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Ordinal exponentiation ( $α^{β}$ )

Ordinal exponentiation is defined recursively by

α^{0} = 1, α^{β + 1} = α^{β} α, α^{λ} = sup_{ξ < λ} α^{ξ}

for nonzero limit

λ

. Transfinite induction gives

α^{β + γ} = α^{β} α^{γ} .

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