General Associative Law for Ordinal Sum

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Theorem

Let $x$ be a finite ordinal.

Let $\left\langle{a_i}\right\rangle$ be a sequence of ordinals.


Then:

$\displaystyle \sum_{i \mathop = 1}^{x + 1} a_i = a_1 + \sum_{i \mathop = 1}^x a_{i + 1}$


Proof 1

The proof shall proceed by induction on $x$.


Basis for the Induction

If $x = 0$, then:

\(\displaystyle \sum_{i \mathop = 1}^{0 + 1} a_i\) \(=\) \(\displaystyle \sum_{i \mathop = 1}^0 a_i + a_1\) definition of ordinal sum
\(\displaystyle \) \(=\) \(\displaystyle a_1\) by Ordinal Addition by Zero
\(\displaystyle \) \(=\) \(\displaystyle a_1 + \sum_{i \mathop = 1}^0 a_i\) by Ordinal Addition by Zero

This proves the basis for the induction.

$\Box$


Induction Step

Suppose that:

$\displaystyle \sum_{i \mathop = 1}^{x + 1} a_i = a_1 + \sum_{i \mathop = 1}^x a_{i + 1}$

Then:

\(\displaystyle \sum_{i \mathop = 1}^{x + 2} a_i\) \(=\) \(\displaystyle \sum_{i \mathop = 1}^{x + 1} a_i + a_{i + 2}\) Definition of Ordinal Sum
\(\displaystyle \) \(=\) \(\displaystyle \left({a_1 + \sum_{i \mathop = 1}^x a_{i + 1} }\right) + a_{i + 2}\) by Inductive Hypothesis
\(\displaystyle \) \(=\) \(\displaystyle a_1 + \left({\sum_{i \mathop = 1}^x a_{i + 1} + a_{i + 2} }\right)\) Ordinal Addition is Associative
\(\displaystyle \) \(=\) \(\displaystyle a_1 + \sum_{i \mathop = 1}^{x + 1} a_{i + 1}\) Definition of Ordinal Sum

This proves the induction step.

$\blacksquare$


Proof 2

From Ordinal Addition is Associative we have that:

$\forall a, b, c \in \operatorname{On}: a + \left({b + c}\right) = \left({a + b}\right) + c$

The result follows directly from the General Associativity Theorem.

$\blacksquare$