User blog comment:Vel!/Music/@comment-5150073-20140503054608/@comment-25418284-20140505122336

We've been over this, so I'll state this as clearly as I can:


 * \(f_{\omega_1}(n)\) is not defined, and there is no elegant and internally consistent way to define it.

This has nothing to do with the continuum hypothesis or real numbers, actually. It's because \(\omega_1\) is a regular cardinal, and it cannot be expressed as a limit of a countable set of countable ordinals. This means that \(\omega_1\) cannot have a fundamental sequence, where "fundamental sequence" of ordinal \(\alpha\) is defined as an infinite (order type \(\omega\)) sequence of ordinals less than \(\alpha\) and with a limit of \(\alpha\).