Rethinking Orders of Infinity with Nonstandard Analysis: An Algebraic Approach

This paper explores a novel approach to studying asymptotic notation and orders of infinity using nonstandard analysis. Traditional analysis relies on complex epsilon-delta arguments to handle orders of infinity. However, nonstandard analysis cleverly hides many quantifiers through the introduction of ultrafilters, transforming the problem into one with a more algebraic nature. The paper demonstrates that within the nonstandard framework, orders of infinity form a totally ordered vector space and possess a completeness property reminiscent of the completeness of real numbers. This algebraic approach simplifies computations with asymptotic notation, especially in symbolic computation, but sacrifices the ability to extract explicit constants.