TITLE: What is an example of two sets which cannot be compared?
QUESTION [13 upvotes]: In set theory, if we do not assume the Axiom of Choice, we cannot prove the Trichotomy Law between cardinals. That is, we cannot prove that for any two sets, there exists an injection from one to the other. But this raises the question: what is an example of a pair of sets between which there does not exist an injection?

REPLY [8 votes]: There is no example in any good and reasonable sense of the word.
Given any two sets $A$ and $B$, there is an extension of the universe, to a larger universe of $\sf ZF$, in which both $A$ and $B$ are countable, and therefore can be compared.
You might want to talk about "definitions" rather than concrete sets. But even that can be difficult. Any reasonable definition for a set which involves bounding the rank of that set (so you can't say something like "the least $V_\alpha$ that cannot be well-ordered" in your definition), is going to fail. The axiom of choice consistently holds up to whatever level you want, and fails in however way you want it to fail afterwards.
The negation of the axiom of choice teaches us nothing about where the axiom of choice fails, or how it fails.
If you want a "concrete" example for two sets, then you can say the following, we know that the axiom of choice fails if and only if there exists some ordinal $\alpha$ such that $\mathcal P(\alpha)$ cannot be well-ordered. Take that power set of $\alpha$, and let $\kappa$ be the least ordinal that cannot be injected into $\mathcal P(\alpha)$. Then $\mathcal P(\alpha)$ and $\kappa$ cannot be compared.
But is this example really helpful? Did you learn anything meaningful from it? No. I haven't told you what is $\alpha$ or what is $\kappa$. And I can't tell you that. Because all you've told me is that the axiom of choice fails. You haven't told me how it fails. Does countable choice holds? Perhaps Dependent Choice holds up to the least measurable cardinal holds? We don't know. We can't know. Unless we assume more about this failure.