Study guide

1
Study guide

• What was Bolzanos contribution to set theory and
  when did he make it.
• What is the cardinal of N,Q,Z? Prove your
  statement.
• Give two formulations of the axiom of choice!
• What is the history of the axiom of choice? Who
  proved what and when about is the relationship
  between the axiom of choice and ZF.
• Formulate the continuum hypothesis.
• What is the history of the continuum hypothesis?
  Who proved what and when about the relationship
  between the continuum hypothesis and ZFC.
• Give Cantors and Freges definition of cardinal.
• Give a definition of ordinals.
• Give a set and a subset of it which are in 1-1
  correspondence to each other.
• What can you say about the cardinals of a set and
  a subset?
• Define a partial order on cardinals!
• What is the relationship between inclusion and lt
  for cardinals?
• Show that there are as many points in the plane
  as on the real line.

2
Study Guide

• Show that the cardinal of R is strictly bigger
  than
• Define addition, multiplication and powers for
  cardinals.
• Argue that c
• Show that
• Give an ordinal which is bigger that the ordinal
  of N but whose cardinal is
• What is the todays verdict on the claim of
  Cantor that all infinite subsets of R are either
  countable or have cardinality c of the continuum?
• What is the todays verdict on the claim of
  Cantor the cardinals are well ordered? What other
  names does this claim have.
• Formulate Russels Paradox and discuss it!
• Which axiom of ZF postulates the natural numbers?
  Explain!
• Which axiom of ZF forbids the forming of sets as
  the one that appears in Russels paradox?
  Explain!
• What has to be assumed to prove that the partial
  order on cardinals is an order?
• Show that w3gtw
• Discuss Hilberts Hotel at infinity, i.e. show
  , !