TITLE: How to rewrite mathematics constructively?
QUESTION [43 upvotes]: Many mathematical subfields often use the axiom of choice and proofs by contradiction. I heard from people supporting constructive mathematics that often one can rewrite the definitions and theorems so that both the axiom of choice and proofs by contradiction aren't needed anymore.
An example is the theory of locales. This is a reformulation of topology that doesn't need the axiom of choice to prove analogues of results that classically need some form of choice, such as Tychonoff's theorem.
I wonder: Are there some tricks behind this "reformulation"? When I have a theorem with a classical proof, how can I reformulate it so that it is provable constructively?
In case this question is too broad, I would be interested in the following example: The usual proof of Gödel's completeness theorem by Henkin isn't constructive. Is there some easy way to reformulate the theorem and the proof constructively? Or is it necessary to define, with much non-trivial work, categorical semantics and state the theorem in this new context?

REPLY [5 votes]: Are you aware that the (semantic) completeness theorem for countable first-order languages is equivalent over RCA0 (a weak subsystem of second-order arithmetic) to WKL0? Essentially the non-constructive part of this lies in the fact that you want to obtain an infinite path from an infinite binary tree. This principle is truly non-constructive, since there is a computable infinite tree with no computable infinite path. So even in the countable realm, which arguably is what truly constructive mathematics should be about, the completeness theorem is non-constructive. But it is not that bad; we can obtain a model of any consistent computable FOL theory that is computable from the first Turing jump.
If you are interested in the use of AC in the uncountable case, the issue lies entirely with whether the theory can be well-ordered. The completeness theorem holds for any theory over a well-orderable language. In general you can find that almost every theorem that invokes AC or Zorn's lemma can be rephrased as theorems about well-ordered stuff whose proofs need no choice. For example, every well-orderable field has an algebraic closure, and every vector space with a well-orderable spanning set has a basis. So even in the usual setting of Zermelo set theory you can easily extract the 'more constructive' version just by adding well-orderability conditions.