User blog comment:Syst3ms/YAUD/@comment-30754445-20180810132619/@comment-32697988-20180810162330

The definition above does not say that cof(α) means the cofinality of α and α[β] means the βth term of the fundamental sequence of α. The words "cofinality" and "fundamental sequence" does not even appear in the definition. (It does use the word "cofinality" in the explanation, but it is not a part of the definition.) So, you can't assume that cof(α) means the cofinality of α and α[β] means the βth term of the fundamental sequence of α.