Hobson’s Conception of Definable Numbers

History and Philosophy of Logic 41 (2):128-139 (2020)
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In this paper, I explore an intriguing view of definable numbers proposed by a Cambridge mathematician Ernest Hobson, and his solution to the paradoxes of definability. Reflecting on König’s paradox and Richard’s paradox, Hobson argues that an unacceptable consequence of the paradoxes of definability is that there are numbers that are inherently incapable of finite definition. Contrast to other interpreters, Hobson analyses the problem of the paradoxes of definability lies in a dichotomy between finitely definable numbers and not finitely definable numbers. To bypass this predicament, Hobson proposes a language dependent analysis of definable numbers, where the diagonal argument is employed as a means to generate more and more definable numbers. This paper examines Hobson’s work in its historical context, and articulates his argument in detail. It concludes with a remark on Hobson’s analysis of definability and Alan Turing’s analysis of computability.



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Zhao Fan
Kobe University

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References found in this work

On Computable Numbers, with an Application to the Entscheidungsproblem.Alan Turing - 1936 - Proceedings of the London Mathematical Society 42 (1):230-265.
Mathematical Logic as Based on the Theory of Types.Bertrand Russell - 1908 - American Journal of Mathematics 30 (3):222-262.
Principia mathematica.A. N. Whitehead & B. Russell - 1910-1913 - Revue de Métaphysique et de Morale 19 (2):19-19.
On some difficulties in the theory of transfinite numbers and order types.Bertrand Russell - 1905 - Proceedings of the London Mathematical Society 4 (14):29-53.
Remarks before the Princeton Bicentennial Conference on Problems in Mathematics.Kurt Gödel - 1946 - In Solomon Feferman, John Dawson & Stephen Kleene (eds.), Kurt Gödel: Collected Works Vol. Ii. Oxford University Press. pp. 150--153.

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