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 $0^{(α)}$ and resemble $α$-generics while leaving us almost completely free to specify the homeomorphism class of $[T_0]$. While a version of this method for $α= ω$ previously appeared in print we give the general construction for an arbitrary ordinal notation $α$. 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 $α$-generics for all $α< ω_1^{CK}$. Finally, we verify a number of corollaries including solutions to problems 57$^{*}$ , 62, 63 (McLaughlin's conjecture), 65 and 71 from Friedman's famous "One Hundred and Two Problems in Mathematical Logic."