Interface:First-order linear order defined via strict inequality

A linear order or total order has an ordering predicate which obeys the familiar properties shown below. Here we present a first-order theory of linear orders (that is, all variables indicate one of the items being ordered, and there is no general way to form sets of items or variables which represent elements of some structure other than the field). This is as opposed to building the linear order within set theory, where the linear order consists of a set plus a relation for the ordering predicate.

Our first-order axioms of linear orders are taken from a book by Margaris.

We start by importing propositional and predicate logic.

 param (PROPOSITIONAL Interface:Classical_propositional_calculus ) param (FIRSTORDER Interface:First-order_logic_with_quantifiability (PROPOSITIONAL) ) 

Strict order
There is an order which obeys the usual equality axioms.  var (object x y z x0 y0 z0 x1 y1 z1) var (variable v v0 v1 v2)

term (formula (< object object)) stmt (LessThanBuilder  (((x0 = x1) ∧ (y0 = y1)) → ((x0 < y0) ↔ (x1 < y1)))) 

It is irreflexive, transitive and total. The totality axiom looks like the trichotomy law, but it merely states that at least one of,  , or   hold; trichotomy holds that exactly one of the three holds (trichotomy can be proved from these three axioms, however ).  stmt (LessThanIrreflexivity  (¬ (x < x))) stmt (LessThanTransitivity  (((x < y) ∧ (y < z)) → (x < z))) stmt (LessThanTotality  (((x < y) ∨ (x = y)) ∨ (y < x))) 

Terms versus variables
An astute reader comparing the axioms above with the ones in Margaris will notice that the above ones are stated in terms of terms rather than variables. For example, Margaris' states irreflexivity as  (where   is a variable), whereas we state it above as , where   is a term. The axioms using variables would suffice; we could convert them to their term counterparts using  combined with the equality axioms. We state the axioms using terms merely to avoid a number of repetitive proofs of this sort.

A similar rationale applies to the distinct variable constraints. In a work like Margaris, the axioms are stated using variables and variables which are given different letters cannot refer to the same variable.

Cited works

 * Margaris, Angelo (1990), First Order Mathematical Logic, ISBN 978-0-486-66269-5