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    The least upper bound of the ordinal indices appearing in... — Carmelics
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    The least upper bound of the ordinal indices appearing in the finitist autonomous progression is the ordinal epsilon_0

    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.Kreisel identified a specific autonomous progression of theories for finitism
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    • 2.Kreisel determined the supremum of ordinal indices in this finitist hierarchy
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    • 3.Epsilon_0 is also the proof-theoretic ordinal of Peano Arithmetic
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
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    • 1.Tait's analysis of finitism identifies primitive recursive arithmetic (PRA) as the correct formalization, not Kreisel's autonomous progression.
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    • 2.PRA's proof-theoretic ordinal is omega^omega, which falls strictly below epsilon_0, undermining the claimed supremum.
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    Reason against 2 of 2
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    • 1.Feferman argued that Kreisel's autonomy condition is epistemically circular: recognizing a progression as finitistically legitimate already presupposes the ordinal comprehension it purports to generate.
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    • 2.If the autonomy condition is viciously circular, the progression cannot legitimately be said to converge on epsilon_0 as a finitistically meaningful bound.
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    Truth & KnowledgeProof of definition segments

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    Related

    Epsilon_0 is also the proof-theoretic ordinal of Peano ArithmeticFeferman argued that Kreisel's autonomy condition is epistemically circular: rec...If the autonomy condition is viciously circular, the progression cannot legitima...Kreisel determined the supremum of ordinal indices in this finitist hierarchy
    +3 moreShow less
    Kreisel identified a specific autonomous progression of theories for finitismPRA's proof-theoretic ordinal is omega^omega, which falls strictly below epsilon...Tait's analysis of finitism identifies primitive recursive arithmetic (PRA) as t...

    Similar

    The ordinals less than epsilon_0 are well-founded (there is no infinit...77%Kreisel determined the supremum of ordinal indices in this finitist hi...77%In autonomous progressions, ascent to a theory T_a is permitted only i...76%The definition of the least upper bound involves a quantifier ranging ...75%

    Source

    AI-extracted1/3 agreementValid
    SEP: proof-theory
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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
    Extraction notes

    Validity: Extracted via Max plan + API grounding/validity checks

    Details

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