Sets of Real Numbers
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In Chapter 3 we defined natural numbers and their ordering and indicated how arithmetic operations on natural numbers can be defined. The next logical step in the development of foundations for mathematics is to define integers and rational numbers. The guiding idea in both cases is to make an arithmetic operation that is only partially defined on natural numbers (subtraction in the case of integers, division in the case of rationals) into a total operation, and belongs more properly in the realm of algebra than set theory. We thus limit ourselves to outlining the main ideas, and leave out almost all proofs. Those can be found in most textbooks on abstract algebra. Better still, the reader may work out some or all of them as exercises.