On the logical strength of Nash-Williams' theorem on transfinite sequences

Abstract

We show that Nash-Williams' theorem asserting that the countable transfinite sequences of elements of a better-quasi-ordering ordered by embeddability form a better-quasi-ordering is provable in the subsystem of second order arithmetic Pi^1_1-CA_0 but is not equivalent to Pi^1_1-CA_0. We obtain some partial results towards the proof of this theorem in the weaker subsystem ATR_0 and we show that the minimality lemmas typical of wqo and bqo theory imply Pi^1_1-CA_0 and hence cannot be used in such a proof.