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  1.  6
    Short Proofs for Slow Consistency.Anton Freund & Fedor Pakhomov - 2020 - Notre Dame Journal of Formal Logic 61 (1):31-49.
    Let Con↾x denote the finite consistency statement “there are no proofs of contradiction in T with ≤x symbols.” For a large class of natural theories T, Pudlák has shown that the lengths of the shortest proofs of Con↾n in the theory T itself are bounded by a polynomial in n. At the same time he conjectures that T does not have polynomial proofs of the finite consistency statements Con)↾n. In contrast, we show that Peano arithmetic has polynomial proofs of Con)↾n, (...)
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  2.  5
    Proof Lengths for Instances of the Paris–Harrington Principle.Anton Freund - 2017 - Annals of Pure and Applied Logic 168 (7):1361-1382.
  3.  2
    Predicative Collapsing Principles.Anton Freund - forthcoming - Journal of Symbolic Logic:1-20.
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  4.  11
    Type-Two Well-Ordering Principles, Admissible Sets, and -Comprehension.Anton Freund - 2018 - Bulletin of Symbolic Logic 24 (4):460-461.
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  5.  6
    A Note on Iterated Consistency and Infinite Proofs.Anton Freund - 2019 - Archive for Mathematical Logic 58 (3-4):339-346.
    Schmerl and Beklemishev’s work on iterated reflection achieves two aims: it introduces the important notion of \-ordinal, characterizing the \-theorems of a theory in terms of transfinite iterations of consistency; and it provides an innovative calculus to compute the \-ordinals for a range of theories. The present note demonstrates that these achievements are independent: we read off \-ordinals from a Schütte-style ordinal analysis via infinite proofs, in a direct and transparent way.
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  6.  9
    Slow Reflection.Anton Freund - 2017 - Annals of Pure and Applied Logic 168 (12):2103-2128.
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  7. Computable Aspects of the Bachmann–Howard Principle.Anton Freund - forthcoming - Journal of Mathematical Logic:2050006.
    We have previously established that [Formula: see text]-comprehension is equivalent to the statement that every dilator has a well-founded Bachmann–Howard fixed point, over [Formula: see text]. In this paper, we show that the base theory can be lowered to [Formula: see text]. We also show that the minimal Bachmann–Howard fixed point of a dilator [Formula: see text] can be represented by a notation system [Formula: see text], which is computable relative to [Formula: see text]. The statement that [Formula: see text] (...)
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