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That doesn't seem possible with the reals. An example from programming that comes to mind is NaN ≹ NaN (in some languages).

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The floating point specification mandates that nan does not compare to nan in any way, so it should be all languages. If you want to know nam, use isnan()

Isn't this what happens with infs (in maths and in many programming languages)?

Edit: Not in many programming languages. In IEEE-754 inf == inf. In SymPy too oo == oo, although it's a bit controversial. Feels sketchy.

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