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> 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)

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>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.

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