 Barwise argues that in fact the fixed point analysis is more flexible and consequently more general than the hierarchical account. This may surprise readers in light of Proposition 2.18, which shows that Barwise’s fixed point definition is equivalent to the hierarchical account. Indeed, while Barwise (1988, 1989) proves a result showing that the fixed point account implies the hierarchical account and gives examples that satisfy the common knowledge hierarchy but fail to be fixed points, a number of authors who have written after Barwise have given various proofs of the equivalence of the two definitions, as was shown in Proposition 2.18. In fact, as (Heifetz 1999) shows, the hierarchical and fixed-point accounts are equivalent for all finite levels of iteration, while fixed-point common knowledge implies the conjunction of mutual knowledge up to any transfinite order, but it is never implied by any such conjunction.