The Triviality Thesis

The basic construction and the Jones tower are generalised to this new setting and the first examples of sub-W*-bundles are constructed.

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He concludes that the Turing machine formalism, despite its extreme simplicity, is powerful enough to capture all humanly executable mechanical procedures over symbolic configurations. What this means is not always so clear, but the basic idea is usually that computation operates over discrete configurations.

Subsequent discussants have almost universally agreed. By comparison, many historically important algorithms operate over continuously variable configurations.

For example, the familiar grade-school algorithms describe how to compute addition, multiplication, and division.

Until the early twentieth century, mathematicians relied upon informal notions of computation and algorithm without attempting anything like a formal analysis.

For example, Euclidean geometry assigns a large role to , which manipulate geometric shapes.

For any shape, one can find another that differs to an arbitrarily small extent.Turing motivates his approach by reflecting on idealized human computing agents.Citing finitary limits on our perceptual and cognitive apparatus, he argues that any symbolic algorithm executed by a human can be replicated by a suitable Turing machine.Symbolic configurations manipulated by a Turing machine do not differ to arbitrarily small extent.Turing machines operate over discrete strings of elements (digits) drawn from a finite alphabet.Turing’s model works as follows: Turing translates this informal description into a rigorous mathematical model.For more details, see the entry on Turing machines.This thesis collates, extends and applies the abstract theory of W*-bundles.Highlightsinclude the standard form for W*-bundles, a bicommutant theorem for W*-bundles, andan investigation of completions, ideals, and quotients of W*-bundles.Developments in the foundations of mathematics eventually impelled logicians to pursue a more systematic treatment.Alan Turing’s landmark paper “On Computable Numbers, With an Application to the Entscheidungsproblem” (Turing 1936) offered the analysis that has proved most influential.