We show that the set of ultimately true sentences in Hartry Field's Revenge-immune solution model to the semantic paradoxes is recursively isomorphic to the set of stably true sentences obtained in Hans Herzberger's revision sequence starting from the null hypothesis. We further remark that this shows that a substantial subsystem of second-order number theory is needed to establish the semantic values of sentences in Field's relative consistency proof of his theory over the ground model of the standard natural numbers: -CA0 (...) (second-order number theory with a -comprehension axiom scheme) is insufficient. We briefly consider his claim to have produced a solution to the semantic paradoxes by introducing this conditional. We remark that the notion of a operator can be introduced in other settings. (shrink)

We consider various concepts associated with the revision theory of truth of Gupta and Belnap. We categorize the notions definable using their theory of circular definitions as those notions universally definable over the next stable set. We give a simplified account of varied revision sequences-as a generalised algorithmic theory of truth. This enables something of a unification with the Kripkean theory of truth using supervaluation schemes.

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.

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

Using infinite time Turing machines we define two successive extensions of Kleene’s ${\mathcal{O}}$ and characterize both their height and their complexity. Specifically, we first prove that the one extension—which we will call ${\mathcal{O}^{+}}$ —has height equal to the supremum of the writable ordinals, and that the other extension—which we will call ${\mathcal{O}}^{++}$ —has height equal to the supremum of the eventually writable ordinals. Next we prove that ${\mathcal{O}^+}$ is Turing computably isomorphic to the halting problem of infinite time Turing computability, (...) and that ${\mathcal{O}^{++}}$ is Turing computably isomorphic to the halting problem of eventual computability. (shrink)

We introduce an analogue of the theory of Borel equivalence relations in which we study equivalence relations that are decidable by an infinite time Turing machine. The Borel reductions are replaced by the more general class of infinite time computable functions. Many basic aspects of the classical theory remain intact, with the added bonus that it becomes sensible to study some special equivalence relations whose complexity is beyond Borel or even analytic. We also introduce an infinite time generalization of the (...) countable Borel equivalence relations, a key subclass of the Borel equivalence relations, and again show that several key properties carry over to the larger class. Lastly, we collect together several results from the literature regarding Borel reducibility which apply also to absolutely $\Delta^1_2$ reductions, and hence to the infinite time computable reductions. (shrink)

We consider the following problem for various infinite-time machines. If a real is computable relative to a large set of oracles such as a set of full measure or just of positive measure, a comeager set, or a nonmeager Borel set, is it already computable? We show that the answer is independent of ZFC for ordinal Turing machines with and without ordinal parameters and give a positive answer for most other machines. For instance, we consider infinite-time Turing machines, unresetting and (...) resetting infinite-time register machines, and α-Turing machines for countable admissible ordinals α. (shrink)