A total order means any two things can be ranked against each other
Pick any two real numbers and one is always less than or equal to the other. Pick any two letters and the dictionary settles which comes first. Mathematicians call an arrangement like this a total order, and the idea underpins everything from counting to a famous tool called Zorn's lemma.
Plenty of orderings fail the test. In a partial order some pairs simply cannot be compared, the way two sets may each contain elements the other lacks. A total order, also called a linear order, adds one demand: every pair must be comparable. Take any slice of a totally ordered set and the slice inherits the same property, so the natural numbers, integers and rationals all come totally ordered from the reals.
Each of those familiar number systems is the model case of some property. The natural numbers are the basic totally ordered set with no upper end, the integers have no end in either direction, the rationals are the standard densely packed order, and the reals are the unbounded order with no gaps. Every ordered field is totally ordered by definition, and any one that is complete in Dedekind's sense turns out to be the real numbers.
Finite total orders are especially tame. Any non-empty finite one has a smallest element, which makes it a well order, and a total order on k items lines up exactly with the first k counting numbers. That is why finite ordered lists can always be numbered first, second, third without loss.
Inside a larger partially ordered collection, a totally ordered piece is called a chain. Zorn's lemma says that if every chain has an upper bound, the collection has at least one maximal element, a result used to prove that every vector space has a basis and that rings have maximal ideals. Chains of subspaces also measure dimension: a vector space's dimension equals the longest chain of nested subspaces it contains.
Source: Total order
