Example of a non empty set with no order relation on it

Question

Is there an example on which you cannot define any order relation ?

Any order relation means neither total order or partial order.

Answer 1

Any set can be given a partial order. Indeed, the relation $=$ is a partial order on any set.

Any set can also be given a total order, but this is much harder to prove, and in fact requires the axiom of choice. Indeed, the axiom of choice is known to be equivalent to the statement that every set can be given a well order, which is a special kind of total order. The statement that every set can be given a total order can thus be considered a weak form of the axiom of choice. See https://mathoverflow.net/questions/37272/are-all-sets-totally-ordered for some discussion.