12 found
Order:
  1.  2
    Computable Aspects of the Bachmann–Howard Principle.Anton Freund - 2019 - Journal of Mathematical Logic 20 (2):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] (...)
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark   4 citations  
  2.  3
    A Note on Ordinal Exponentiation and Derivatives of Normal Functions.Anton Freund - 2020 - Mathematical Logic Quarterly 66 (3):326-335.
    No categories
    Direct download  
     
    Export citation  
     
    Bookmark   2 citations  
  3.  8
    Derivatives of Normal Functions in Reverse Mathematics.Anton Freund & Michael Rathjen - 2021 - Annals of Pure and Applied Logic 172 (2):102890.
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  4.  9
    Well Ordering Principles and -Statements: A Pilot Study.Anton Freund - 2021 - Journal of Symbolic Logic 86 (2):709-745.
    In previous work, the author has shown that $\Pi ^1_1$ -induction along $\mathbb N$ is equivalent to a suitable formalization of the statement that every normal function on the ordinals has a fixed point. More precisely, this was proved for a representation of normal functions in terms of Girard’s dilators, which are particularly uniform transformations of well orders. The present paper works on the next type level and considers uniform transformations of dilators, which are called 2-ptykes. We show that $\Pi (...)
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  5.  11
    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, (...)
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  6.  7
    Proof Lengths for Instances of the Paris–Harrington Principle.Anton Freund - 2017 - Annals of Pure and Applied Logic 168 (7):1361-1382.
  7.  3
    What is Effective Transfinite Recursion in Reverse Mathematics?Anton Freund - 2020 - Mathematical Logic Quarterly 66 (4):479-483.
    No categories
    Direct download  
     
    Export citation  
     
    Bookmark   1 citation  
  8.  2
    How Strong Are Single Fixed Points of Normal Functions?Anton Freund - 2020 - Journal of Symbolic Logic 85 (2):709-732.
    In a recent paper by M. Rathjen and the present author it has been shown that the statement “every normal function has a derivative” is equivalent to $\Pi ^1_1$ -bar induction. The equivalence was proved over $\mathbf {ACA_0}$, for a suitable representation of normal functions in terms of dilators. In the present paper, we show that the statement “every normal function has at least one fixed point” is equivalent to $\Pi ^1_1$ -induction along the natural numbers.
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  9.  11
    Slow Reflection.Anton Freund - 2017 - Annals of Pure and Applied Logic 168 (12):2103-2128.
    Direct download (5 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  10.  16
    Type-Two Well-Ordering Principles, Admissible Sets, and -Comprehension.Anton Freund - 2018 - Bulletin of Symbolic Logic 24 (4):460-461.
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  11.  6
    Predicative Collapsing Principles.Anton Freund - 2020 - Journal of Symbolic Logic 85 (1):511-530.
    We show that arithmetical transfinite recursion is equivalent to a suitable formalization of the following: For every ordinal \alpha there exists an ordinal /beta such that 1 + \beta \cdot (\beta + \alpha) admits an almost order preserving collapse into \beta. Arithmetical comprehension is equivalent to a statement of the same form, with \beta \cdot \alpha at the place of \beta \cdot (\beta + \alpha). We will also characterise the principles that any set is contained in a countable coded ω-model (...)
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  12.  7
    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.
    No categories
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark