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    Challenges→Many-one reducibility implies Turing reducibility (A ≤_m B implies A ≤_T B)

    In constructive and intuitionistic frameworks (following Brouwer and Bishop), the existence of a computable f does not automatically yield an effective procedure without additional choice principles that are not constructively neutral.

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    Key Terms

    Bishop(history of mathematics)
    Errett Bishop was an American mathematician who developed constructive analysis, a stricter version of math that only accepts proofs you can actually build or compute.
    Brouwer
    Brouwer was a Dutch mathematician and philosopher (1881-1966) who fundamentally changed how mathematicians think about proof and logic. He argued that math shouldn't just accept something as true because it follows logically; instead, mathematicians should be able to actually construct or demonstrate mathematical objects to prove they exist. His ideas challenged the traditional approach to mathematics and influenced debates about the foundations of mathematical reasoning that continue today.
    Choice principles(as logical tools)
    Rules in logic that allow you to make selections from groups of things; the most famous is the 'axiom of choice,' which says you can always pick one item from each group, even if there are infinitely many groups.
    Constructive/Constructivism(as a mathematical and logical framework)

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    A approach to math and logic that only accepts something as true if you can actually build or demonstrate it step-by-step, rather than just proving it must exist in theory.
    Constructively neutral(as a property of logical principles)
    Something that doesn't contradict constructive thinking or require assumptions that constructivists reject as unproven.
    Effective procedure(as the practical outcome being discussed)
    A clear set of instructions that actually works in practice to solve a problem or accomplish a task.
    Intuitionistic(as a logical framework)
    A system of logic that rejects the idea that something must be either true or false if we can't actually verify which one it is; closely related to constructivism.
    computable(As employed in the technical literature discussed in this passage)
    Computable by an effective method

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    Truth & Knowledge1 linkedModality & Possibility1 linked

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    Many-one reducibility implies Turing reducibility (A ≤_m B implies A ≤_T B)

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