Constructing the Integers—Naturally

We can create mathematical constructions to represent the integers in terms of the naturals. These constructions are called models of the integers. But why would we want to do that? And what is a model, precisely?

In a model of something new, like our integers, we’re trying to show that there’s a way of making objects that will obey the axioms that we’ve defined. To do that, you take the things that you already know exist and use them as building blocks. With those building blocks, you build objects that will follow the axioms of the new system. For example, to construct the integers, we’re going to start with the natural numbers, which are objects that we already know about and understand. Then we’re going ...

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