30 1. Logic and foundations Indeed, from the principle of mathematical induction, the hypotheses of Proposition 1.11.1 force A to be the entire set of natural numbers, and so Proposition 1.11.1 is logically equivalent to the assertion that the set of natural numbers is infinite. In the rigorous world of formal mathematics, Proposition 1.11.1 is of course uncontroversial, and is easily proven by a simple “no-self-defeating object” argument: Proof. Suppose for contradiction that A were finite. As A is non-empty (it contains 0), it must therefore have a largest element n (as can be seen by a routine induction on the cardinality of A). But then by hypothesis, n + 1 would also have to lie in A, and n would thus need to be at least as large as n + 1, a contradiction. But if one allows for vagueness in how one specifies the set A, then Proposition 1.11.1 can seem to be false, as observed by the ancient Greeks with the sorites paradox. To use their original example, consider the ques- tion of how many grains of sand are required to make a heap of sand. One or zero grains of sand are certainly not sufficient, but clearly if one places enough grains of sand together, one would make a heap. If one then “de- fines” A to be the set of all natural numbers n such that n grains of sand are not sufficient to make a heap, then it is intuitively plausible that A obeys both Hypothesis 1 and Hypothesis 2 of the above proposition, since it is intuitively clear that adding a single grain of sand to a non-heap cannot convert it to a heap. On the other hand, it is just as clear that the set A is finite, thus providing a counterexample to Proposition 1.11.1. The problem here is the vagueness inherent in the notion of a “heap”. Given a pile of sand P, the question “Is P a heap?” does not have an absolute truth value what may seem like a heap to one observer may not be so to another. Furthermore, what may seem like a heap to one observer when presented in one context, may not look like a heap to the same observer when presented in another context for instance, a large pile of sand that was slowly accumulated over time may not seem14 as large as a pile of sand that suddenly appeared in front of the observer, even if both piles of sand were of identical size in absolute terms. There are many modern variants of the sorites paradox that exploit vagueness of definition to obtain conclusions that apparently contradict rig- orous statements such as Proposition 1.11.1. One of them is the interesting 14The well-known (though technically inaccurate) boiling frog metaphor is a particularly graphic way of depicting this phenomenon, which ultimately arises from the fact that people usually do not judge quantities in absolute terms, but instead by using relative measurements that compare that quantity to nearby reference quantities.
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