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Partly as a motivating framework, partly propaganda from a practicing logician, we will discuss the importance of axiomatic systems in the upper level courses and their particular relevance to students in secondary education. On a simple level, the axiomatic definitions of algebra give a framework to compare and contrast differences between objects like the integers and rationals. But they also explain some often confounding questions, like why can't we divide by zero? A nuanced understanding of number systems can only come from considering their salient features.