Why is the well ordering principle counter-intuitive?

Question

I read here that while 'The Axiom of Choice agrees with the intuition of most mathematicians; the Well Ordering Principle is contrary to the intuition of most mathematicians'. I don't understand why this is so. According to Wikipedia, 'the well-ordering principle states that every non-empty set of positive integers contains a least element'. This is quite a straightforward definition. Is it the way you derive the equivalence of the Axiom of Choice and well ordering principle that makes it counter-intuitive or the definition of the least element?

Answer 1

The well-ordering principle just refers to $\mathbb{N}$. What is being referred to is the well-ordering theorem, which says that every set admits a well-order. That's the thing that seems a little bit crazy, if you consider how you might achieve this even for a "basic" set like $\mathbb{R}$.

As for choice, a very non-objectionable version is "the product of an arbitrary family of non-empty sets is non-empty". It's equally difficult to think of what a counter-example might look like.

If I would be so bold as to endorse a book on the topic, it would definitely be Horst Herrlich's Axiom of Choice. It will simultaneously convince you of the statement, of the statement's negation, and of neither. Cognitive dissonance eat your heart out.

Answer 2

The well ordering theorem is equivalent too "for every set, this is a strict total order, such that every subset has a minimum element." Think about how you would order $\mathbb{R}$ such that that is true (the normal ordering wouldn't work for sure.)

Answer 3

People object to the idea that every set can be well-ordered because they often think of sets alongside a natural structure imposed on them, and they initially expect that a well-order of the set would somehow respect that natural structure.

For example, people don't think of $\Bbb R$ as just a set, they think about it as an ordered set, with binary operations. If you expect that a well-ordering of $\Bbb R$ would respect the natural order of $\Bbb R$ then you run into problems: what is the successor of $0$? What is the minimal element of the order?

However, it is true that a well-ordering need not be compatible with that order of $\Bbb R$. The reason this is an interesting objection is that people have no qualms with the claim that $\Bbb Q$ is a countable set, and that there is a bijection from $\Bbb N$ onto $\Bbb Q$. This bijection induces a well-ordering of $\Bbb Q$ and it too doesn't respect the natural order of the rational numbers. Not even one bit.

Another objection people may have, and you see this objection often with cranks who argue that all the mathematics based on infinities is inconsistent, is that if a set is well-ordered then we must have a name for every element in the set, because then you can ask "what is the minimal element that cannot be named?", and so you can name that element. Since the set of names is countable well-ordering of a set would have to imply it is a countable set.

However the problem with this argument is that "the set of unnameable objects" is not a first-order definition, and so we can't really define that set in the universe of set theory. So this sort of objection is wrong just as well.

Finally, there is a reasonable objection that people may have, and that that the assumption of the axiom of choice induces all sort of intangible objects that we can't construct by any reasonable means. These people feel that philosophically we need to avoid such mathematics, and they prefer to work in a different framework that requires things to be explicitly constructed, and that's fine. This is the only valid objection I heard so far. Although there are people who voice these sort of opinions along with other near-crank claims against the axiom of choice.