Example of an algorithm that lacks a proof of correctness
Here is an algorithm for the identity function:


Input: $n$
Check if the $n$th binary string encodes a proof of $0 &gt; 1$ in ZFC, and if so, output $n+1$
Otherwise, output $n$


Most people suspect this algorithm computes the identity function, but we don't know, and we can't prove it in the commonly accepted framework for mathematics, ZFC.