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Good Math by Mark C. Chu-Carroll

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Building Natural Numbers

All the way back in 1, Natural Numbers, we defined the natural numbers axiomatically. In most of math, we define objects and their behaviors using axiomatic definitions like the rules of Peano arithmetic. One of the more subtle problems in math is that if you want to define a new mathematic construct, it’s not enough to just define a collection of axioms. A collection of axioms is a logical definition of a kind of object and the way that that kind of object works, but the axioms don’t actually show that an object that fits the definition exists or that it even makes sense to create an object that fits the definition. To make the axiomatic definition work, you need to show that it’s possible for the objects described ...

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