Sets and Size

1
Sets and Size

• Basic Question Compare the size of sets.
• First distinction finite or infinite.
• What is a finite set?
• How can one compare finite sets?
• Number of elements
• Inclusion A proper subset has less elements
• What about infinite sets?
• By definition they all have infinitely many
  elements.
• Does a proper subset have less elements?

2
Basic questions and techniques

• Good tool to compare sets is a 1-1 correspondence
  which is sometimes called a bijection.
• This works well for finite sets and also for
  infinite sets.
• It gives a way to talk about the size, magnitude
  or power of infinite sets.
• Basic question compare the infinite sets of the
  natural numbers N, the rational numbers Q and the
  real numbers R.

3
Set Theory

• Greek philosophers the infinite as a source for
  paradox.
• Aristotle (384-322 BC)
• Zeno (495?-435? B.C.)
• Bernard Bolzano (1781-1848) a first mathematical
  study of the nature of infinite sets.
• Georg Cantor (1845-1918) A comprehensive study
  of infinite sets. The beginning of modern set
  theory. Introduces cardinals, ordinals the
  continuum hypothesis.
• Bertrand Russel (1872-1970). Gives his paradox
  which shows that the naïve approach to set theory
  like that of Frege will not work.
• Ernst Zermelo (1871-1953). Introduces the Axiom
  of Choice and shows that with it all sets are can
  be well ordered. Together with Adolf Fraenkel
  (1891-1965) gives axioms of set-theory.

4
Axioms of choice and the continuum hypothesis

• Cantor introduced the continuum hypothesis and
  believed that he can prove it.
• Zermelo introduced the axiom of choice.
• Zermelo and Fraenkel introduced the axioms of set
  theory ZF and ZFC.
• Kurt Gödel (1906-1978) showed that
• It is impossible to prove the consistency of set
  theory within set theory.
• The axiom of choice is relatively consistent,
  i.e. if ZF is consistent, so is ZFC.
• The continuum hypothesis is relatively consistent
  w.r.t. ZFC.
• Paul Cohen (1934)
• The negation of the continuum hypothesis is
  relatively consistent w.r.t. ZFC, i.e. it is
  independent.
• The axiom of choice is independent of ZF.