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of types), can be reduced to the decision problem for S, and on the other hand that the decision problem for S is unsolvable. Hence the same results now follow for S'" in place of S. A fortiori, the general decision problem for systems in normal form is unsolvable. This provides an example of an unsolvable decision problem which is of very simple form, and which may for that reason be found useful in obtaining proofs of unsolvability of decision problems arising in various special branches of mathematics. Moreover, as'the author points out, if we define a normal set of sequences on the letters Oi, o2, ••• ,af as the set of assertions containing those letters only, in any system in normal form whose primitive symbols include those letters, and a binormal set as a normal set whose complement is normal, then on this basis a new and independent approach to the question of effective calculability is possible, the notion of binormality taking the place of that of recursiveness or of X-definability, and the notion of normality taking the place of that of recursive enumerability. The paper contains also a discussion of the "problem of tag," a decision problem closely related to the decision problem for systems in Post's normal form; and a statement, without proof, of an alternative reduction of systems in canonical form (in place of the reduction to normal form). According to the historical footnote above referred to, the author had in 1921 at least an outline of a proof (using the diagonal method) of the unsolvability of the general decision problem for systems in normal form, and in consequence of this an anticipation of Godel's incompleteness theorem (41&?). These results were not published. ALONZO CHURCH