Reachability for infinite time Turing machines with long tapes

Infinite time Turing machine models with tape length α, denoted T_α, strengthen the machines of Hamkins and Kidder with tape length ω. A new phenomenon is that for some countable ordinals α, some cells annot be halting positions of T_α given trivial input. The main open question in a paper of Rin from 2014 asks about the size of the least such ordinal δ. We answer this by providing various characterizations. For instance, δ is the least ordinal with any of the following properties: • For some ξ < α, there is a T_ξ -writable but not T_α-writable subset of ω. • There is a gap in the T_α-writable ordinals. • α is uncountable in L_λ^α . Here λ_α denotes the supremum of T_α-writable ordinals, i.e. those with a Tα-writable code of length α. We further use the above characterizations, and an analogue to Welch’s submodel characterization of the ordinals λ, ζ and Σ, to show that δ is large in the sense that it is a closure point of the function α → Σ_α, where Σ_α denotes the supremum of the T_α-accidentally writable ordinals.