In [1] Myhill gave a system K in which much of the classical theory of rational and real numbers could be carried out but which was nevertheless complete; this was achieved by sacrificing the notions of negation and universal quantification and introducing instead the ancestral as a primitive idea. He mentioned two ways of dealing with real numbers in K; the first was to use the half-section corresponding to x, i.e. the class of rationals r satisfying r ≤, x; the second was to use the whole-section corresponding to x, i.e. the relation between rationals r, s which holds when r ≤ x and x ≤ s. In [2] he proved various theorems about the relation between these two forms of definition. In particular he proved that every bounded class of half-sections definable in K has a least bound definable in K. In a footnote he said, ‘This theorem risks triviality because it is doubtful whether there exists such a bounded class.’ We show in this note that this is the case; in fact there are not even any definable classes consisting entirely of classes of rationals. It is not difficult to see how this happens; since K has no negation, any statement matrix A (α) defining a class of classes of sequences is satisfied by α = V, where V is the class of all sequences. Thus there are no classes which consist entirely of half-sections. The same is true for whole-sections, showing that Conjecture I of [1], viz. ‘Not every bounded class of whole-sections definable in K has a least bound definable in K’, is trivially false. Since it may be of some intrinsic interest, we give below a more complete description of the classes of classes of sequences which are definable in K (or rather in K1 see below) than is necessary merely in order to prove the above assertion.