There are counter intuitive consequences of the Axiom of Choice. There are counter intuitive consequences without choice [1]. There are weaker versions of the

There are counter intuitive consequences of the Axiom of Choice. There are counter intuitive consequences without choice [1]. There are weaker versions of the Axiom of Choice, like countable choice. Here’s a statement that is equivalent to the Axiom of Choice: The product of non-empty sets is non-empty. It seems clear, naively speaking, that the product of non empty sets should be non empty and believing this means you accept the Axiom of Choice. Whatever you decide to believe with regard to the Axiom of Choice you end up with weird stuff happening when dealing with uncountable sets. The intuition with dealing with countable sets does not extend to uncountable sets in many cases. [1]. https://mathoverflow.net/a/22935 https://mathoverflow.net/a/22935