Background
A class of languages $C$ has Gold's Property if $C$ contains
- a countable infinite number of languages $L_i$ such that $L_i \subsetneq L_{i + 1}$ for all $i > 0$
- a further language $L_\infty$ such that for any $i > 0$, if $x \in L$ then $x \in L_\infty$.
Then, Gold's theorem is:
Any class of languages with Gold's Property is unlearnable
In other words, Gold's Property is a sufficient condition for unlearnability.
Question
What is the weakest (natural) necessary condition? In other words, I want the weakest property $P$ such that:
Any unlearnable class of languages has property $P$
In particular: is Gold's Property such a property? Can Gold's theorem be strengthened to an if and only if?
Alternatively, as @TsuyoshiIto pointed out in the comments:
What is a sufficient condition for a class of languages to be learnable?