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  1. Fraïssé’s Conjecture in [Math]-Comprehension.Antonio Montalbán - 2017 - Journal of Mathematical Logic 17 (2):1750006.
    We prove Fraïssé’s conjecture within the system of Π11-comprehension. Furthermore, we prove that Fraïssé’s conjecture follows from the Δ20-bqo-ness of 3 over the system of Arithmetic Transfinite Recursion, and that the Δ20-bqo-ness of 3 is a Π21-statement strictly weaker than Π11-comprehension.
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  • Reverse Mathematics and Initial Intervals.Emanuele Frittaion & Alberto Marcone - 2014 - Annals of Pure and Applied Logic 165 (3):858-879.
    In this paper we study the reverse mathematics of two theorems by Bonnet about partial orders. These results concern the structure and cardinality of the collection of initial intervals. The first theorem states that a partial order has no infinite antichains if and only if its initial intervals are finite unions of ideals. The second one asserts that a countable partial order is scattered and does not contain infinite antichains if and only if it has countably many initial intervals. We (...)
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  • $${\Pi^1_2}$$ -Comprehension and the Property of Ramsey.Christoph Heinatsch - 2009 - Archive for Mathematical Logic 48 (3-4):323-386.
    We show that a theory of autonomous iterated Ramseyness based on second order arithmetic (SOA) is proof-theoretically equivalent to ${\Pi^1_2}$ -comprehension. The property of Ramsey is defined as follows. Let X be a set of real numbers, i.e. a set of infinite sets of natural numbers. We call a set H of natural numbers homogeneous for X if either all infinite subsets of H are in X or all infinite subsets of H are not in X. X has the property (...)
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  • A Survey of Determinacy of Infinite Games in Second Order Arithmetic.Keisuke Yoshii - 2017 - Annals of the Japan Association for Philosophy of Science 25:35-44.
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  • Weak Axioms of Determinacy and Subsystems of Analysis II.Kazuyuki Tanaka - 1991 - Annals of Pure and Applied Logic 52 (1-2):181-193.
    In [10], we have shown that the statement that all ∑ 1 1 partitions are Ramsey is deducible over ATR 0 from the axiom of ∑ 1 1 monotone inductive definition,but the reversal needs П 1 1 - CA 0 rather than ATR 0 . By contrast, we show in this paper that the statement that all ∑ 0 2 games are determinate is also deducible over ATR 0 from the axiom of ∑ 1 1 monotone inductive definition, but the (...)
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  • Borel Quasi-Orderings in Subsystems of Second-Order Arithmetic.Alberto Marcone - 1991 - Annals of Pure and Applied Logic 54 (3):265-291.
    We study the provability in subsystems of second-order arithmetic of two theorems of Harrington and Shelah [6] about Borel quasi-orderings of the reals. These theorems turn out to be provable in ATR0, thus giving further evidence to the observation that ATR0 is the minimal subsystem of second-order arithmetic in which significant portion of descriptive set theory can be developed. As in [6] considering the lightface versions of the theorems will be instrumental in their proof and the main techniques employed will (...)
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