# Infinite Ordinal can be expressed Uniquely as Sum of Limit Ordinal plus Finite Ordinal

## Theorem

Let $x$ be an ordinal.

Suppose $x$ satisfies $\omega \subseteq x$.

Then $x$ has a unique representation as $\left({y + z}\right)$ where $y$ is a limit ordinal and $z$ is a finite ordinal.

## Proof

Take $K_{II}$ to be the set of all limit ordinals.

Then set $y = \bigcup \left\{{w \in K_{II}: w \le x}\right\}$

The set $\left\{{w \in K_{II}: w \le x}\right\}$ is non-empty because $\omega \subseteq x$.

By Union of Ordinals is Least Upper Bound, $y \in K_{II}$ and $y \le x$.

By Ordinal Subtraction when Possible is Unique, there is a unique $z$ such that $x = \left({y + z}\right)$

Assume $\omega \le z$.

Then, again by Ordinal Subtraction when Possible is Unique:

- $z = \left({\omega + w}\right)$

and so:

\(\displaystyle x\) | \(=\) | \(\displaystyle \left({y + \left({\omega + w}\right)}\right)\) | Equality is Transitive | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \left({\left({y + \omega}\right) + w}\right)\) | Ordinal Addition is Associative |

But $y + \omega$ is a limit ordinal by Limit Ordinals Preserved Under Ordinal Addition:

- $\varnothing < \omega \implies y < y + \omega$

This contradicts the fact that $y$ is the largest limit ordinal smaller than $x$.

Therefore, $z \in \omega$.

Thus we have proven that such a selection of $y$ and $z$ exists.

Suppose $z$ and $w$ both satisfy:

- $\left({y + w}\right) = \left({y + z}\right)$

By Ordinal Addition is Left Cancellable, we have $w = z$.

Thus $z$ is unique.

To prove uniqueness for $y$, suppose that $x = \left({y + u}\right)$ and $x = \left({w + z}\right)$.

Assume further, WLOG, that $y \le w$.

Then:

\(\displaystyle y \le w\) | \(\implies\) | \(\displaystyle \exists n: w = \left({y + n}\right)\) | Ordinal Subtraction when Possible is Unique | ||||||||||

\(\displaystyle \) | \(\implies\) | \(\displaystyle \left({y + u}\right) = \left({\left({y + n}\right) + z}\right)\) | Substitutivity of Equality | ||||||||||

\(\displaystyle \) | \(\implies\) | \(\displaystyle \left({y + u}\right) = \left({y + \left({n + z}\right)}\right)\) | Ordinal Addition is Associative | ||||||||||

\(\displaystyle \) | \(\implies\) | \(\displaystyle u = \left({n + z}\right)\) | Ordinal Addition is Left Cancellable | ||||||||||

\(\displaystyle \) | \(\implies\) | \(\displaystyle \left({n + z}\right) \in \omega\) | by the fact that $u \in \omega$ | ||||||||||

\(\displaystyle \) | \(\implies\) | \(\displaystyle n \in \omega\) | Ordinal is Less than Sum | ||||||||||

\(\displaystyle \) | \(\implies\) | \(\displaystyle n \notin K_{II}\) | by the fact that $\omega$ is the smallest limit ordinal | ||||||||||

\(\displaystyle \) | \(\implies\) | \(\displaystyle n = \varnothing \lor \exists m: n = m^+\) | Definition of Limit Ordinal |

Assume that $\exists m: n = m^+$.

\(\displaystyle n = m^+\) | \(\implies\) | \(\displaystyle w = \left({y + m^+}\right)\) | Definition of $n$ | ||||||||||

\(\displaystyle \) | \(\implies\) | \(\displaystyle w = \left({y + m}\right)^+\) | Definition of Ordinal Addition | ||||||||||

\(\displaystyle \) | \(\implies\) | \(\displaystyle w \notin K_{II}\) | Definition of Limit Ordinal |

This is clearly a contradiction, so $n = \varnothing$ and $w = y$.

$\blacksquare$

## Sources

- 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $\S 8.13$