\( \omega^\omega \)

The ordinal \( \omega^\omega \) is relatively small compared to other countable ordinals, but has some interesting properties. In particular, \( \omega^\omega \) is:

• The least \( \alpha \) so that \( \alpha \) is the \( \alpha \)th limit ordinal.
• The least limit of additive principal ordinals.
• The least ordinal which is, for all \( n < \omega \), an element of the class \( L^n(\mathrm{Ord}) \), where \( L \) is the limit point operator.
• The proof-theoretic ordinal of \(\mathrm{RCA}_0\)
• The proof-theoretic ordinal of \(\mathrm{WKL}_0\).
• The proof-theoretic ordinal of Peano arithmetic, with induction restricted to \( \Sigma^0_1 \)-formulae.
• The proof-theoretic ordinal of primitive recursive arithmetic.