A Note on Hardy's Persistent Numbers

The following contains several straightforward theorems, directly based on work of HARDY [l] and BACHMANN [2 ] . There is the problem of assigning canonical fundamental sequences to limit numbers in CANTOR'S second number class Z ( N ~ ) . It is clear that for comparatively early limit numbers of Z ( K ~ ) , a certain particular choice of fundamental sequences is natural. But the above results point either to the impossibility of a natural assignment for all the limits of Z(K,,) , or t,o a serious defect in our notion of uncountability.