Wellordering vs. Zorn's lemma

Question

Many mathematicians outside mathematical logic dislike wellorderings, ordinals and corresponding transfinite arguments. They use zorn's lemma instead and claim one does not need ordinals at all. Examples are

• Every vector space has a basis
• Every filter can be extended to an ultrafilter
• Hahn-Banach theorem

However many cases are not that easy. I often have difficulties to convert a simple proof that uses ordinals and transfinite induction into one using zorn's lemma instead. For example, how would one proove the following assertions with zorn's lemma?

1. $\mathbb{R}^3$ is the union of pairwise disjoint unit-circles.
2. There is a set of reals of the cardinality of the continuum that has no perfect subset.
3. There is a non-determined set of reals.

Answer 1

Every proof with transfinite induction on a well-ordering can be essentially translated into a proof by Zorn's lemma. However this is an issue of simplicity. One can write a very long and difficult proof that a injective polynomial map from $\Bbb C^n\to\Bbb C^n$ is surjective, or one can use the correct tools from model theory and prove this quickly.

Sometimes things are easier to prove by well-ordering a set and going by induction, and sometimes things are easier to do with Zorn's lemma. Sometimes things are difficult in either case, and sometimes they are easy in either case. The idea is to identify the needed property for the proof and use the most suitable choice principle for that.

Equally you don't see people prove that there is a Bernstein set using the fact that every vector space has a Hamel basis; or using Tychonoff's theorem.

Answer 2

I remember using Zorn's lemma to prove that every vector space has a basis, and I remember my professor at the time giving us a warning that not all mathematicians entirely agree with Zorn's lemma. As to one of your questions, I believe this paper does use Zorn's lemma, though I have not read it fully: http://www.math.chalmers.se/~wastlund/partitions.ps .

I'm not really sure where you got 'many mathematicians' from. I was only aware that the stubborn few did not like it, and not the majority of the community.