A distinction is made between total correctness, which additionally requires that the algorithm terminates, and partial correctness, which simply requires that if an answer is returned it will be correct. Since there is no general solution to the halting problem, a total correctness assertion may lie much deeper.
For example, if we are successively searching through integers 1, 2, 3, … to see if we can find an example of some phenomenon — say an odd perfect number — it is quite easy to write a partially correct program (use integer factorization to check n as perfect or not). But to say this program is totally correct would be to assert something currently not known in number theory.
A proof would have to be a mathematical proof, assuming both the algorithm and specification are given formally. In particular it is not expected to be a correctness assertion for a given program implementing the algorithm on a given machine. That would involve such considerations as limitations on memory.
A deep result in proof theory, the Curry-Howard correspondence, states that a proof of functional correctness in constructive logic corresponds to a certain program in the lambda calculus. Converting a proof in this way is called program extraction.
In auditing, correctness is one of the financial statement assertions that has to be ensured. For example, an auditor has to ensure a transaction is correctly recorded in the books. For recording a purchase, he or she has to check the posting to accounts payable and purchase account, gross amount, amount of discount received from supplier.