The statement A = "There exists a well-ordering of the reals" is independent of ZF. My understanding is that the statement B = "There exists an explicit well-ordering of the reals" is also independent of ZF, yet this seems counterinutitive to me, because of the following line of reasoning: If B were true, then A would be provable in ZF, which is impossible since A is independent of ZF, so B must be false. This line of reasoning seems perfectly clear to me, and I see no reason why it cannot be carried out in ZF itself. But if the line of reasoning could be carried out in ZF, then that would mean that ZF implies that not B, contradicting the claim that B is independent of ZF.
So can anyone clarify how exactly ZF+B is consistent?
Any help would be greatly appreciated.
Thank You in Advance.