A countable limit ordinal $\kappa$ has cofinality $\omega$. One proves this in ZF, say, using the usual trick for representing $\kappa$ as a countable set of reals having closed convex span $[0,1]$ (with the usual order) and then comparing with any increasing sequence in $[0,1]$ converging to 1.

Nevertheless, I suspect independence from ZF for the following uniform version of this claim:

There exists a function $f:\omega \times \omega_1 \rightarrow \omega_1$ such that

1. $f(\alpha,\beta) < \beta$;

2. ${\rm sup}_\alpha f(\alpha,\beta) =\beta $ for $\beta$ a limit ordinal.

(For fixed $\beta$ assume that $f(\alpha,\beta)$ increases with $\alpha$, if you like.)

Briefly, such an $f$ would support, by induction and coding tricks, the construction of an injection from $\omega_1$ to ${\Bbb R}$; that in turn would mean that CH implies the existence of a well-ordering of the reals. (Details on demand.)

Questions:

1. Does this independence come up in the literature?

2. Can someone point me to a model of ZF+CH where the reals have no well-ordering?

3. Is it easy to get directly a model of ZF having no such $f$ by forcing?

4. Is the existence of $f$ equivalent to any well-known consequences of AC?

5. Are there toposes where even the original cofinality statement (on some reasonable interpretation) fails (for lack, say, of a global bijection between $\kappa$ and the natural number object)?