# Ordinal Exponentiation of Terms

From ProofWiki

## Theorem

Let $x$, $y$, and $z$ be ordinals.

Let $n$ be a natural number.

Let $x$ be a limit ordinal.

Let $y$, $z$ and $n$ be greater than $0$.

Then:

- $\left({ x^y \times n }\right) ^z = x^{y \mathop \times z} \times n$ if $z$ is not a limit ordinal

- $\left({ x^y \times n }\right) ^z = x^{y \mathop \times z}$ if $z$ is a limit ordinal

## Proof

The proof shall proceed by Transfinite Induction on $z$.

### Basis for the Induction

The hypothesis requires that $z \ne 0$, so the induction starts at $z = 1$.

\(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \left({ x^y \times n }\right) ^1\) | \(=\) | \(\displaystyle \) | \(\) | \(\displaystyle \) | \(\displaystyle x^y \times n\) | \(\displaystyle \) | \(\displaystyle \) | definition of ordinal exponentiation | ||

\(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \) | \(\) | \(\displaystyle \) | \(\displaystyle x^{y \mathop \times 1} \times n\) | \(\displaystyle \) | \(\displaystyle \) | Ordinal Multiplication by One |

This proves the basis for the induction.

$\Box$

### Induction Step

\(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \left({ x^y \times n }\right) ^z\) | \(=\) | \(\displaystyle \) | \(\) | \(\displaystyle \) | \(\displaystyle x^{y \mathop \times z} \times n\) | \(\displaystyle \) | \(\displaystyle \) | Inductive Hypothesis | ||

\(\displaystyle \) | \(\displaystyle \implies\) | \(\displaystyle \) | \(\displaystyle \left({ x^y \times n }\right) ^{z^+}\) | \(=\) | \(\displaystyle \) | \(\) | \(\displaystyle \) | \(\displaystyle \left({ x^y \times n }\right) ^z \times x^y \times n\) | \(\displaystyle \) | \(\displaystyle \) | definition of ordinal exponentiation | ||

\(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \) | \(\) | \(\displaystyle \) | \(\displaystyle x^{y \mathop \times z} \times n \times x^y \times n\) | \(\displaystyle \) | \(\displaystyle \) | Inductive Hypothesis | ||

\(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \) | \(\) | \(\displaystyle \) | \(\displaystyle x^{y \mathop \times z} \times x^y \times n\) | \(\displaystyle \) | \(\displaystyle \) | Natural Number Times Ordinal | ||

\(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \) | \(\) | \(\displaystyle \) | \(\displaystyle x^{y \mathop \times z + y} \times n\) | \(\displaystyle \) | \(\displaystyle \) | Ordinal Sum of Powers | ||

\(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \) | \(\) | \(\displaystyle \) | \(\displaystyle x^{y \mathop \times z^+} \times n\) | \(\displaystyle \) | \(\displaystyle \) | definition of ordinal multiplication |

This proves the induction step.

$\Box$

### Limit Case

Suppose that this statement holds for all $w \in z$ where $z$ is a limit ordinal.

Then:

\(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle x^{y \mathop \times z}\) | \(=\) | \(\displaystyle \) | \(\) | \(\displaystyle \) | \(\displaystyle \left({x^y}\right)^z\) | \(\displaystyle \) | \(\displaystyle \) | Ordinal Power of Power | ||

\(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\le\) | \(\displaystyle \) | \(\) | \(\displaystyle \) | \(\displaystyle \left({x^y \times n}\right)^z\) | \(\displaystyle \) | \(\displaystyle \) | Subset Right Compatible with Ordinal Exponentiation | ||

\(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \) | \(\) | \(\displaystyle \) | \(\displaystyle \bigcup_{w \in z} \left({x^y \times n}\right)^w\) | \(\displaystyle \) | \(\displaystyle \) | definition of ordinal exponentiation | ||

\(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\le\) | \(\displaystyle \) | \(\) | \(\displaystyle \) | \(\displaystyle \bigcup_{w \in z} x^{y \mathop \times w} \times n\) | \(\displaystyle \) | \(\displaystyle \) | Proof by Cases where $w \in K_{I}$ or $w \in K_{II}$ | ||

\(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\le\) | \(\displaystyle \) | \(\) | \(\displaystyle \) | \(\displaystyle \bigcup_{w \in z} x^{y \mathop \times w + 1}\) | \(\displaystyle \) | \(\displaystyle \) | Membership is Left Compatible with Ordinal Exponentiation | ||

\(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\le\) | \(\displaystyle \) | \(\) | \(\displaystyle \) | \(\displaystyle \bigcup_{w \in z} x^{y \mathop \times w^+}\) | \(\displaystyle \) | \(\displaystyle \) | Membership is Left Compatible with Ordinal Multiplication and Membership is Left Compatible with Ordinal Exponentiation | ||

\(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \) | \(\) | \(\displaystyle \) | \(\displaystyle \bigcup_{w \in z} \left({ x^y }\right)^{w^+}\) | \(\displaystyle \) | \(\displaystyle \) | Ordinal Power of Power | ||

\(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \) | \(\) | \(\displaystyle \) | \(\displaystyle \left({ x^y }\right)^z\) | \(\displaystyle \) | \(\displaystyle \) | definition of ordinal exponentiation |

This proves the limit case.

$\blacksquare$

## Sources

- Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*(1971): $\S 8.47$