Intersection-closed full AFL and the recursively enumerable languages

A study is made of conditions on a language L which ensure that the smallest intersection-closed full AFL containing L (written ℱ ^ ∩ ( L ) ) does or does not contain all recursively enumerable languages. For example, it is shown that if L = {ani/j⩾0} and limi→∞ inf(ni+1/ni) > 1, then ℱ ^ ∩ ( L ) contains all recursively enumerable languages. On the other hand, it is shown that if L ⊆ a* and the ratio of the number of words in L of length less than n to n goes to 1 as n → ∞, then ℱ ^ ∩ ( L ) does not contain all recursively enumerable languages.