 The natural numbers form the most elementary number system. (Some mathematicians count \(0\) as a natural number as well, but some others do not.) \(1\) is a natural number. For any natural number \(n\), \(n+1\)—the successor of \(n\)—is also a natural number. The natural numbers—\(1, 2, 3, \dots\)—are closed under addition: if \(n_1\) and \(n_2\) are natural numbers, then so is \(n_1+n_2\). And they are closed under multiplication: if \(n_1\) and \(n_2\) are natural numbers, then so is \(n_1 \cdot n_2\). We use the natural numbers for counting ‘how many’ of something there are, though they clearly fail when applied to infinite sets, for instance the set of squares of natural numbers or the set of natural numbers themselves. This is what ‘infinities of counting’ will extend, in section  3.3.