Alonzo Church suggested to the writer that a certain problem of Thue [6] might be proved unsolvable by the methods of [5]. We proceed to prove the problem recursively unsolvable, that is, unsolvable in the sense of Church [1], but by a method meeting the special needs of the problem.

Thue's (general) problem is the following. Given a finite set of symbols α1, α2, …, αμ, we consider arbitrary strings (Zeichenreihen) on those symbols, that is, rows of symbols each of which is in the given set. Null strings are included.

In previous papers we developed two functional calculi of first order based on strict implication which we called S2 and S4. In the present paper, these systems will be extended to include a functional calculus of second order with the purpose of introducing the relation of identity of individuals.

Primitive symbols. ̂ {the abstraction operator, the blank space to be replaced by an appropriate variable}.