Abstract

Leibniz and Newton, both independently credited as inventors of calculus, relied on the concept of an infinitesimal (nonzero “numbers ” that were “infinitely small”) in their development. Our standard rigorous treatment of calculus involves an “arbitrary epsilon ” limit definiton. There’s an alternative rigorous study of calculus beyond the limits of real analysis. In 1961, Robinson constructed the “hyperreal line ” as a direct consequence of the compactness theorem of first order logic. We will examine some typical proofs of known statements in advanced calculus and extend the nonstandard framework to other mathematical fields. 1 Introduction to First-Order Logic 1.1 First-Order Languages Define a first-order language to be a set of symbols, as a base containing a symbol for the logical NAND, quantifiers ∃ and ∀, equality (=), grouping parenthesies/brackets, and variables (as many as are needed). Though NAND is all

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