{"history":[],"article":{"id":"1b92ffe2-ccd2-4fda-94ff-80f7438c38d0","title":"Pseudo-order","slug":"pseudo-order","content":":::summary\n:::\n\n# \nIn [constructive mathematics](https://bliptext.com/articles/constructivism-mathematics), **pseudo-order** is a name given to certain [binary relations](https://bliptext.com/articles/binary-relation) appropriate for modeling continuous orderings.\nIn classical mathematics, its axioms constitute a formulation of a [strict total order](https://bliptext.com/articles/strict-total-order) (also called linear order), which in that context can also be defined in other, equivalent ways.\n\n# Examples\nThe [constructive theory of the real numbers](https://bliptext.com/articles/constructive-analysis) is the prototypical example where the pseudo-order formulation becomes crucial.\nA real number is less than another if [there exists](https://bliptext.com/articles/there-exists) (one can construct) a rational number greater than the former and less than the latter.\nIn other words, here x < y holds if there exists a rational number z such that x < z < y.\nNotably, for the continuum in a constructive context, the usual [trichotomy law](https://bliptext.com/articles/trichotomy-mathematics) does not hold, i.e. it is not automatically provable.\nThe 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.\n\n# Definition\nA pseudo-order is a binary relation satisfying the three conditions:\nx < y\nthen either\nx < z\nor\nz < y\n. That is, for all x, y and z,\n\n## Auxiliary notation\nThere are common constructive reformulations making use of contrapositions and the [valid equivalences](https://bliptext.com/articles/intuitionistic-logic) as well as.\nThe negation of the pseudo-order x

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