Set theory

Set theory is the basic part of mathematics. The whole of mathematics as it is taught today is usually formulated in the language of set theory and is based on the axioms of set theory. Most mathematical objects that are treated in some areas such as algebra, calculus, geometry, stochastics or topology, to name only a few, can be defined as sets. Measured by this set theory is a relatively young science; only after overcoming the foundational crisis of mathematics at the beginning of the 20th century set theory could take its present, central and fundamental place in mathematics.

History

19th century

Set theory was founded by Georg Cantor in the years 1874-1897. Instead of the term amount he used initially words such as " quintessence " or " diversity "; of sets and set theory he said later. In 1895 he formulated the following set definition:

" By a" set " we understand any collection M of specific well-distinguished objects m of our intuition or our thought (which the " " are called by M into a whole. Items ) "

Cantor classified the quantities, in particular the infinite, according to their thickness. Finite amount which is the number of its elements. He called two quantities equivalent ( same cardinality ) if they can be mapped bijectively to each other. The cardinality or cardinal number of a set M is according to Cantor, the equivalence class of equivalent to M ( the same powerful ) quantities. He watched probably the first, that there are different infinite widths. The set of natural numbers and all of the same powerful countable sets are called according to Cantor, all other infinite sets are called uncountable.

• The amounts of the natural, the rational ( Cantor's first diagonal argument ) and the algebraic numbers are countable and thus equally powerful.
• The set of real numbers has greater power than that of the natural numbers, is therefore nichtabzählbar ( Cantor's second diagonal argument).
• The set of all subsets of a set M has ( its power set ) is always greater power than M, which is also known as a set of Cantor.
• From two sets is at least an equally powerful to a subset of the other. This is proved with the help of the detail treated by Cantor well-ordering.
• There are uncountably many widths.

Cantor called the continuum problem: Is there a thickness between that of the set of natural numbers and that of the set of real numbers? He even tried to solve it, but was unsuccessful. Later it turned out that the issue is not in principle decidable.

In addition to Cantor and Richard Dedekind was an important pioneer of set theory. He spoke of systems instead of quantities and in 1872 he developed a set-theoretical construction of the real numbers and 1888 a verbal set-theoretic axiomatization of the natural numbers. He formulated here as the first extensionality of set theory.

Designated Giuseppe Peano, the amounts as classes already created in 1889 the first formal class logic calculus as the basis for its arithmetic with the Peano axioms, which he first formulated in a precise set-theoretic language. He thus developed the basis for today's formal language of set theory, and led many in use today symbols, especially the element sign that says " is an element of" as is verbalized.

Another set-theoretic justification of arithmetic tried Gottlob Frege little later in his calculus of 1893. Discovered in this Bertrand Russell 1902 is a contradiction, which became known as Russell's antinomy. This contradiction and other contradictions arise from an unlimited amount of education, which is why the early form of set theory was later called naive set theory. But Cantor set definition does not intend such a naive set theory, as his proof of the universal class is a non- quantity by the second Cantor's antinomy.

Cantor's set theory was hardly known to his contemporaries in their meaning and in no way considered a revolutionary advance, but was met with some mathematicians, such as Leopold Kronecker rejected. Even more they fell into disrepute, were known as antinomies, so that about Henri Poincaré scoffed: " The logic is no longer sterile - it now reflects contradictions. "

20th century

In the 20th century, Cantor's ideas translated by more and more; at the same time took place within the evolving mathematical logic one axiomatization of set theory by which previously prevailing contradictions could be overcome.

1903/1908 developed Bertrand Russell 's theory of types, always have in the amounts of a higher type than their elements so problematic amount of training would be impossible. He rejected the first way out of the contradictions and showed in the Principia Mathematica of 1910-1913 also a piece of the performance of the applied theory of types. Ultimately, however, it proved to be inadequate for Cantor's set theory and could not prevail because of their complexity.

Handy and contrast successful was the 1907 developed by Ernst Zermelo axiomatic set theory, which he created specifically for non-contradictory reasoning of the set theory of Cantor and Dedekind. Abraham Fraenkel remarked in 1921 that in addition to its replacement axiom is needed. Zermelo it fitted into his Zermelo -Fraenkel system of 1930, which he called ZF system shortly. He designed it for Elements that are not quantities but as set elements come into question and take into account Cantor " objects of our intuition ." Today's Zermelo -Fraenkel set theory, however, is by Fraenkel idea a pure set theory whose objects are exclusively quantities.

Many mathematicians translated but instead on a consistent axiomatization to a pragmatic set theory, avoided the problem sets, such as the often issued amount teachings of Felix Hausdorff in 1914 or by Erich Kamke from 1928. Gradually, it became increasingly aware of mathematicians that the set theory is an indispensable basis for the structuring of mathematics. The IF system proved successful in practice, so it is now recognized as the basis of modern mathematics by the majority of mathematicians; no contradictions could be derived from the IF system more. However, the consistency could only be detected for the set theory with finite sets, but not for the whole IF system that includes Cantor's set theory with infinite sets; according to Gödel's incompleteness theorem of 1931 is such evidence of consistency in principle not possible. Gödel's discoveries infected only Hilbert's program to put the mathematics and set theory to a proven consistent axiomatic basis, a limit, but prevented the success of the theory in any way, so that spoke of a foundational crisis of mathematics, of the followers of intuitionism, in reality There was nothing to feel.

The final recognition of ZF set theory in practice, however, withdrew even for prolonged periods. The mathematician Nicolas Bourbaki group with pseudonym contributed significantly to this recognition; she wanted the math based on set theory represent uniformly, and emphasized this in 1939 into central areas mathematics successfully. In the 1960s, it became widely known that the ZF set theory is the basis of mathematics. There was even a temporary period in which the set theory was treated in the primary school.

Parallel to the success story of set theory, however, the discussion of the amount of axioms in the professional world remained to date. It also emerged alternative axiomatic set doctrines, about 1937, not to Cantor or Zermelo - Fraenkel, but on the theory of types exploratory set theory by Willard Van Orman Quine from the New Foundations ( NF), 1940, the Neumann - Bernays - Gödel set theory, ZF generalized to classes, or 1955, the Ackermann set theory, who continued to re- Cantor set definition.

Definitions

In pure set theory is the element predicate ( ie is an element of ) the only necessary basic relation. All set-theoretic concepts and statements be defined from it by logical operators of predicate logic.

Laws

The amount with respect to the relation partially ordered, as is true for all:

• Reflexivity:
• Antisymmetry: Off and follows
• Transitivity: For and follows

The set operations union and intersection are commutative, associative, and mutually distributive:

• Associative law: and
• Commutative law: and
• Distributive: and
• De Morgan's law: and
• Absorption law: and

For the difference of the following rules apply:

• Associative laws: and
• Distributive: and and and

For the symmetric difference following rules apply:

• Associative law:
• Commutative:
• Distributive: