Math 2.0

Mathematical logic and foundations Model theory: types, categoricity, stability Example

Dense linear order

Builds on Total orderRational numbersOrder isomorphismŁoś–Vaught testQuantifier elimination

Example

A linear order (D,<)(D, <) is dense if for all a<ba < b in DD there is c∈Dc \in D with a<c<ba < c < b. The theory DLO\mathrm{DLO} of dense linear orders without endpoints is a theory in the signature {<}\set{<}. It says that << is a dense linear order on a set with at least two elements. It also says that there is no least and no greatest element. The rational numbers and the real numbers with their usual order are models.

DLO\mathrm{DLO} is ℵ0\aleph_0-categorical: any two countable models are order isomorphic. Check, by Cantor's back-and-forth argument: list the elements of two countable models AA and BB, and build a chain of finite order-preserving partial maps. At odd steps, send the next element of AA to an element of BB in the same position relative to the finite image. This element exists because BB is dense and has no endpoints. At even steps, do the same from BB back to AA. The union of the chain is an isomorphism from AA onto BB.

Remarks

Since DLO\mathrm{DLO} has no finite models, the Łoś–Vaught test shows that it is complete; it is also decidable. It has quantifier elimination. It is not κ\kappa-categorical for any uncountable κ\kappa. For κ=2ℵ0\kappa = 2^{\aleph_0}, compare ℝ\R with the order ℝ+ℚ\R + \Q, a copy of ℝ\R followed by a copy of ℚ\Q. Both are dense without endpoints and have size 2ℵ02^{\aleph_0}. They are not isomorphic. In ℝ\R every nonempty set with an upper bound has a least upper bound. In ℝ+ℚ\R + \Q the copy of ℝ\R is bounded above but has no least upper bound.

Reading path

To understand this page from the ground up, read these 13 pages in order. The foundations come first.

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Cardinal number, concept, no evidence levelCardinal numberconceptCategorical theory, concept, no evidence levelCategorical theoryconceptComplete theory, concept, no evidence levelComplete theoryconceptElementary substructure, concept, no evidence levelElementary substructu…conceptOrder isomorphism, concept, no evidence levelOrder isomorphismconceptQuantifier elimination, concept, no evidence levelQuantifier eliminationconceptRational numbers, object, no evidence levelRational numbersobjectSignature, concept, no evidence levelSignatureconceptStructure, concept, no evidence levelStructureconceptTotal order, concept, no evidence levelTotal orderconceptLöwenheim–Skolem theorem, statement, no evidence levelLöwenheim–Skolem theo…statementŁoś–Vaught test, statement, no evidence levelŁoś–Vaught teststatementDense linear order, example, no evidence levelDense linear orderexample
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Cardinal number concept
  2. Categorical theory concept
  3. Complete theory concept
  4. Elementary substructure concept
  5. Order isomorphism concept
  6. Quantifier elimination concept
  7. Rational numbers object
  8. Signature concept
  9. Structure concept
  10. Total order concept
  11. Löwenheim–Skolem theorem statement
  12. Łoś–Vaught test statement
  13. Dense linear order example

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Dense_order
About this page
Tag2W4S2H4VYA
URIm2://2W4S2H4VYA
Kindexample
WritingW0 stub
MSC03C35, 03C10, 06A05
Instance ofCategorical theory
UsesTotal order, Rational numbers, Order isomorphism, Łoś–Vaught test, Quantifier elimination
Changed2 November 2026

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