Wikipedia 10K Redux

Reconstructed by Reagle from Starling archive; see blog post for context.

Set

A set is a collection of objects. For example, one can define the set S = {Sn: Sn is a sibling of the Larry M. Sanger, who is the Editor-in-Chief of Nupedia}.

We require that sets be well-defined. Given an object Sn, we must be able to determine if Sn belongs to S.----

What, there are no recursively enumerable sets?

Mathematicians are prone to present sets of numbers as examples:

  • Natural Numbers which are used for counting the members of sets.
  • Integers which appear as solutions to equations like x+a=b.
  • Rational Numbers which appear as solutions to equations like a+bx=c.
  • Algebraic Numbers which can appear as solutions to polynomial equations (with rational coefficients) and may involve radicals.
  • Real Numbers which include Transcendental Numbers (which can't appear as solutions to polynomial equations with rational coefficents) as well as the Algebraic Numbers
  • Complex Numbers which provide solutions to equations such as x^2+1=0.

    • Statistical Theory is built on the base of Set Theory and Probabillity Theory.

    Care must be taken with verbal descriptions of sets. One can describe in words a set which has no members. If one assumes it has members, an apparent Paradox or Antinomy may occur.

    Wrap you mind around the barber who shaves everyone in town and only those who do not shave themselves. Now, ask yourself, "Who shaves the barber?"