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    Carmelics

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    Made withinDC&Austin
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    Home/Original/inverse
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    Inverse View

    It is not the case that Autonomous progressions of theories properly internalize the general concept of progressions of theories

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    Reasons For

    2 perspectives
    Reason for 1 of 2
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    • 1.The boot-strapping process still presupposes a background metatheory capable of recognizing constructive ordinals, smuggling in external commitments.
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    • 2.Feferman's own analysis of predicativity shows that any ordinal recognition procedure relies on prior comprehension principles not internal to the progression.
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    • 3.A genuinely autonomous system cannot appeal to O without an antecedent definition of O that transcends the theories being generated.
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    Reason for 2 of 2
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    • 1.Kreisel demonstrated that the informal notion of 'constructive ordinal' resists full formalization within any single accepted theory, making the ascending condition semantically indeterminate.
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    • 2.If the criterion for legitimate ascent cannot be made fully precise inside the progression, the claim of proper internalization reduces to an idealized fiction rather than a rigorous logical achievement.
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    Reasons Against

    1 perspective
    Reason against
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    • 1.In autonomous progressions, ascent to a theory T_a is permitted only if a prior accepted theory T_b has already demonstrated that the ordinal index a belongs to the set of constructive ordinals O
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    • 2.This condition ensures the hierarchy is generated by a self-justifying boot-strapping process rather than by externally imposed ordinals
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