These kinds of non-constructive results are almost always present when you invoke choice (when it's actually relevant), and this is a well-known example of the

These kinds of non-constructive results are almost always present when you invoke choice (when it's actually relevant), and this is a well-known example of the non-intuitiveness of choice. If you don't like this, you're probably a constructivist and may also disagree with the law of the excluded middle. A classic quote/joke on the topic: "The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?"