|Abstract||Using an axiomatization of second-order arithmetic (essentially second-order Peano Arithmetic without the Successor Axiom), arithmetic's basic operations are defined and its fundamental laws, up to unique prime factorization, are proven. Two manners of expressing a system's consistency are presented - the "Godel" consistency, where a wff is represented by a natural number, and the "real" consistency, where a wff is represented as a second-order sequence, which is a stronger notion. It is shown that the system can prove at least its Godel consistency and that closely allied systems can prove their real consistency|
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Dan E. Willard (2006). On the Available Partial Respects in Which an Axiomatization for Real Valued Arithmetic Can Recognize Its Consistency. Journal of Symbolic Logic 71 (4):1189 - 1199.
Leszek Aleksander Kołodziejczyk (2006). On the Herbrand Notion of Consistency for Finitely Axiomatizable Fragments of Bounded Arithmetic Theories. Journal of Symbolic Logic 71 (2):624 - 638.
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