Forum:Are there ordinal hierarchies without fundamental sequences?

Define an ordinal hierarchy as \(F: \omega_1 \mapsto (\mathbb{N} \mapsto \mathbb{N})\) where for all \(\alpha < \beta\), \(F(\alpha)\) eventually outgrows \(F(\beta)\). Define a fundamental sequence system as \(S: \omega_1 \mapsto (\omega_1)^\omega\) (where \((\omega_1)^\omega\) is the set of all \(\omega\)-sequences in \(\omega_1\)) such that for all \(\alpha\) we have \(\sup\{b: b \in S(a)\} = \alpha\). If we take the axiom of choice, ordinal hierarchies and fundamental sequence systems must exist, so let's scale it back to ZF. Is there a model of ZF where ordinal hierarchies exist but fundamental sequence systems don't? it's vel time 07:21, October 4, 2014 (UTC)