User:Jamesholbert
Definition of Set[edit]
Naively, a set is a collection of distinct objects (also called elements or members). Widely considered the founder of set theory, Georg Cantor published a journal article in 1895, defining a set, or aggregate (Menge),
"Unter einer 'Menge' verstehen wir jede Zusammenfassung M von bestimmten wohluntershiedenen Objecten m unsrer Anschauung oder unseres Denkens (welche die 'Elemente' von M genannt werden) zu einem Ganzen."[1]
An English translation is
"By a 'Set', we understand any Collection M of certain well-distinguished Objects m of our intuition or our thought (which are called the 'Elements' of M) into a Whole."
Also, see Philip Jourdain's translation in the image to the right.[2]
Cantor's definition is, however, inadequate for formal mathematics. In a letter to Gottlob Frege in 1902, Bertrand Russell pointed out his eponomyous paradox, which immediately follows from Cantor's inadequate conception of a set.[3] Russell's paradox is simply the contradiction that arises from using Cantor's definition and considering the set R of all things that are not members of themselves. Thus, R is a member of itself if and only if R is not a member of itself. That is, R is both a member and not a member of itself.
Patrick Suppes rigorously defines a set in his Axiomatic Set Theory (1960) as "something which has members or is the empty set", that is,
- y is a set if and only if (∃x)(x ∈ y ∨ y = 0).[4]
This definition is in the context of a formal logic system based on an article by John von Neumann in 1927 and detailed in a book by Alonzo Church in 1956.[5] [6] The system has five classifications of symbols — constants, variables, sentential connectives, quantifiers, and grouping punctuation — along with two primitive constants, 0 denoting the empty set and ∈ denoting the membership relation.
Mathematicians often avoid a rigorous definition of a set. Paul Halmos starts his Naive Set Theory with
One thing that the development will not include is a definition of sets. The situation is analogous to the familiar axiomatic approach to elementary geometry.[7]
He continues, "A pack of wolves, a bunch of grapes, or a flock of pigeons are all examples of sets.... An element of a set may be a wolf, a grape, or a pigeon.... [or another] set itself."[8]
Today, most working mathematicians outside of mathematical logic or category theory, accept a set as a primitive notion, that is, an undefined concept, like points in Euclidean geometry. This is the view taken in the most commonly accepted foundation of mathematics, the axiomatic set theory known as Zermelo-Fraenkel set theory, which the ZFC axioms establish.
References[edit]
- ^ Cantor, Georg. Beiträge zur Begründung der transfiniten Mengenlehre. (Erster Artikel.) Mathematische Annalen, Vol. 46, Issue 4, pp 481–512. (1895).
- ^ Cantor, Georg. Contributions to the Foundation of the Theory of Transfinite Numbers. Translated by Jourdain, Philip E. B. Dover. (1915).
- ^ Jean van Heijenoort (1967).
- ^ Suppes, Patrick. Axiomatic Set Theory. D. Van Nostrand, p. 19. (1960).
- ^ Neumann, John von. Zur Hilbertschen Beweistheorie. Math. Zeitschrift, Vol. 26, pp. 1–46. (1927).
- ^ Church, Alonzo. Introduction to Mathematical Logic. Princeton University Press. (1956).
- ^ Halmos, Paul R. Naive Set Theory. D. Van Nostrand, p. 1. (1960).
- ^ Ibid.