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A norm is a surjective function from the Baire space \(\mathbb{R}\) onto an ordinal. Given two norms \(\varphi, \psi\) we write \(\varphi \leq_{\mathrm{N}} \psi\) if \(\varphi\) continuously reduces to \(\psi\). Then \({\leq_{\mathrm{N}}}\) is a preordering and so passing to the set of corresponding equivalence classes yields a partial order, the hierarchy of norms.
Assuming the axiom of determinacy (\(\textsf{AD}\)) the hierarchy of norms is a wellorder. The length \(\Sigma\) of the hierarchy of norms was investigated by Löwe in [1]; he determined that \(\Sigma \geq \Theta^2\) (where \(\Theta := \sup\{\alpha \mid \text{There is a surjection from}\ \mathbb{R} \ \text{onto}\ \alpha \}\)). In his talk ``Multiplication in the hierarchy of norms'', given at the ASL 2011 North American Meeting in Berkeley, Löwe presented a binary operation \({\boxtimes}\) on the hierarchy of norms such that for wellchosen norms \(\varphi, \psi\) the ordinal rank of \(\varphi\boxtimes \psi\) in the hierarchy of norms is at least as big as the product of the ordinal ranks of \(\varphi\) and \(\psi\), which implies that \(\Sigma\) is closed under ordinal multiplication and so \(\Sigma \geq \Theta^{\omega}\).
In this talk I will note that in fact for wellchosen norms \(\varphi, \psi\) the ordinal rank of \(\varphi\boxtimes \psi\) is exactly the product of the ranks of \(\varphi\) and \(\psi\) with an intermediate factor of \(\omega_1\). Furthermore using a stratification of the hierarchy of norms into initial segments closed under the \({\boxtimes}\)-operation I will show that \(\Sigma \geq \Theta^{\left( \Theta^{\Theta} \right)}\).
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