---
tag: FMFGTVVR4S
kind: concept
name: Order type
msc: [06A05, "03E10"]
sources:
  - url: https://en.wikipedia.org/wiki/Order_type
same_as:
  wikidata: Q620006
  wikipedia: Order type
---

## Summary

The order type of a totally ordered set is what it shares with every set that is [[order isomorphic|0Y74KN99A5]] to it.
Two ordered sets have the same order type exactly when there is an order isomorphism between them.
Standard order types are $\omega$ for the natural numbers, $\eta$ for the rationals and $\lambda$ for the reals.
The order types of well-ordered sets are the [[ordinal numbers|0RDJK7EQYY]].

## Explanations

### L1 human

The order type forgets what the elements are and keeps only how they are arranged.
The natural numbers $0, 1, 2, \dots$ and the even numbers $0, 2, 4, \dots$ are different sets, but both form one infinite line going up from a first element.
So they have the same order type, $\omega$.
The integers are different in shape: they have no first element, so their order type is not $\omega$.