Ordinal Exponentiation of Terms

From ProofWiki
Jump to: navigation, search

Theorem

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

Let $n$ be a finite ordinal.

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 $z = 1$.


\(\displaystyle \left({ x^y \times n }\right) ^1\) \(=\) \(\displaystyle x^y \times n\)          definition of ordinal exponentiation          
\(\displaystyle \) \(=\) \(\displaystyle x^{y \mathop \times 1} \times n\)          Ordinal Multiplication by One          

This proves the basis for the induction.

$\Box$

Induction Step

\(\displaystyle \left({ x^y \times n }\right) ^z\) \(=\) \(\displaystyle x^{y \mathop \times z} \times n\)          Inductive Hypothesis          
\(\displaystyle \implies\) \(\displaystyle \left({ x^y \times n }\right) ^{z^+}\) \(=\) \(\displaystyle \left({ x^y \times n }\right) ^z \times x^y \times n\)          definition of ordinal exponentiation          
\(\displaystyle \) \(=\) \(\displaystyle x^{y \mathop \times z} \times n \times x^y \times n\)          Inductive Hypothesis          
\(\displaystyle \) \(=\) \(\displaystyle x^{y \mathop \times z} \times x^y \times n\)          Finite Ordinal Times Ordinal          
\(\displaystyle \) \(=\) \(\displaystyle x^{y \mathop \times z + y} \times n\)          Ordinal Sum of Powers          
\(\displaystyle \) \(=\) \(\displaystyle x^{y \mathop \times z^+} \times n\)          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:

\(\displaystyle x^{y \mathop \times z}\) \(=\) \(\displaystyle \left({x^y}\right)^z\)          Ordinal Power of Power          
\(\displaystyle \) \(\le\) \(\displaystyle \left({x^y \times n}\right)^z\)          Subset Right Compatible with Ordinal Exponentiation          
\(\displaystyle \) \(=\) \(\displaystyle \bigcup_{w \in z} \left({x^y \times n}\right)^w\)          definition of ordinal exponentiation          
\(\displaystyle \) \(\le\) \(\displaystyle \bigcup_{w \in z} x^{y \mathop \times w} \times n\)          Proof by Cases where $w \in K_{I}$ or $w \in K_{II}$          
\(\displaystyle \) \(\le\) \(\displaystyle \bigcup_{w \in z} x^{y \mathop \times w + 1}\)          Membership is Left Compatible with Ordinal Exponentiation          
\(\displaystyle \) \(\le\) \(\displaystyle \bigcup_{w \in z} x^{y \mathop \times w^+}\)          Membership is Left Compatible with Ordinal Multiplication and Membership is Left Compatible with Ordinal Exponentiation          
\(\displaystyle \) \(=\) \(\displaystyle \bigcup_{w \in z} \left({ x^y }\right)^{w^+}\)          Ordinal Power of Power          
\(\displaystyle \) \(=\) \(\displaystyle \left({ x^y }\right)^z\)          definition of ordinal exponentiation          

This proves the limit case.

$\blacksquare$


Sources