmetic from his of 1889. Acquaintance was the younger contradictory set-theoretic calculus Gottlob Frege's arithmetic of 1893, as Russell 1902, the Russell antinomy proved in him. These two early volumes calculi are therefore certain to be classified as naive quantity teachings, even though they are the first attempts to formalize set theory and axiomatic to clarify.
The fact that Cantor's non- axiomatic set theory is inconsistent, however, is not provable, since its definition alone does not generate a contradiction. Cantor himself did not change its definition when he discovered his antinomies, but separated quantities as consistent multiplicities, their interaction composure Will is to " one thing" possible inconsistent multiplicities, where this is not the case. He obviously understood the term " multiplicity " in the sense of today's general class term; his inconsistent multiplicities thus corresponds to the modern concept of the real class. Cantor set definition and set theory so do not be naive in historically close inspection; they are more likely to be characterized as "open", there is not enough grip without clear axioms of set term. The problem with Cantor is thus only the blur of the crowd concept, one can interpret meaningful or meaningless. Only the senseless interpretation produces antinomies or paradoxes here, including so-called semantic paradoxes, where the unclear statements syntax is utilized to unacceptable amount formations; known examples are:
- The set of all finite definable decimal results in the Richardian paradox of 1905.
- The set of all natural numbers definable finally yields the Berry paradox of 1908.
- The set of all heterological words (they call a feature that they themselves do not possess ) generates the Grelling -Nelson's antinomy of 1908.
The transition from the naive set theory to a generally accepted axiomatic set theory was a long historical process with different approaches. Ernst Zermelo created in 1907 the first axiomatic set theory with the goal of preventing both types of paradoxes; these Zermelo set theory allows one hand to the amount training only " definite " statements that arise from the equality and the element predicate by logically combining, on the other hand regulates them the amount of education by axioms that are so narrow that the antinomian quantities can no longer be formed, and so far, that all necessary for the derivation of Cantor's set theory quantities can be formed. This goal Zermelo reached but only the advanced, predicate logic clarified Zermelo -Fraenkel set theory ( ZFC ). She sat down in the 20th century gradually and became widely recognized foundation of modern mathematics.
Conflicting naive volumes were previously in ZFC no longer be formed because Zermelo's axiom only allows a limited amount of education. Proved the consistency is only for set theory with finite sets ( ZFC without axiom of infinity ), but not with endless amounts of Godel's incompleteness theorem due. This also applies to extensions of ZFC set theory to the class logic in which the universal class, the ordinal class or the Russell class can be formed as proper classes, but not as sets.