Quantum Mathematics

Abstract
This paper explores the development of mathematics on a quantum logical base when mathematical postulates are taken as necessary truths. First it is shown that first-order Peano arithmetic formulated with quantum logic has the same theorems as classical first-order Peano arithmetic. Distribution for first-order arithmetical formulas is a theorem not of quantum logic but rather of arithmetic. Second, it is shown that distribution fails for second-order Peano arithmetic without extensionality. Third, it is shown that distribution holds for second-order Peano arithmetic (second-order quantum logic) with extensionality. Some remarks about extensions to quantum set theory are made.
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J. L. Bell (1986). A New Approach to Quantum Logic. British Journal for the Philosophy of Science 37 (1):83-99.
J. F. Pabion (1982). Saturated Models of Peano Arithmetic. Journal of Symbolic Logic 47 (3):625-637.
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