The Tree Pulldown Method: McLaughlin's Conjecture and Beyond
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912337360846848 |
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| author | Harrington, Leo A. Gerdes, Peter M. |
| author_facet | Harrington, Leo A. Gerdes, Peter M. |
| contents | 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." |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_14323 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Tree Pulldown Method: McLaughlin's Conjecture and Beyond Harrington, Leo A. Gerdes, Peter M. Logic 03D60, 03D30 (Primary), 03C70 (Secondary) 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." |
| title | The Tree Pulldown Method: McLaughlin's Conjecture and Beyond |
| topic | Logic 03D60, 03D30 (Primary), 03C70 (Secondary) |
| url | https://arxiv.org/abs/2504.14323 |