Pseudo-order

In constructive mathematics, pseudo-order is a name given to certain binary relations appropriate for modeling continuous orderings.

In classical mathematics, its axioms constitute a formulation of a strict total order (also called linear order), which in that context can also be defined in other, equivalent ways.

Examples

The constructive theory of the real numbers is the prototypical example where the pseudo-order formulation becomes crucial. A real number is less than another if there exists (one can construct) a rational number greater than the former and less than the latter. In other words, here x < y holds if there exists a rational number z such that x < z < y.

Notably, for the continuum in a constructive context, the usual trichotomy law does not hold, i.e. it is not automatically provable. The axioms in the characterization of orders like this are thus weaker (when working using just constructive logic) than alternative axioms of a strict total order, which are often employed in the classical context.

Definition

A pseudo-order is a binary relation satisfying the three conditions:

It is not possible for two elements to each be less than the other. That is, for all

x

{\displaystyle x}

and

y

{\displaystyle y}

,

¬

(

x

<

y

∧

y

<

x

)

{\displaystyle \neg (x<y\land y<x)}

Every two elements for which neither one is less than the other must be equal.