User blog comment:Vel!/Music/@comment-5150073-20140503054608/@comment-5150073-20140505120315

I should be pleased to find such maps. We can define then $$f_{\omega_1}(n) = f_{\omega_1[\alpha]}(n)$$, where n is the real number and $$\alpha$$ is the countable ordinal which maps to n. Unfortunately, somewhere in GWiki I saw a proof which says that it is impossible in principle. As for existence $$f_{\omega_1}(n)$$ for natural n, it depends on Continuum Hypothesis. There are $$2^{\aleph_0}$$ natural-to-natural functions, and $$\omega_1$$ countable ordinals... if CH is true, $$2^{\aleph_0}$$ is no larger than $$\omega_1$$ and there are exactly $$\omega_1$$ N-to-N functions, thus $$f_{\omega_1}(n)$$ doesn't exist.