Ordinal Exponentiation via Cantor Normal Form/Corollary

Theorem

Let $x$ and $y$ be ordinals.

Let $x$ be a limit ordinal and let $y > 0$.

Let $\sequence {a_i}$ be a sequence of ordinals that is strictly decreasing on $1 \le i \le n$.

Let $\sequence {b_i}$ be a sequence of natural numbers.

Then:

$\ds \paren {\sum_{i \mathop = 1}^n x^{a_i} \times b_i}^{x^y} = x^{a_1 \mathop \times x^y}$

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Proof

By the hypothesis, $x^y$ is a limit ordinal by Limit Ordinals Closed under Ordinal Exponentiation.

The result follows from Ordinal Exponentiation via Cantor Normal Form/Limit Exponents.

$\blacksquare$

Also see

• Cantor Normal Form

Sources

• 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 8.50$