Set Theory with the Axiom of Choice
In this chapter, the authors look at the relationship between axiom of choice (AC) and the well-ordering principle, namely that every set can be well-ordered. They also look at the impact of AC on cardinal arithmetic. Using AC the authors can define arithmetic operations on cardinals. The authors state and prove attractive equalities involving cardinals, fully justifying Georg Cantor's description of cardinals as possessing an arithmetic. They explore what is probably the major problem posed by Cantor but not resolved during his lifetime, the continuum problem. For Cantor and his contemporaries, a 'continuum' was essentially any non-trivial segment of the real une and the continuum was the segment consisting of the real line itself. Hubert's problems, presented at the Paris International Congress of Mathematicians in 1900, both set and reflected much of the agenda for mathematical research in the 20th century.