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    Gödel, Turing, and Church independently arrived at extens... — Carmelics
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    Challenges→The justification for classifying specific problems as undecidable can be no stronger than the confidence placed in Church's Thesis.

    Gödel, Turing, and Church independently arrived at extensionally equivalent formalizations, and this invariance across disparate formalisms gives the thesis a quasi-definitional status rather than a contingent one.

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    Key Terms

    Church(in the history of logic and computation)
    Alonzo Church (1903-1995), an American logician and mathematician who created formal systems to define what problems can be solved by mathematical procedures.
    Disparate formalisms(referring to the distinct methods developed by Gödel, Turing, and Church)
    Very different mathematical or logical systems or approaches.
    Formalizations(referring to different mathematical systems created by Gödel, Turing, and Church)
    Precise, rule-based mathematical or logical systems that define something rigorously, without ambiguity.
    Gödel(as a historical figure in mathematical logic)
    Kurt Gödel was a 20th-century mathematician and logician who proved that any consistent formal system (a set of logical rules) is incomplete—meaning there are true statements it can't prove.

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    Quasi-definitional status(describing how the Church-Turing thesis is treated as essentially true rather than just a hypothesis)
    Having a status that is almost like a definition—accepted as a fundamental truth rather than something that needs to be proven.
    Turing
    # Turing Alan Turing was a British mathematician and scientist (1912-1954) who is considered the father of computer science and artificial intelligence. He invented the "Turing Machine," a theoretical device that helped define what computers could and couldn't do, and created the "Turing Test," a famous challenge to determine whether a machine can exhibit intelligent behavior indistinguishable from a human. His groundbreaking work during World War II on code-breaking, combined with his pioneering ideas about thinking machines, made him one of the most influential thinkers of the 20th century.
    contingent(De Interpretatione 12–13)
    Equated with 'possible'; on the two-sided interpretation, contingency excludes necessity (possibility implies non-necessity).
    extensionally equivalent(Applied to Church's thesis and Turing's thesis regarding functions of positive integers)
    Two theses are extensionally equivalent when they are about one and the same class of functions
    invariance(Woodward's criterion within the manipulability account of causation)
    A measure of the extent to which a relationship between two variables satisfying the manipulation condition remains stable or unchanged as various other changes are made in the background of that relationship

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    The justification for classifying specific problems as undecidable can be no str...

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