We describe the solution of the Limit Rule Problem of Revision Theory and discuss the philosophical consequences of the fact that the truth set of Revision Theory is a complete 1 2 set.

9.9.05 1245am NY time Do human persons hypercompute? Or, as the doctrine of computationalism holds, are they information processors at or below the Turing Limit? If the former, given the essence of hypercomputation, persons must in some real way be capable of infinitary information processing. Using as a springboard Gödel’s little-known assertion that the human mind has a power “converging to infinity, ” and as an anchoring problem Rado’s (1963) Turing-uncomputable “busy beaver ” (or Σ) function, we present in this short paper a new argument that, in fact, human persons can hypercompute. The argument is intended to be formidable, not conclusive: it brings Gödel’s intuition to a greater level of precision, and places it within a sensible case against computationalism. 1

We look at various notions of a class of denability operations that generalise inductive operations, and are characterised as \revision operations ". More particularly we: (i) characterise the revision theoretically denable subsets of a countable acceptable structure; (ii) show that the categorical truth set of Belnap and Gupta's theory of truth over arithmetic using fully varied revision sequences yields a complete 1 3 set of integers; (iii) the set of stably categorical sentences using their revision operator is similarly 1 3 and which is complete in Godel's universe of constructible sets L; (iv) give an alternative account of a theory of truth - realistic variance that simplies full variance, whilst at the same time arriving at Kripkean xed points. We should like to thank the Mathematics Department of UC Berkeley for its hospitality during the period when this research was conducted, and to the Institut fur Formale Logik in Vienna for a Gastprofessur where this paper was nally written. Keywords: revision theory, denability theory, admissibility theory, descriptive set theory. AMS Classications: 03A05,03D70,03E15,03F35 1 1

Abstract. Revision is a method to deal with non-monotonic processes. It has been used in theory of truth an an answer to semantic paradoxes as the liar, but the idea is universal and resurfaces in many areas of logic and applications of logic. In this survey, we describe the general idea in the framework of pointer semantics and point out that beyond the formal semantics given by Gupta and Belnap, the process of revision itself and its behaviour may be the central features that allow us to model our intuitions about truth, and is applicable to a lot of other areas like belief, rationality, and many more. 1

Abstract not found

Abstract. Wegner and Eberbach[Weg04b] have argued that there are fundamental limitations to Turing Machines as a foundation of computability and that these can be overcome by so-called superTuring models. In this paper we contest their claims for interaction machines and the πcalculus. 1

We discuss the question of Ralf-Dieter Schindler whether for infinite time Turing machines P f = NP f can be true for any function f from the reals into ω1. We show that “almost everywhere ” the answer is negative. 1

• We produce a classification of the pointclasses using infinite time turing machines with 1-tape. The reason for choosing this formalism is that it apparently yields a smoother classification of classes defined by algorithms that halt at limit ordinals. • We consider some relations of such classes with other similar notions, such as arithmetical quasi-inductive definitions. • It is noted that the action of ω many steps of such a machine can correspond to the double jump operator (in the usual Turing sense): a− → a ′ ′. • The ordinals beginning gaps in the “clockable ” ordinals are admissible ordinals, and the length of such gaps corresponds to the degree of reflection those ordinals enjoy. 1

Abstract. In the Cellular Automata (CA) literature, discrete lines inside (discrete) space-time diagrams are often idealized as Euclidean lines in order to analyze a dynamics or to design CA for special purposes. In this article, we present a parallel analog model of computation corresponding to this idealization: dimensionless signals are moving on a continuous space in continuous time generating Euclidean lines on (continuous) space-time diagrams. Like CA, this model is parallel, synchronous, uniform in space and time, and uses local updating. The main difference is that space and time are continuous and not discrete (i.e. R instead of Z). In this article, the model is restricted to Q in order to remain inside Turing-computation theory. We prove that our model can carry out any Turing-computation through two-counter automata simulation and provide some undecidability results.

Abstract. • We produce a classification of the pointclasses of sets of reals produced by infinite time turing machines with 1-tape. The reason for choosing this formalism is that it apparently yields a smoother classification of classes defined by algorithms that halt at limit ordinals. • We consider some relations of such classes with other similar notions, such as arithmetical quasi-inductive definitions. • It is noted that the action of ω many steps of such a machine can correspond to the double jump operator (in the usual Turing sense): a−→ a ′ ′. • The ordinals beginning gaps in the “clockable ” ordinals are admissible ordinals, and the length of such gaps corresponds to the degree of reflection those ordinals enjoy. 1