Assume that $L$ is decidable. Is the following language decidable? $$ L' = \{w | \text{ there is } u \in L \text{ such that u and w are equal to at most two positions.}\}$$
For example, let $u=abcd$. The following words are equal to $u$ at most two positions: $\{aba, agg, ab, zzz, dsa, ...\}$, but not $\{ abc, zbcd \}$
My approach:
A language is decidable iff some enumerator enumerates the language in lexicographic order.
I use above theorem.
So, $L$ is decidable. Therefore, let's take enumerator $E$ for it. Let We construct a $TM'$ for $L'$. Let enumerator $E$ be connected to $TM'$. For every input $w$ given to $TM'$ the $TM'$ compare it position by position with every string $v$ generated by $E$ in lexicographical order till $|v|\le|w|$ If number of the same positions $>2$ reject $w$. Otherwise, accept.
Ok?