The Lost Melody Theorem for Infinite Time Blum-Shub-Smale Machines
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866911692742459392 |
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| author | Carl, Merlin |
| author_facet | Carl, Merlin |
| contents | We consider recognizability for Infinite Time Blum-Shub-Smale machines, a model of infinitary computability introduced in Koepke and Seyfferth [KS]. In particular, we show that the lost melody theorem (originally proved for ITTMs in Hamkins and Lewis [HL]), i.e. the existence of non-computable, but recognizable real numbers, holds for ITBMs, that ITBM-recognizable real numbers are hyperarithmetic and that both ITBM-recognizable and ITBM-unrecognizable real numbers appear at every level of the constructible hierarchy below $L_{ω_{1}^{\text{CK}}}$ at which new real numbers appear at all. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2009_10582 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | The Lost Melody Theorem for Infinite Time Blum-Shub-Smale Machines Carl, Merlin Logic We consider recognizability for Infinite Time Blum-Shub-Smale machines, a model of infinitary computability introduced in Koepke and Seyfferth [KS]. In particular, we show that the lost melody theorem (originally proved for ITTMs in Hamkins and Lewis [HL]), i.e. the existence of non-computable, but recognizable real numbers, holds for ITBMs, that ITBM-recognizable real numbers are hyperarithmetic and that both ITBM-recognizable and ITBM-unrecognizable real numbers appear at every level of the constructible hierarchy below $L_{ω_{1}^{\text{CK}}}$ at which new real numbers appear at all. |
| title | The Lost Melody Theorem for Infinite Time Blum-Shub-Smale Machines |
| topic | Logic |
| url | https://arxiv.org/abs/2009.10582 |