For any MSO formula over the infinite binary tree whose existential witness is well-founded, the minimal ordinal rank bound is either strictly below ω² or equal to ω1, and it is decidable which holds.
Bradfield, Jacques Duparc, and Sandra Quickert
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A Dichotomy Theorem for Ordinal Ranks in MSO
For any MSO formula over the infinite binary tree whose existential witness is well-founded, the minimal ordinal rank bound is either strictly below ω² or equal to ω1, and it is decidable which holds.