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Award ID contains: 2054379

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  1. ; (, Canadian Mathematical Bulletin)
    Abstract For which choices of$$X,Y,Z\in \{\Sigma ^1_1,\Pi ^1_1\}$$does no sufficiently strongX-sound andY-definable extension theory prove its ownZ-soundness? We give a complete answer, thereby delimiting the generalizations of Gödel’s second incompleteness theorem that hold within second-order arithmetic. 
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    Free, publicly-accessible full text available July 24, 2026
  2. ; (, 14th Panhellenic Logic Symposium)
    For n ∈ N and ε > 0, given a sufficiently long sequence of events in a probability space all of measure at least ε, some n of them will have a common intersection. A more subtle pattern: for any 0 < p < q < 1, we cannot find events Ai and Bi so that μ (Ai ∩ Bj ) ≤ p and μ (Aj ∩ Bi ) ≥ q for all 1 < i < j < n, assuming n is sufficiently large. This is closely connected to model-theoretic stability of probability algebras. We survey some results from our recent work in [7] on more complicated patterns that arise when our events are indexed by multiple indices. In particular, how such results are connected to higher arity generalizations of de Finetti’s theorem in probability, structural Ramsey theory, hypergraph regularity in combinatorics, and model theory. 
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    Accepted Manuscript Full Text Available
  3. ; ; ; (, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences)
    We show that the Priess-Crampe & Ribenboim fixed point theorem is provable in R C A 0 . Furthermore, we show that Caristi’s fixed point theorem for both Baire and Borel functions is equivalent to the transfinite leftmost path principle, which falls strictly between A T R 0 and Π 1 1 - C A 0 . We also exhibit several weakenings of Caristi’s theorem that are equivalent to W K L 0 and to A C A 0 . This article is part of the theme issue ‘Modern perspectives in Proof Theory’. 
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