2
votes
1answer
179 views

Can we get uncountable ordinal numbers through constructive method?

As we know, $2^{\aleph_0}$ is a limit ordinal number, however, it is greater than $\omega$, $\omega+\omega$, $\omega \cdot \omega$, $\omega^\omega$, $\omega\uparrow\uparrow\omega$, and even $\omega ...
6
votes
0answers
157 views

Computable increasing function from $\omega_1^{CK} \to \mathbb{R}$

If $f:\omega_1 \to [0,1]$ such that $f(0)=0$ and such that $f(\alpha)<1$ and $\alpha<\beta$ imply $f(\alpha)<f(\beta)$ then there is a $\gamma$ such that $f(\gamma)=1$. Non-constructive ...