A couple of problems that is impossible to prove from the constructivism angle:

1) Addition of the natural numbers 2) equality of two real numbers

When you restrict your tools to perceptron based feed forward networks with high parallelism and no real access to 'common knowledge', the solution set is very restricted.

Basically what Gödel proved that destroyed Russel's plans for the Mathmatica Principia applies here.

Programmers can decide what is sufficient if not perfect in models.