The Art of Proof pp 13-23 | Cite as

# Natural Numbers and Induction

## Abstract

**Before You Get Started**. From previous mathematics, you are accustomed to the symbol <. If we write 7 < 9 you read it as “7 is less than 9,” and if we write m < n you read it as “m is less than n.” But what should, for example, “n greater than 0” mean? If you look back over what we have done so far, you will notice that we have not ordered the integers: even the statement 0 < 1 does not appear. Here we impose another axiom on **Z** to handle these questions. This axiom will specify which integers are to be considered positive. What should it mean for an integer to be positive? Try to come up with an axiomatic way to describe positive integers. And then think about how we could use positive integers to define the symbol <. Another question: Is **Z** an infinite set? And what does this mean? Without Axiom 2.1 below we have no answer. We deal with this in Chapter 13.

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