# Why is the axiom of choice controversial? [duplicate]

In other words, what are the arguments for ZF over ZFC, and what philosophical issues have people raised against including it as a standard axiom of set theory?

## marked as duplicate by Noah Schweber, Asaf Karagila♦ axiom-of-choice StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Jun 25 '16 at 22:29

Some more things: The axiom of choice is indeed an extremely useful axiom in many areas of math. However, it gives rise to the Banach-Tarski Paradox and the existence of nonmeasurable subsets of the real numbers. Also, the axiom of choice is equivalent to the statement that any set can be well-ordered, i.e., every nonempty set can be endowed with a total order such that every nonempty subset has a least element. Therefore, this means that $\mathbb{R}$ can be well-ordered. However, no one has ever been able to explicitly state how one does this, and if I'm not mistaken, it may be impossible to do so with the current axioms we have available. I also want to add that one person in the comments section mentioned, we need AC in order to talk about cardinalities of infinite sets.