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Richard Dedekind
Richard Dedekind In 1856, German Mathematician Richard Dedekind began a series of lectures on Group Theory -- using the word "domain" to refer to sets and groups. Dedekind will spend the next several decades abstracting algebra, and will develop a "set theory" as a framework for his research. -
Georg Cantor
Georg Cantor In contrast to much of Dedekind's work involving discrete sets, Cantor begins lecturing and researching on continuous sets. -
Cantor and Dedekind
Georg Cantor and Richard Dedekind meet for the first time and initiate an on-again off-again correspondence. They would collaborate several times over the following decades. -
Cantor -- R is uncountable
Cantor first establishes that the set of Real Numbers is not countable. Later, algebraic numbers are countable and transcendental numbers are not countable. -
Cantor - Cardinality
Cantor defines cardinality as the size of a given set. Cantor further declares the cardinality of N is aleph-naught and that the cardinality of all infinite sets is not equal. (The cardinality of all countably infinite sets is also aleph-null.) Aleph-numbers -
Richard Dedekind -- Substitutions
Richard Dedekind releases a text on "substitutions." Dedekind publishes a book on the subject of substitutions, by which he means a proccess which transforms "elements a, b, c, ..." of one "domain" into elements "a', b', c', ..." of another "domain." Dedekind goes on on the define an "equivalence" of these domains that is realized when the transformation preserves the original domain's size and structure. -
John Venn -- Venn Diagrams
Venn Diagrams A visualization of the contents of sets -- along with their intersecteing and disjointed subsets. The name Venn Diagram would come about later; Venn referred to his creation as Eulerian Circles, as it was based on Euler Diagrams developed 100 years earlier. Proper Venn Diagrams differ from Euler Diagrams in that all possible intersections are shown. In Euler Diagrams, disjoint sets are depicted as such. -
Cantor -- Proof that R is uncountable
Cantor publishes his famous, elegant proof concerning the non-countability of R. -
Giuseppe Peano - Notation
Peano's text "Formulario Mathematico" established much of the standard set theoretical notation common today. Notation included:
- intersection and union,
- inclusion,
- element of, and
- differences. -
Cantor - Cantor's Paradox
Georg Cantor discovers Cantor's paradox, a first indication of the inadequacy of the axioms of naive set theory. -
Russell's Paradox
Burtrand Russell discovers a paradox inherent in Naive Set Theory, "the set of all sets that do not contain themselves" establishing a need for an axiomatic set theory. -
ZFC Completed
ZFC Axioms Zermelo–Fraenkel Set Theory with the Axiom of Choice is finally established. A set of nine axioms ensure that no paradoxical sets can be built. These axioms include definitions of equality of sets, prohibitions agains sets containing themselves, etc.
