# Is it an open problem whether fast-growing hierarchies can be defined without fundamental sequences?

Googleology Wiki says this, concerning the relation between fast-growing hierarchies defined for all countable ordinals, and the existence of a system of assigning a canonical fundamental sequence to each countable limit ordinal:

It is an open problem whether an ordinal hierarchy can be defined without using fundamental sequences. More explicitly, the problem concerns whether there is a model of $$ZF$$ such that there exists an $$F : \omega_1 \rightarrow (\mathbb{N} \rightarrow \mathbb{N})$$ where for all $$\alpha > \beta, F(\alpha)$$ eventually outgrows $$F(\beta)$$, but there does not exist an $$S: \omega_1 \cap \text{Lim} \rightarrow (\mathbb{N} \rightarrow \omega_1)$$ such that for all $$\alpha, \sup(R) = \alpha$$ where $$R$$ is the range of $$S(\alpha)$$.

My question is, is it true that the question of whether there exists such a model of $$ZF$$ is an open problem?

• Can't you just define F in L? If F is valid in L, I think it would also be valid in V, right? – PyRulez Nov 28 '19 at 21:43