The Lost Melody Theorem for Infinite Time Blum-Shub-Smale Machines
Logic
2026-05-19 v2
Abstract
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 at which new real numbers appear at all.
Keywords
Cite
@article{arxiv.2009.10582,
title = {The Lost Melody Theorem for Infinite Time Blum-Shub-Smale Machines},
author = {Merlin Carl},
journal= {arXiv preprint arXiv:2009.10582},
year = {2026}
}