User blog comment:KthulhuHimself/Library Function/@comment-25601061-20151114192719

Suggestion:

For a second-order Lib(n) function, we need a second-order language, which will be based off of Definer Language, and will be called Second-Order Definer Language. This time, we allow expressions that include the Lib(n) function, but we express this function as Lib2(n), and since this is a different function, it removes the paradoxes. So  Lib2(n) is defined as:  "The smallest integer larger than any finite natural number expressed in all the symbols after the last blank in any decent interpretation of any Q-well-defined or well-defined, non-empty string   with a finite number of decent interpretations in Second-Order Definer language with n or less being the sum of all symbols or almanacs in the expression."