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| An '''ordering''' is a theoretical commitment to perform some action regarding all the members of a [[set]], one by one.
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| This is a somewhat unintuitive definition. It works because if you perform a single action at a time, then some of the actions you take will be before others, and some will be after others. [TODO I don't like it]
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| What if it's like... an ordering is a relationship between the members of a [[set]] and some actions. Hmm
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| Many sets of actions are automatically ordered. Actions or events. Many sets of
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| Ordering seems like maybe it has something to do with time? At least one type of ordering does, namely time-ordering. Like, I think that the concepts of [["before" and "after"]] are irreducible primaries. We just ''perceive'' that some events happen before or after other events. That probably comes first, and then by analogy we can start talking about other things being before or after.
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| There are two other significant uses of the verb "to order." One is the act of giving commands (orders) to other people, as in "he ordered a pizza." The other is the act of organizing (bringing order to) something, as in "he ordered his room." Both of these usages are ''related to'' the mathematical usage of the term; it's not a coincidence that they are all described by the same word. But getting into that would take us too far afield.
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| == Examples ==
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| Library books are ordered according to the Dewey Decimal System.
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| A single-file line of people could be ordered by height, with the shortest person in the front, and the tallest person in the back.
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| [[Natural numbers]] have a default ordering (by construction), where n+1 comes after n.
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| The marks on a clock are ordered clockwise.
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| To learn math, one must proceed in a certain order: some things must be learned before other things.
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| The alphabet is ordered. Little kids learn to remember this ordering by singing the alphabet song.
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| In feudal Europe, there was an order of succession (who inherits the throne after whom) which was largely agreed upon. The king's eldest son inherited the throne after the king died. If the king's eldest son was also dead, then ''his'' eldest son would inherit the throne. If the king's eldest son was dead and had no sons, then the king's second-eldest son would inherit the throne. Etc etc.
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| === Non-examples ===
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| The books on my bookshelf are not ordered: they are in random positions with respect to one another.
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| Most single-file lines of people are not ordered by height.
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| === Examples of "order" (n) which are not examples of "ordering" [TODO move] ===
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| The law of causality states that things act in accordance with their nature. One formulation of this law is that the universe has order.
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| If a riot happens, the police are supposed to come in and restore order to the streets.
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| ----The Order of the Phoenix.
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| == The traditional concept ==
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| Traditional mathematics would define this as a A set S is said to have the structure of a partial order is some binary relation R among members of the set (<math>R : S \times S \rightarrow \{0,1\}</math>), which satisfying some properties. [TODO I don't want to get into partial ordering B.S. how to explain]
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