Abstract
This paper shows the equivalence for provability between two infinitary systems with the ω-rule. One system is the positive one-sided fragment of Peano arithmetic without Exchange rules. The other system is two-sided Heyting Arithmetic plus the law of Excluded Middle for \(\Sigma^0_1\)-formulas, and it includes Exchange. Thus, the logic underlying positive Arithmetic without Exchange, a substructural logic, is shown to be a logic intermediate between Intuitionism and Classical Logic, hence a subclassical logic. As a corollary, the authors derive the equivalence for positive formulas among provability in those two systems and validity in two apparently unrelated semantics: Limit Computable Mathematics, and Game Semantics with 1-backtracking.
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Berardi, S., Tatsuta, M. (2007). Positive Arithmetic Without Exchange Is a Subclassical Logic. In: Shao, Z. (eds) Programming Languages and Systems. APLAS 2007. Lecture Notes in Computer Science, vol 4807. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-76637-7_18
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DOI: https://doi.org/10.1007/978-3-540-76637-7_18
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-76636-0
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