[yakubin]

> I'm not talking about Gödel or unprovable statements, because even if something is not provable it might be constructable.

What precisely do you mean by that statement? What is the thing that is not provable but constructible? A conjecture? If so, then what do you mean when you say that a conjecture is constructible? What does it mean to construct it?

As far as I know, "construction" in maths usually means a finite process. And proofs are finite sequences of strings, satisfying certain rules specified by a proof framework (such as sequent calculus). So I'm curious in what way something could be constructible, but not provable. To be honest, I don't understand how those words could apply to the same things, because proving usually applies to conjectures, statements, formulas etc. But constructing usually applies to definitions and proofs, which cannot said to be true or false at all.

[raverbashing]

> What is the thing that is not provable but constructible?

Just look at the grandparent message for an example: "An actual practical statement that is unprovable in Peano arithmetic is Goodstein's theorem"

Or proving the Collatz Conjecture. You need only basic algebra to build it, but something much more complex to prove it (if it is provable)

[guimplen]

>I'm not talking about Gödel or unprovable statements, because even if something is not provable it might be constructable. >Or proving the Collatz Conjecture. You need only basic algebra to build it, but something much more complex to prove it (if it is provable)

It seems like your first statement is about constructibility of the same object (that should be a proof). In your second statement you talk about constructibility of an object and provability of a certain statement about an object. I may be be judgemental, but being so imprecise is a big no-no in metamathematics.