Cantor normal form: Difference between revisions
Jump to navigation
Jump to search
Content added Content deleted
(Created page with "'''Cantor normal form''' is a standard form of writing ordinals. 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 smaller than \( \alpha \). Thus, Cantor normal form can be iterated to form a notatio...") |
CreeperBomb (talk | contribs) mNo edit summary |
||
| (2 intermediate revisions by 2 users not shown) | |||
| Line 1: | Line 1: | ||
{{stub}} |
|||
| ⚫ | |||
| ⚫ | '''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. |
||
| ⚫ | |||
| ⚫ | |||
Latest revision as of 03:36, 24 March 2024
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 \( \varepsilon_0 \).