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    Made withinDC&Austin
    Statements
    321,452
    Perspectives
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    42
    Home/Original/inverse
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    Inverse View

    It is not the case that The Polynomial Hierarchy collapses if either the Buss hierarchy of theories collapses at some level or S_2 (or T_2) is finitely axiomatizable

    ?Set your confidence on the premises below to see your aggregate.

    Reasons For

    2 perspectives
    Reason for 1 of 2
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    • 1.The correspondence between provably total functions of S^i_2 and Σ^p_i relies on Buss's witnessing theorem, which itself assumes consistency of the theories involved.
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    • 2.Gödel's second incompleteness theorem entails that S_2 cannot prove its own consistency, so the collapse inference operates outside the system it purports to characterize.
      ?

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    • 3.A hierarchy collapse demonstrated only in a metatheory stronger than S_2 may be epistemically inaccessible to the very computational agents whose complexity it claims to describe.
      ?

      Think about whether this reason is strong or weak

    Reason for 2 of 2
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    • 1.Finite axiomatizability of S_2 or T_2 is an open problem, and modal epistemologists following Williamson hold that possibility claims require more than the absence of known refutations.
      ?

      Think about whether this reason is strong or weak

    • 2.The conditional structure 'if X collapses then PH collapses' conveys no information about the actual modal status of PH unless the antecedent condition is independently motivated as genuinely possible.
      ?

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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.The provably total functions of S^i_2 correspond to the i-th level of the Polynomial Hierarchy
      ?

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    • 2.If there exists i such that T^i_2 = S^{i+1}_2, the corresponding levels of the Polynomial Hierarchy coincide
      ?

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    • 3.If S_2 or T_2 is finitely axiomatizable, the union hierarchy collapses to a finite level, collapsing the Polynomial Hierarchy correspondingly
      ?

      Think about whether this reason is strong or weak

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