Study guide What was Bolzano’s 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 Frege’s 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 < for cardinals? Show that there are as many points in the plane as on the real line.

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 today’s 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 today’s verdict on the claim of Cantor the cardinals are well ordered? What other names does this claim have. Formulate Russel’s 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 Russel’s paradox? Explain! What has to be assumed to prove that the partial order on cardinals is an order? Show that w+3>w Discuss Hilbert’s Hotel at infinity, i.e. show , !