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Ordinal Arithmetic and Normal Form

Ordinal Arithmetic#

Addition, multiplication, exponentiations are all defined in the canonical manner (initial case, successor case), but with limits.

Definition. (Addition). For all ordinals \(\alpha\),

  1. \(\alpha + 0 = \alpha\).
  2. \(\alpha+(\beta+1) = (\alpha+\beta)+1\) for all \(\beta\).
  3. \(\alpha+\beta = \lim_{\xi\to\beta}\alpha+\beta\), for all limit \(\beta > 0\).

Definition. (Multiplication). For all ordinals \(\alpha\),

  1. \(\alpha\cdot 0 = 0\).
  2. \(\alpha\cdot(\beta + 1) = \alpha\cdot\beta + \beta\) for all \(\beta\).
  3. \(\alpha\cdot\beta = \lim_{\xi\to\beta} \alpha\cdot\xi\) for all limit \(\beta > 0\).

Definition. (Exponentiation). For all ordinals \(\alpha\),

  1. \(\alpha^0 = 1\).
  2. \(\alpha^{\beta+1} = \alpha^\beta\cdot\alpha\) for all \(\beta\).
  3. \(\alpha^\beta = \lim_{\xi\to\beta}\alpha^\xi\) for all limit \(\beta>0\).

By induction it can be proved that \(+\) and \(\cdot\) are associative, but neither of them are commutative.

Ordinal sums and products can be defined geometrically as can sums and products of arbitrary linear orders. We can further prove the following:

  1. If \(\beta<\gamma\) then \(\alpha+\beta < \alpha + \gamma\). (by induction)
  2. If \(\alpha<\beta\) then there is a unique \(\delta\) such that \(\alpha+\delta =\beta\). (let \(\delta\) be the order type of the set \(\{\xi:\alpha\leq\xi<\beta\}\))
  3. If \(\beta<\gamma\) and \(\alpha>0\) then \(\alpha\cdot \beta < \alpha \cdot \gamma\). (by induction)
  4. If \(\alpha>0\) and \(\gamma\) is arbitrary then there is a unique \(\beta\) and a unique \(\rho<\alpha\) s.t. \(\gamma = \alpha\cdot\beta +\rho\). (let \(\beta\) be the greatest ordinal s.t. \(\alpha\cdot\beta\leq \gamma\), important)
  5. If \(\beta<\gamma\) and \(\alpha>1\) then \(\alpha^\beta < \alpha^\gamma\). (by induction)

Cantor's Normal Form Theorem#

Theorem. Every ordinal \(\alpha > 0\) can be represented uniquely in the form

\[\alpha = \omega^{\beta_1}\cdot k_1 + \ldots + \omega^{\beta_n}\cdot k_n\]

where \(n\geq 1, \alpha \geq \beta_1 > \ldots > \beta_n\) and \(k_1,\ldots,k_n\) are nonzero natural numbers.

Note that in the normal form it is possible to have \(\alpha = \omega^\alpha\), the least ordinal with this property is called \(\epsilon_0\).