Cantor normal form

Cantor normal form is a standard form of writing ordinals. Cantor's normal form theorem states that every ordinal \( \alpha \) can be written uniquely as \( \omega^{\beta_1} + \omega^{\beta_2} + \dots + \omega^{\beta_k} \), where \( \beta_1 \ge \beta_2 \ge \dots \ge \beta_k \) and \( k \ge 0 \) is an integer.

When \( \alpha \) is smaller than \( \varepsilon_0 \), the exponents \( \beta_1 \) through \( \beta_k \) are all strictly smaller than \( \alpha \). Thus, Cantor normal form can be iterated to form an ordinal notation system for ordinals less than \( \varepsilon_0 \).