consider a countably infinite list of infinite strings.. such that each string has an ordinal of $ \ \bf ɷ.2 \ $, and the entire list also has an ordinal of $ \ \bf ɷ.2 \ $. Can we use cantor's diaonalization on this list in this manner ?
I think we should be able to do that, and the diagonalized string would also have an ordinal of $ \ \bf ɷ.2 \ $, but can't find any particular such example on internet...
You certainly can do that. We may regard each string as a function $f : \omega \cdot 2 \to S$, where $|S| \geq 2$ is the set of characters which the strings take on. We may enumerate the set of strings $\{f_\alpha : \alpha < \omega \cdot 2\}$, and define the diagonalisation $g : \omega \cdot 2 \to 2$ by stipulating that $g(\alpha) \neq f_\alpha(\alpha)$ for all $\alpha < \omega \cdot 2$. Then $g \neq f_\alpha$ for all $\alpha$.