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Array theory combines APL with set theory, transfinite arithmetic, and operationally transformed functions to produce an axiomatic theory in which the theorems hold for all arrays having any finite number of axes of arbitrary countable ordinal lengths. The items of an array are again arrays. The treatment of ordinal numbers and letters is similar to Quine's treatment of individuals in set theory. The theory is developed first as a theory of lists. This paper relates the theory to the eight axioms of Zermelo–Fraenkel set theory, describes the structure of arrays, interprets empty arrays in terms of vector spaces, presents a system of axioms for certain properties of operations related to the APL function of reshaping, deduces a few hundred theorems and corollaries, develops an algebra for determining the behavior of operations applied to empty arrays, begins the axiomatic development of a replacement operator, and provides an informal account of unions. Cartesian products, Cartesian arrays, and outer, positional, separation, and reduction transforms.
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