Naive set theory

The concept of naive set theory emerged at the beginning of the 20th century for the set theory of the 19th century , in which an unregulated or unrestricted set formation was practiced. Because of contradictions that arise in it, it was later replaced by the axiomatic set theory , in which the formation of sets is regulated by axioms . “Naive set theory” therefore primarily describes this early form of unregulated set theory and is to be understood as a contrast to axiomatic set theory. Not infrequently, however, in mathematical literature after 1960, a descriptive set theory is also described as naive; therefore this name can also be used to denote an unformalized axiomatic set theory or an axiomatic set theory without metalogical considerations.

Problem

Georg Cantor's definition of sets is often cited for the intention of unrestricted naive set formation : By a “set” we mean every combination M of certain well-differentiated objects m of our perception or our thinking (which are called the “elements” of M) into a whole . On closer inspection, however, this is not conclusive (see below). A set theory with unrestricted set formation can be found in other mathematicians of the late 19th century: Richard Dedekind and Gottlob Frege . It is therefore quite typical of early set theory. From the point of view of the mathematicians of the 20th century, it was called naive set theory, because it leads to contradictions in certain extreme set formations. Well-known antinomies , which are also called logical paradoxes , are for example the following in naive set theory:

• The set of all ordinal numbers leads to the Burali-Forti paradox of 1897 (first published antinomy).
• The set of all cardinal numbers produces Cantor's first antinomy of 1897.
• The set of all things or sets produces Cantor's second antinomy of 1899.
• The set of all sets that do not contain themselves as elements results in Russell's antinomy of 1902.

Such genuine logical contradictions can only be proven if naively assumed axioms fix the existence of all sets for any properties. This applies, for example, to the set theory on which Dedekind based his arithmetic in 1888, since there he declared all systems (classes) to be things and elements. Known was the younger contradictory set-theoretic calculus of Frege arithmetic of 1893, since in him Russell 1902 proved Russell's antinomy. These early set calculi are therefore certainly to be classified as naive set theory, although they are precisely the first attempts to make set theory more precise axiomatically.

The fact that Cantor's set theory is contradictory cannot be proven, however, since his set definition alone does not create a contradiction. Without clear axioms, his concept of sets is too open and not sufficiently catchy. It can be interpreted meaningfully or meaninglessly. Only the meaningless interpretation generates antinomies or paradoxes, including so-called semantic paradoxes, in which the unclear statement syntax is exploited to form impermissible sets; well-known examples are:

• The set of all finitely definable decimal numbers results in Richard's paradox of 1905.
• The set of all finitely definable natural numbers results in the Berry Paradox of 1908.
• The set of all heterological words (they name a characteristic that they themselves do not have) produces the Grelling-Nelson antinomy of 1908.

Cantor himself separated sets as consistent multiplicity, which “can be combined into 'one thing'”, from inconsistent multiplicity, which is not the case, in letters in which he described his antinomies. He evidently understood the concept of multiplicity in the sense of today's more general class concept; its inconsistent multiplicity therefore corresponds to the modern concept of the real class . In those letters there are also Cantor's set axioms , which he did not publish, but which clearly demonstrate his non-naive point of view.

Main solution

The transition from naive set theory to a generally recognized axiomatic set theory was a lengthy historical process with different approaches. Ernst Zermelo published an axiomatic set theory for the first time in 1908 with the aim of preventing both kinds of paradoxes; On the one hand, this Zermelo set theory allows only definite statements about the formation of sets , which arise from the equality and the element predicate through logical connection, on the other hand it regulates the set formation through axioms that are so narrow that the antinomic sets can no longer be formed, and so on far that all sets necessary for the derivation of Cantor's set theory can be formed. However, Zermelos only achieved this goal with the expanded, predicate-logic precise Zermelo-Fraenkel set theory (ZFC). It gradually gained acceptance in the 20th century and became the widely accepted foundation of modern mathematics.

Up to now, contradicting naive quantities could no longer be formed in ZFC, because Zermelo's axiom of elimination only allows a limited formation of quantities. However, the consistency can only be proven for set theory with finite sets (ZFC without the axiom of infinity ), but not with infinite sets because of Gödel's incompleteness theorem . This also applies to extensions of the ZFC set theory to class logic , in which the universal class, the ordinal number class or the Russell class can be formed as real classes , but not as sets.

Individual evidence

1. Felix Hausdorff : Basic features of the set theory , Leipzig 1914, page 1f “naive set concept”.
2. z. E.g .: Paul R. Halmos: Naive set theory , Göttingen 1968 (unformalized ZF set theory)
3. z. E.g .: Walter Felscher: Naive quantities and abstract numbers I-III , Mannheim, Vienna, Zurich, 1978/1979.
4. Contributions to the foundation of transfinite set theory , Mathematische Annalen , Vol. 46, p. 481. online
5. Richard Dedekind: What are and what are the numbers? , Braunschweig 1888, §1.2
6. ^ Gottlob Frege: Basic Laws of Arithmetic , Volume 1, Jena 1893, reprint Hildesheim 1966; in Volume 2, Jena 1903. In the epilogue pp. 253–261, Frege discusses the antinomy.
7. Russell's letter to Frege of June 16, 1902 in: Gottlob Frege: Correspondence with D. Hilbert, E. Husserl, B. Russell , ed. G. Gabriel, F. Kambartel, C. Thiel, Hamburg 1980, p. 59f.
8. ^ Letter from Cantor to Dedekind of August 3, 1899 in: Georg Cantor, Briefe, ed. H. Meschkowski and W. Nilson, Berlin, Heidelberg, New York 1999, p. 407. In the excerpt from p. 440 he said that he had already taken into account the thingness of the sets in his set definition in the summary "to a whole".
9. Ernst Zermelo: Investigations on the fundamentals of set theory , 1907, in: Mathematische Annalen 65 (1908), pp. 261, 264.