# Ordinal Multiplication via Cantor Normal Form/Limit Base

## Theorem

Let $x$ and $y$ be ordinals.

Let $x$ be a limit ordinal.

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 finite ordinals.

Then:

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

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## Proof

The proof shall proceed by finite induction on $n$:

For all $n \in \N_{\le 0}$, let $\map P n$ be the proposition:

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

Since $x$ is a limit ordinal, it follows that $x^y$ is a limit ordinal by Limit Ordinals Closed under Ordinal Exponentiation.

### Basis for the Induction

$\map P 1$ is the statement:

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

\(\ds \sum_{i \mathop = 1}^1 \paren {x^{a_i} b_i} \times x^y\) | \(=\) | \(\ds \paren {x^{a_i} \times b_i} \times x^y\) | Definition of Ordinal Sum | |||||||||||

\(\ds \) | \(=\) | \(\ds x^{a_i} \times \paren {b_i \times x^y}\) | Ordinal Multiplication is Associative | |||||||||||

\(\ds \) | \(=\) | \(\ds x^{a_i} \times x^y\) | Finite Ordinal Times Ordinal | |||||||||||

\(\ds \) | \(=\) | \(\ds x^{a_i \mathop + y}\) | Ordinal Sum of Powers |

This is our basis for the induction.

$\Box$

### Induction Hypothesis

Now we need to show that, if $\map P k$ is true, where $k \ge 1$, then it logically follows that $\map P {k + 1}$ is true.

So this is our induction hypothesis:

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

Then we need to show:

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

### Induction Step

This is our induction step:

By Upper Bound of Ordinal Sum, it follows that:

- $\ds \sum_{i \mathop = 1}^n \paren {x^{a_{i \mathop + 1} } b_{i \mathop + 1} } < x^{a_1}$

By Membership is Left Compatible with Ordinal Multiplication:

\(\ds x^{a_1} b_1\) | \(\le\) | \(\ds x^{a_1} b_1 + \sum_{i \mathop = 1}^k \paren {x^{a_{i \mathop + 1} } b_{i \mathop + 1} }\) | Ordinal is Less than Sum | |||||||||||

\(\ds \) | \(\le\) | \(\ds x^{a_1} b_1 + x^{a_1}\) | Membership is Left Compatible with Ordinal Addition | |||||||||||

\(\ds \leadsto \ \ \) | \(\ds x^{a_1} \times b_1 \times x^y\) | \(\le\) | \(\ds \sum_{i \mathop = 1}^{k \mathop + 1} \paren {x^{a_i} b_i} \times x^y\) | Subset is Right Compatible with Ordinal Multiplication and General Associative Law for Ordinal Sum | ||||||||||

\(\ds \) | \(\le\) | \(\ds x^{a_1} \times \paren {b_1 + 1} \times x^y\) | Subset is Right Compatible with Ordinal Multiplication |

But:

\(\ds x^{a_1} \times b_1 \times x^y\) | \(=\) | \(\ds x^{a_1} \times x^y\) | Finite Ordinal Times Ordinal | |||||||||||

\(\ds \) | \(=\) | \(\ds x^{a_1 \mathop + y}\) | Ordinal Sum of Powers | |||||||||||

\(\ds x^{a_1} \times \paren {b_1 + 1} \times x^y\) | \(=\) | \(\ds x^{a_1} \times x^y\) | Finite Ordinal Times Ordinal | |||||||||||

\(\ds \) | \(=\) | \(\ds x^{a_1 \mathop + y}\) | Ordinal Sum of Powers |

Therefore:

\(\ds x^{a_1 \mathop + y}\) | \(\le\) | \(\ds \sum_{i \mathop = 1}^n \paren {x^{a_i} b_i} \times x^y\) | Substitutivity of Class Equality | |||||||||||

\(\ds \) | \(\le\) | \(\ds x^{a_1 \mathop + y}\) | Substitutivity of Class Equality | |||||||||||

\(\ds \leadsto \ \ \) | \(\ds \sum_{i \mathop = 1}^n \paren {x^{a_i} b_i} \times x^y\) | \(=\) | \(\ds x^{a_1 \mathop + y}\) | Definition 2 of Set Equality |

$\blacksquare$

## Also see

## Sources

- 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $\S 8.46$