In a previous note it was shown that if a bound f(i) is placed on degrees of elements in some basis of an ideal Ai in polynomial ring k[IX1, . . . 9 Xn] over an explicitly given field k, i = 0, 1, 2, * , then a bound can be (and was) constructed for length of a strictly ascending chain AO < A1 < .. . . This result is now obtained using a strictly finitist argument. A corollary is a finitist version of Hilbert's theorem on ascending chains. An early high point in tradition of constructive mathematics often associated with name of Kronecker is paper [1] of Hermann, which treats various ideal-theoretic notions in polynomial rings. Although paper contains errors and obscurities, still it does make considerable contributions to problems posed. According to Hermann, the assertion that a computation can be carried through in a finite number of steps shall mean that an upper bound for number of operations needed for computation can be given. Thus it does not suffice, for example, to give a procedure for which one can theoretically verify that it leads to goal in a finite number of operations, so long as no upper bound for number of these operations is known. This is obscure, really, though intention seems clear enough in situations actually dealt with. In [41 we posed following problem: A bound f (i) is placed on degrees of elements in some basis of an ideal Ai in polynomial ring k[Xl .. , Xn] over field k, i = 0,1, 2, . .-: place a bound on length of a strictly ascending chain A 0 < A 1 < K . . . It will be convenient to regard / as having been given multi-recursively. In [41 we give a bound which is multi-recursively defined in terms of data / and n. Our bound, unlike those given by Hermann, does not appear to be primitive recursive, but that is probably due to problem being more complex than any dealt with by Hermann; essence of matter is, we say, that a multi-recursively defined bound on number of Received by editors November 16, 1971. AMS (MOS) subject classifications (1970). Primary 02E99, 13F20, 13E05.