Characterisations of Variant Transfinite Computational Models:Infinite Time Turing, Ordinal Time Turing, and Blum-Shub-Smale machines

We consider how changes in transfinite machine architecture can sometimes alter substantially their capabilities. We approach the subject by answering three open problems touching on: firstly differing halting time considerations for machines with multiple as opposed to single heads, secondly space requirements, and lastly limit rules. We: 1) use admissibility theory, Σ2-codes and Π3-reflection properties in the constructible hierarchy to classify the halting times of ITTMs with multiple independent heads; the same for Ordinal Turing Machines which have On length tapes; 2) determine which admissible lengths of tapes for transfinite time machines with long tapes allow the machine to address each of their cells – a question raised by B. Rin; 3) characterise exactly the strength and behaviour of transfinitely acting Blum–Shub–Smale machines using a Liminf rule on their registers – thereby establishing there is a universal such machine. This is in contradistinction to the machine using a ‘continuity’ rule which fails to be universal

Some observations on truth hierarchies

AbstractWe show how in the hierarchies${F_\alpha }$of Fieldian truth sets, and Herzberger’s${H_\alpha }$revision sequence starting from any hypothesis for${F_0}$(or${H_0}$) that essentially each${H_\alpha }$(or${F_\alpha }$) carries within it a history of the whole prior revision process.As applications (1) we provide a precise representation for, and a calculation of the length of, possiblepath independent determinateness hierarchiesof Field’s (2003) construction with a binary conditional operator. (2) We demonstrate the existence of generalized liar sentences, that can be considered as diagonalizing past the determinateness hierarchies definable in Field’s recent models. The ‘defectiveness’ of such diagonal sentences necessarily cannot be classified by any of the determinateness predicates of the model. They are ‘ineffable liars’. We may consider them a response to the claim of Field (2003) that ‘the conditional can be used to show that the theory is not subject to “revenge problems”.’</jats:p