Show that a language is decidable iff some enumerator enumerates the language in decreasing order
If there is an enumerator in decreasing order then the language is finite and hence decidable.
But that means any infinite language wont have such an enumerator, and in particular there is some infinite language that is decidable (such as $\Sigma^*$) and it wont have such an enumerator.
So, this theorem is incorrect