The question is the title. 


Working in ZF, is it true that: for every nonempty set X, there exists a total order on X ? 


If it is false, do we have an example of a nonempty set that has no total order?


Thanks



Are all sets totally ordered ?
In the paper [Dense orderings,
partitions and weak forms of choice, by Carlos G. Gonzalez FUNDAMENTA MATHEMATICAE 147 (1995)](http://matwbn.icm.edu.pl/ksiazki/fm/fm147/fm14712.pdf), the author states the following theorem, where AC is the Axiom of Choice, DO is the assertion that every infinite set has a dense linear order, O is the assertion that every set has a linear order, and DPO is the assertion that every infinite set has a (nontrivial) dense partial order. 


**Theorem 1.** AC implies DO implies O implies DPO. Moreover, none of the implications
is reversible in ZF and DPO is independent of ZF.


Thus, in particular, the assertion that every set has a total order is strictly weaker than AC. 


(Also, it would seem that Gonzalez means to assume Con(ZF) for the latter claims of his theorem.)


