User blog comment:Scorcher007/About Cofinality, sipmly/@comment-34422464-20190722143442/@comment-35470197-20190722150320

> \(\alpha = \{x_1,x_2,\ldots,x_n,\ldots\}_{n<\beta}\)

You need \(\sup\) at the beginning of the right hand side above. Moreover, it is not recursive because there is no canonical choice of \(\beta\). Therefore you need \(\inf\) at the beginning of the right hand side of the definition of cof.