Thanks for updating your chart, [Scott](https://forum.azimuthproject.org/discussion/comment/16390/#Comment_16390), and thanks for giving me a heads-up. I've updated my own comment accordingly.

You originally were imagining a thing called an "unordered set":

> So those that obey the rule:

> * _**unordered**_: for all x and y, there exists no x,y such that x≤y.

One could make up this notion, but it would be a useless notion: one is saying "take a set and equip it with a relation \\(\le\\), but also decree that this relation never holds: that is, \\(x \le y\\) is never true".

This is sort of like defining an adjective "cruncy" by saying

> **Cruncy** (_adj.)_: a property of plants that never holds: no plant is ever cruncy.

One could do it, but it's not much use.

More generally, mathematicians have learned over time, and especially since the invention of category theory, that's it's vastly better to make definitions contain only "positive" properties: that is, properties that don't involve "not" or "no". There are deep reasons for this, but it would take more category theory to explain them!

You originally were imagining a thing called an "unordered set":

> So those that obey the rule:

> * _**unordered**_: for all x and y, there exists no x,y such that x≤y.

One could make up this notion, but it would be a useless notion: one is saying "take a set and equip it with a relation \\(\le\\), but also decree that this relation never holds: that is, \\(x \le y\\) is never true".

This is sort of like defining an adjective "cruncy" by saying

> **Cruncy** (_adj.)_: a property of plants that never holds: no plant is ever cruncy.

One could do it, but it's not much use.

More generally, mathematicians have learned over time, and especially since the invention of category theory, that's it's vastly better to make definitions contain only "positive" properties: that is, properties that don't involve "not" or "no". There are deep reasons for this, but it would take more category theory to explain them!