A decidable well-order on the natural numbers is a well-order whose comparison relation is computable. Comparing two codes is an effective finite task; the proof that the relation has no infinite descending chain is a separate mathematical assertion. Computable Cantor normal form notation gives such an order of type epsilon zero.
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A well-order is a type of ordering on a set, with specific properties that make it particularly useful in various areas of mathematics, particularly in set theory and number theory.