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The signature of orders has no constants or functions, and one binary relation symbols ≤. (It is of course possible to use ≥, < or > instead as the basic relation, with the obvious minor changes to the axioms.) We define x ≥ y, x < y, x > y as abbreviations for y ≤ x, x ≤ y ∧¬y ≤ x, y < x, Some first-order properties of orders: Transitive: ∀x ∀y ∀z x ≤ y∧y ≤ z → x ≤ z Reflexive: ∀x x ≤ x Antisymmetric: ∀x ∀y x ≤ y ∧ y ≤ x → x = y Partial: Transitive∧Reflexive∧Antisymmetric; Linear (or total): Partial ∧ ∀x ∀y x≤y ∨ y≤x Dense ∀x ∀z x < z → ∃y x < y ∧ y < z ("Between any 2 distinct elements there is another element") There is a smallest element: ∃x ∀y x ≤ y There is a largest element: ∃x ∀y y ≤ x Every element has an immediate successor: ∀x ∃y ∀z x < z ↔ y ≤ z The theory DLO of dense linear orders without endpoints (i.e. no smallest or largest element) is complete, ω-categorical, but not categorical for any uncountable cardinal. There are 3 other very similar theories: the theory of dense linear orders with a: Smallest but no largest element; Largest but no smallest element; Largest and smallest element. Being well ordered ("any non-empty subset has a minimal element") is not a first-order property; the usual definition involves quantifying over all subsets. |
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