Indecomposable

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File:Https://upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Omega-exp-omega-labeled.svg/832px-Omega-exp-omega-labeled.svgAn indecomposable ordinal is an ordinal that can't be expressed as the sum of two smaller numbers.

Form

An indecomposable ordinal is in the form \(\omega^n\).

Proof

Any ordinal that multiplies with a finite amount or added with anything can be expressed as the sum of two smaller ordinals. To avoid this, start with the smallest indecomposable ordinal and multiply it by \(\omega\) every time. This is the best way of finding all indecomposable ordinals. Why? It is because \(\omega^n\) is equal to \(\omega\) sums of \(\omega^(n-1)\).


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