Cofinality of countable ordinals in ZF, and in toposes

Question

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)?

Answer 1

Yes, this comes up. For example, Raisonnier showed that the existence of an injection $\omega_1 \to \mathbb{R}$ implies the existence of a subset of $\mathbb{R}$ which is not Lebesgue measurable over ZF + DC.

Jean Raisonnier, A mathematical proof of S. Shelah’s theorem on the measure problem and related results, Israel J. Math. 48 (1984), no. 1, 48--56.

Important examples of models where this property fails are:

• The Feferman-Levy model where $\mathbb{R}$ is a countable union of countable sets. [See Theorem 10.6 in Thomas J. Jech, The Axiom of Choice, Studies in Logic and the Foundations of Mathematics, Vol. 75. North-Holland Publishing Co., Amsterdam-London; Amercan Elsevier Publishing Co., Inc., New York, 1973. xi+202 pp.]

• The Solovay model where every subset of $\mathbb{R}$ is Lebesgue measurable. [Robert M. Solovay, A model of set-theory in which every set of reals is Lebesgue measurable, Ann. of Math. (2) 92 (1970), 1--56.]

Both of these model constructions use an inaccessible cardinal. This is necessary for if $\omega_1$ is not inaccessible in $L$ then there is a real $x$ such that $\omega_1^{L[x]} = \omega_1$ and then the required function can be constructed in $L[x]$ (which always satisfies AC).

Raisonnier's Theorem in a topos-theoretic context was discussed my answer this MathOverflow question:

Alex Simpson, How strong is “all sets are Lebesgue Measurable” in weaker contexts than ZF?