Set Versus Collection | StudyTution

Set theory as a foundattion for mathematics

  • A set is a collection of items
  • Use set theory to build up all of mathematics
  • Georg Cantor, Richard Dedekind 1870s
  • Natural numbers can be “defined” as follows
  • 0 corresponds to the empty set ∅
  • 1 is the set {0, {0}} = {∅, {∅}}
  • 2 is the set {1, {1}}
  • . . .
  • j + 1 is the set {j, {j}}
  • Define arithmetic operations in terms of set building

Russell’s Paradox

  • Set theory assumes the emptyset ∅ and basic set building operations
  • Union ∪, Intersection ∩, Cartesian product ×, . . .
  • Set comprehension — subset that satisfies a condition
  • Is every collection a set? Is there a set of all sets?
  • Consider S, all sets that do not contain themselves
  • S is a set, by set comprehension
  • Does S belong to S?
  • Yes? But elements of S do not contain themselves
  • No? Any set that does not contain itself should be in S
  • Russell’s Paradox — also discovered by Ernst Zermelo
    Cannot have “set of all sets”

Sets and collections

  • Russell’s Paradox tells us that not every collection can be called a set
  • Collection that is not a set is sometimes called a class
  • The paradox had a major impact on set theory as a logical foundation of mathematics
  • For us, just be sure that we always build new sets from existing sets
  • Don’t manufacture sets “out of thin air” — “set” of all sets
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