Hopefully someone better educated than me can answer this - several of the definitions in the link feel constructivist, i.e. they describe constructions of of real numbers. It seems easy to think of methods of constructing non-rational numbers, by e.g. using infinite sequences, by taking roots, or whatever.

It seems harder to prove that every real number can be constructed via such a method.

Is there a construction-based method that can produce ALL real numbers between, say, 0 and 1? This seems unlikely to me, since the method of construction would probably be based on some sort of enumeration, meaning that you would only end up with countably many numbers. But maybe someone else can help me become un-confused.

> several of the definitions in the link feel constructivist, i.e. they describe constructions of of real numbers.

If you are a constructivist, then you will supply direct proofs for your results as you reject indirect proof, proof by contradiction, law of excluded middle, and things of this nature.

The converse does not necessarily hold. Providing a direct construction of an object satisfying the field and completeness axioms (e.g. the Dedekind construction) does not necessarily mean that one is a constructivist. Indeed, one can use the Dedekind construction and still go on to prove many more results on top of it that still do rely on indirect proof and reductio ad absurdum.

The definitions provided appear as though they are constructive, but they are not actually constructive, they are set-theoretic existence claims that quantify over all sequences, in particular over undefinable sets. Specifically, the description that appears constructive doesn't actually define any particular real number, it only defines the universe in which the real numbers live.

Another subtle detail is that while it's true that every real number corresponds to (and can be represented by) a Cauchy sequence of rationals, the very sequence itself might be undefinable.

Constructivist basically means being able to be explicit. Dedekind cuts and Cauchy sequences are not necessarily constructivist though something described by one of them can be explicitly descriptive for some applications. Any approach which produces all real numbers as commonly accepted will fail to be explicit in all cases as such explicitness presumably implies the real number has been expressed uniquely with finite strings and finite alphabets which can describe at most a countable number of them.

The decimal numbers, for example, can be viewed as an infinite converging sum of powers of ten. Theoretically one could produce a description, but only a countable number of those could be written down in finite terms (some kind of finite recipe). So those finite ones could fall in a constructivist camp, but the ones requiring an infinite string to describe would, as far as I understand constructivism, not fall under being constructivist. To be clear, the finite string doesn't have other be explicit about how to produce the numbers, just that it is naming the thing and it can be derived from that. So square root of 2 names a real number and there is a process to compute out the decimals so that exists in a constructivist sense. But "most" real numbers could not be named.

Your original intuition, that only a countable subset of real numbers can be described or used in any way, is correct. The rest are just “there.” They exist, but we can’t really use them for anything.

It gets weirder. What is a set? For finite sets, we know it intuitively. But consider the Axiom of Choice. There is a consistent mathematics in which a choice set is a set, and one in which the same meta-mathematical object is not a set. (Unless, of course, ZF is inconsistent.)

I may be misunderstanding your concern, but I believe this is what is meant by "Categoricity for the real numbers"