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    Autonomous progressions of theories properly internalize ... — Carmelics
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    Autonomous progressions of theories properly internalize the general concept of progressions of theories

    Proof of definition segmentsTruth & Knowledge
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    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
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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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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 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 against 2 of 2
    ?
    • 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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    Related

    A genuinely autonomous system cannot appeal to O without an antecedent definitio...Feferman's own analysis of predicativity shows that any ordinal recognition proc...If the criterion for legitimate ascent cannot be made fully precise inside the p...In autonomous progressions, ascent to a theory T_a is permitted only if a prior ...
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    Kreisel demonstrated that the informal notion of 'constructive ordinal' resists ...The boot-strapping process still presupposes a background metatheory capable of ...This condition ensures the hierarchy is generated by a self-justifying boot-stra...

    Similar

    Kreisel identified a specific autonomous progression of theories for f...78%In autonomous progressions, ascent to a theory T_a is permitted only i...78%If theories are sets of sentences in a particular language, then stati...77%Mathematical theories are integral parts of scientific theories77%

    Source

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    In the foregoing progressions the ordinals remained external to the theory. Autonomous progressions of theories are the proper internalization of the general concept of progressions. In the autonomous case one is allowed to ascend to a theory \(\bT_a\) only if one already has shown in a previously accepted theory \(\bT_b\) that \(a\in{\cO}\). This idea of generating a hierarchy of theories via a boot-strapping process appeared for the first time in Kreisel 1960, where it was proposed as a way of
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    Details

    Type
    claim
    Perspectives
    3 (1 for, 2 against)
    Edits
    1 edit