Set Theory

Abstract

12. An ordinal written as ωαn1 + . . . + ω αnk, where α1 > . . . > αk are ordinals (and k and n1, . . . , nk are non-zero natural numbers), is said to be in Cantor Normal Form. Show that every non-zero ordinal has a unique Cantor Normal Form. What is the Cantor Normal Form for the ordinal ǫ0? 13. What is the smallest fixed point of α 7→ ω? The next smallest? And the next smallest? Show that the fixed points are unbounded, and explain why this means that we may index the fixed points by the ordinals. Is there a countable ordinal α such that α is the α-th fixed point?