Gödel’s Master Argument: what is it, and what can it do?

This text is expository. We explain Gödel’s ‘Master Argument’ for incompleteness as distinguished from the 'official' proof of his 1931 paper, highlight its attractions and limitations, and explain how some of the limitations may be transcended by putting it in a more abstract form that makes no reference to truth

On the Semantics of Intensionality and Intensional Recursion

Intensionality is a phenomenon that occurs in logic and computation. In the most general sense, a function is intensional if it operates at a level finer than (extensional) equality. This is a familiar setting for computer scientists, who often study different programs or processes that are interchangeable, i.e. extensionally equal, even though they are not implemented in the same way, so intensionally distinct. Concomitant with intensionality is the phenomenon of intensional recursion, which refers to the ability of a program to have access to its own code. In computability theory, intensional recursion is enabled by Kleene's Second Recursion Theorem. This thesis is concerned with the crafting of a logical toolkit through which these phenomena can be studied. Our main contribution is a framework in which mathematical and computational constructions can be considered either extensionally, i.e. as abstract values, or intensionally, i.e. as fine-grained descriptions of their construction. Once this is achieved, it may be used to analyse intensional recursion.Comment: DPhil thesis, Department of Computer Science & St John's College, University of Oxfor

Computable analysis on the space of marked groups

We investigate decision problems for groups described by word problem algorithms. This is equivalent to studying groups described by labelled Cayley graphs. We show that this corresponds to the study of computable analysis on the space of marked groups, and point out several results of computable analysis that can be directly applied to obtain group theoretical results. Those results, used in conjunction with the version of Higman's Embedding Theorem that preserves solvability of the word problem, provide powerful tools to build finitely presented groups with solvable word problem but with various undecidable properties. We also investigate the first levels of an effective Borel hierarchy on the space of marked groups, and show that on many group properties usually considered, this effective hierarchy corresponds sharply to the Borel hierarchy. Finally, we prove that the space of marked groups is a Polish space that is not effectively Polish. Because of this, many of the most important results of computable analysis cannot be applied to the space of marked groups. This includes the Kreisel-Lacombe-Schoenfield-Ceitin Theorem and a theorem of Moschovakis. The space of marked groups constitutes the first natural example of a Polish space that is not effectively Polish.Comment: 46 pages, Theorem 4.6 was false as stated, it appears now, having been corrected, as Theorem 5.

New definitions in the theory of Type 1 computable topological spaces

Motivated by the two remarks, that the study of computability based on the use of numberings -- Type 1 computability -- does not have to be restricted to countable sets equipped with onto numberings, and that computable topologies have been in part developed with the implicit hypothesis that the considered spaces should be computably separable, we propose new definitions for Type 1 computable topological spaces. We define computable topological spaces without making reference to a basis. Following Spreen, we show that the use of a formal inclusion relation should be systematized, and that it cannot be avoided if we want computable topological spaces to generalize computable metric spaces. We also compare different notions of effective bases. The first one, introduced by Nogina, is based on an effective version of the statement "a set $O$ is open if for any point in $O$, there is a basic set containing that point and contained in $O$''. The second one, associated to Lacombe, is based on an effective version of "a set $O$ is open if it can be written as a union of basic open sets''. We show that neither of these notions of basis is completely satisfactory: Nogina bases do not permit to define computable topologies unless we restrict our attention to countable sets, and the conditions associated to Lacombe bases are too restrictive, and they do not apply to metric spaces unless we add effective separability hypotheses. We define a new notion of basis, based on an effective version of the Nogina statement, but adding to it several classically empty conditions, expressed in terms of formal inclusion relations. Finally, we obtain a new version of the theorem of Moschovakis which states that the Lacombe and Nogina approaches coincide on countable recursive Polish spaces, but which applies to sets equipped with non-onto numberings, and with effective separability as a sole hypothesis.Comment: 50 pages, 2 figure

Computing Measure as a Primitive Operation in Real Number Computation

We study the power of BSS-machines enhanced with abilities such as computing the measure of a BSS-decidable set or computing limits of BSS-computable converging sequences. Our variations coalesce into just two equivalence classes, each of which also can be described as a lower cone in the Weihrauch degrees. We then classify computational tasks such as computing the measure of ???-set of reals, integrating piece-wise continuous functions and recovering a continuous function from an L?([0, 1])-description. All these share the Weihrauch degree lim