# Statman's Hierarchy Theorem

@article{Westerbaan2017StatmansHT, title={Statman's Hierarchy Theorem}, author={Bram Westerbaan and Bas Westerbaan and Rutger Kuyper and Carst Tankink and Remy Viehoff and HENK P. Barendregt}, journal={ArXiv}, year={2017}, volume={abs/1711.05497} }

In the Simply Typed $\lambda$-calculus Statman investigates the reducibility relation $\leq_{\beta\eta}$ between types: for $A,B \in \mathbb{T}^0$, types freely generated using $\rightarrow$ and a single ground type $0$, define $A \leq_{\beta\eta} B$ if there exists a $\lambda$-definable injection from the closed terms of type $A$ into those of type $B$. Unexpectedly, the induced partial order is the (linear) well-ordering (of order type) $\omega + 4$.
In the proof a finer relation $\leq_{h…

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