5
$\begingroup$

Is there a constructive proof for Heine Borel Theorem, where the Heine Borel Theorem refers to every open covering of a closed bounded interval has a finite sub-cover? The proof I know use proof by contradiction. However, is it good to have a constructive proof (i.e., really give a way to construct the finite sub-cover from arbitrary open covering)? And are there such proofs?

$\endgroup$

1 Answer 1

4
$\begingroup$

The answer to this depends in part on whether you would consider Markov's principle as being valid. If you trust Markov's principle, then you can simply do an exhaustive search in your infinite open cover for a finite subcover: Classical logic tells you it is absurd that it does not exist, and given a finite list of rational intervals it is decidable whether or not it covers the given interval.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.