The main result of the paper states that for any natural number \(h\) first-order Peano arithmetic PA does not prove the following sentence: For all \(K\) there exists an \(M\) which bounds the lengths \(n\) of all strictly descending sequences \(\langle \alpha_{0}, \ldots , \alpha_{n} \rangle\) of ordinals less than \(\varepsilon_{0}\) which satisfy the condition that the norm \(N \alpha_{i}\) of the \(i\)-th term \(\alpha_{i}\) is bounded by \(K + | i| \times | i| _{h}\), where the symbols have the following meaning: for \(\alpha\) less than \(\varepsilon_{0}\) the symbol \(N \alpha\) denotes the number of occurrences of \(\omega\) in the Cantor normal form of \(\alpha\), further \(| n| \) denotes the binary length of a natural number \(n\), \(| n| _{h}\) denotes the \(h\)-times iterated binary length of \(n\) and inv(\(n\)) is the least \(h\) such that \(| n| _{h} \leq 2\). It is also shown that, e.g., primitive recursive arithmetic PRA proves that for all \(K\) there is an \(M\) which bounds the length \(n\) of any strictly descending sequence \(\langle \alpha_{0}, \ldots , \alpha_{n} \rangle\) of ordinals less than \(\varepsilon_{0}\) which satisfies the condition that the norm \(N \alpha_{i}\) of the \(i\)-th term \(\alpha_{i}\) is bounded by \(K + | i| \times \text{inv}(i)\). The proofs are based on results from proof theory and techniques from asymptotic analysis of Polya-style enumerations.