12 found
Order:
  1.  8
    Well Ordering Principles and -Statements: A Pilot Study.Anton Freund - forthcoming - Journal of Symbolic Logic:1-35.
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  2.  1
    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   3 citations  
  3.  5
    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   1 citation  
  4.  7
    Proof Lengths for Instances of the Paris–Harrington Principle.Anton Freund - 2017 - Annals of Pure and Applied Logic 168 (7):1361-1382.
  5.  9
    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.  9
    Slow Reflection.Anton Freund - 2017 - Annals of Pure and Applied Logic 168 (12):2103-2128.
    Direct download (5 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  7.  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  
  8.  1
    A Note on Ordinal Exponentiation and Derivatives of Normal Functions.Anton Freund - 2020 - Mathematical Logic Quarterly 66 (3):326-335.
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  9.  1
    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  
  10.  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.
    No categories
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark  
  11.  4
    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.  2
    What is Effective Transfinite Recursion in Reverse Mathematics?Anton Freund - 2020 - Mathematical Logic Quarterly 66 (4):479-483.
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark