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    Autonomous progressions of theories can characterize fini... — Carmelics
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    Home/Modality & Possibility
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    Autonomous progressions of theories can characterize finitism and predicativism in a mathematically precise way

    Modality & PossibilityProof of definition segments
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    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
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    • 1.The boot-strapping process generates a hierarchy of theories starting from an accepted base theory T_0 via an extension procedure T -> T' that leads from a correct theory to a more encompassing correct theory
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    • 2.The language of the theories contains a formula Acc(x) such that provability of Acc(a-bar) in a correct theory entails that a belongs to the constructive ordinals O
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
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    • 1.Weyl's predicativism was grounded in an informal conceptual commitment to definability from natural numbers, not in a formal ordinal-indexed hierarchy of systems.
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    • 2.Translating predicativism into autonomous progressions up to Γ₀ imposes a set-theoretic framework that Weyl would have rejected as itself impredicative in spirit.
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    • 3.A mathematically precise characterization that distorts the foundational motivation it purports to capture fails as a faithful account of the philosophical position.
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    Reason against 2 of 2
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    • 1.The notion of 'correctness' in the boot-strapping procedure presupposes an antecedent grasp of the very ordinal notions the progression is meant to characterize.
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    • 2.Feferman himself acknowledged that the autonomous condition requires recognizing which ordinal notations are 'legitimate,' a judgment that cannot be formalized without circularity.
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    Related

    A mathematically precise characterization that distorts the foundational motivat...Feferman himself acknowledged that the autonomous condition requires recognizing...The boot-strapping process generates a hierarchy of theories starting from an ac...The language of the theories contains a formula Acc(x) such that provability of ...
    +3 moreShow less
    The notion of 'correctness' in the boot-strapping procedure presupposes an antec...Translating predicativism into autonomous progressions up to Γ₀ imposes a set-th...Weyl's predicativism was grounded in an informal conceptual commitment to defina...

    Similar

    Kreisel identified a specific autonomous progression of theories for f...86%All mathematics used in physical theory can be recaptured in a predica...78%Autonomous progressions of theories properly internalize the general c...77%Computing theory is finitistic and constructivist in nature75%

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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