Basis Representation Theorem for Ordinals

Theorem
Let $x$ and $y$ be ordinals.

Let $x > 1$ and $y > 0$.

Then there exist unique finite sequences of ordinals:
 * $\left\langle{a_i}\right\rangle, \ \left\langle{b_i}\right\rangle$

both of unique length $n$ such that:


 * $(1): \quad \left\langle{a_i}\right\rangle$ is a strictly decreasing sequence for $1 \le i \le n$


 * $(2): \quad 0 < b_i < x$ for all $1 \le i \le n$


 * $(3): \quad \displaystyle y = \sum_{i \mathop = 1}^n x^{a_i} b_i$

Proof
The proof shall proceed by Transfinite Induction (Strong Induction) on $y$.

The inductive hypothesis states that for all $v < y$, there exist unique finite sequences of ordinals:
 * $\left\langle{c_i}\right\rangle, \ \left\langle{d_i}\right\rangle$

both of unique length $n$ such that:


 * $(1): \quad \left\langle{c_i}\right\rangle$ is strictly decreasing sequence for $1 \le i \le n$


 * $(2): \quad 0 < d_i < x$ for all $1 \le i \le n$.


 * $(3): \quad \displaystyle v = \sum_{i \mathop = 1}^n x^{c_i} d_i$

Since $x > 1$, it follows by Unique Ordinal Exponentiation Inequality that:


 * $x^z \le y < x^{z+1}$

for some unique $z$.

By the Division Theorem for Ordinals:


 * $y = x^z w + v$ and $v < x^z$

for some unique $z$, $w$, and $v$.

Thus:

So $v < y$.

If $v = 0$, then the statement is proven.

If $v \ne 0$, then we may apply the inductive hypothesis.

By the inductive hypothesis,


 * $\displaystyle v = \sum_{i \mathop = 1}^n x^{c_i} d_i$

Therefore, $\displaystyle y = x^z w + \sum_{i \mathop = 1}^n x^{c_i} d_i$

Set:


 * $a_1 = z$


 * $a_{i + 1} = c_i$ for $1 \le i \le n$


 * $b_1 = w$


 * $b_{i + 1} = d_i$ for $1 \le i \le n$

Since $w \ne 0$ and $d_i \ne 0$, it follows that $b_i \ne 0$ for all $1 \le i \le n+1$.

Moreover, since $z > c_1$ and $\left\langle{c_i}\right\rangle$ is strictly decreasing, it follows that also $\left\langle{a_i}\right\rangle$ is strictly decreasing.

The equation for the first lemma can be rewritten:


 * $\displaystyle y = x^{a_1} b_1 + \sum_{i \mathop = 1}^n x^{a_{i + 1} } b_{i + 1}$

By General Associative Law for Ordinal Sum, it follows that:


 * $\displaystyle y = \sum_{i \mathop = 1}^{n + 1} x^{a_i} b_i$

Thus, existence is proven.

Furthermore, since $z$ and $\left\langle{c_i}\right\rangle$ are unique, and $w$ and $\left\langle{a_i}\right\rangle$ are unique, then $\left\langle{a_i}\right\rangle$ and $\left\langle{b_i}\right\rangle$ are unique.

Also see

 * Definition:Cantor Normal Form