The prospects for mathematical logic in the twenty-first century

The four authors present their speculations about the future developments of mathematical logic in the twenty-first century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.Comment: Association for Symbolic Logi

Computability and analysis: the legacy of Alan Turing

We discuss the legacy of Alan Turing and his impact on computability and analysis.Comment: 49 page

A Primer on the Tools and Concepts of Computable Economics

Computability theory came into being as a result of Hilbert's attempts to meet Brouwer's challenges, from an intuitionistc and constructive standpoint, to formalism as a foundation for mathematical practice. Viewed this way, constructive mathematics should be one vision of computability theory. However, there are fundamental differences between computability theory and constructive mathematics: the Church-Turing thesis is a disciplining criterion in the former and not in the latter; and classical logic - particularly, the law of the excluded middle - is not accepted in the latter but freely invoked in the former, especially in proving universal negative propositions. In Computable Economic an eclectic approach is adopted where the main criterion is numerical content for economic entities. In this sense both the computable and the constructive traditions are freely and indiscriminately invoked and utilised in the formalization of economic entities. Some of the mathematical methods and concepts of computable economics are surveyed in a pedagogical mode. The context is that of a digital economy embedded in an information society

Computability and Algorithmic Complexity in Economics

This is an outline of the origins and development of the way computability theory and algorithmic complexity theory were incorporated into economic and finance theories. We try to place, in the context of the development of computable economics, some of the classics of the subject as well as those that have, from time to time, been credited with having contributed to the advancement of the field. Speculative thoughts on where the frontiers of computable economics are, and how to move towards them, conclude the paper. In a precise sense - both historically and analytically - it would not be an exaggeration to claim that both the origins of computable economics and its frontiers are defined by two classics, both by Banach and Mazur: that one page masterpiece by Banach and Mazur ([5]), built on the foundations of Turing’s own classic, and the unpublished Mazur conjecture of 1928, and its unpublished proof by Banach ([38], ch. 6 & [68], ch. 1, #6). For the undisputed original classic of computable economics is Rabinís effectivization of the Gale-Stewart game ([42];[16]); the frontiers, as I see them, are defined by recursive analysis and constructive mathematics, underpinning computability over the computable and constructive reals and providing computable foundations for the economist’s Marshallian penchant for curve-sketching ([9]; [19]; and, in general, the contents of Theoretical Computer Science, Vol. 219, Issue 1-2). The former work has its roots in the Banach-Mazur game (cf. [38], especially p.30), at least in one reading of it; the latter in ([5]), as well as other, earlier, contributions, not least by Brouwer.

Computational reverse mathematics and foundational analysis

Reverse mathematics studies which subsystems of second order arithmetic are equivalent to key theorems of ordinary, non-set-theoretic mathematics. The main philosophical application of reverse mathematics proposed thus far is foundational analysis, which explores the limits of different foundations for mathematics in a formally precise manner. This paper gives a detailed account of the motivations and methodology of foundational analysis, which have heretofore been largely left implicit in the practice. It then shows how this account can be fruitfully applied in the evaluation of major foundational approaches by a careful examination of two case studies: a partial realization of Hilbert's program due to Simpson [1988], and predicativism in the extended form due to Feferman and Sch\"{u}tte. Shore [2010, 2013] proposes that equivalences in reverse mathematics be proved in the same way as inequivalences, namely by considering only $\omega$-models of the systems in question. Shore refers to this approach as computational reverse mathematics. This paper shows that despite some attractive features, computational reverse mathematics is inappropriate for foundational analysis, for two major reasons. Firstly, the computable entailment relation employed in computational reverse mathematics does not preserve justification for the foundational programs above. Secondly, computable entailment is a $\Pi^1_1$ complete relation, and hence employing it commits one to theoretical resources which outstrip those available within any foundational approach that is proof-theoretically weaker than $\Pi^1_1\text{-}\mathsf{CA}_0$.Comment: Submitted. 41 page