The Tree Pulldown Method: McLaughlin's Conjecture and Beyond

Abstract

This paper finally fully elaborates the tree pulldown method used by one of us (Harrington) to settle McLaughlin's conjecture. This method enables the construction of a computable tree $T_0$ whose paths are incomparable over $0^{(\hat{I}\pm )}$ and resemble $\hat{I}\pm$-generics while leaving us almost completely free to specify the homeomorphism class of $[T_0]$. While a version of this method for $\hat{I}\pm =\ddot{I}‰$ previously appeared in print we give the general construction for an arbitrary ordinal notation $\hat{I}\pm$. We also demonstrate this method can be applied to a `non-standard' ordinal notation to establish the existence of a computable tree whose paths are hyperarithmetically incomparable and resemble $\hat{I}\pm$-generics for all $\hat{I}\pm <\ddot{I}{‰}_1^{CK}$. Finally, we verify a number of corollaries including solutions to problems 57${}^{\ast }$ , 62, 63 (McLaughlin's conjecture), 65 and 71 from Friedman's famous "One Hundred and Two Problems in Mathematical Logic."