4 let s be the set of all natural numbers that are

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4. Let S be the set of all natural numbers that are describable in English words using no more than 50 characters (so, 240 is in S since we can describe it as “two hundred forty”, which requires fewer than 50 characters). Assuming that we are allowed to use only the 27 standard characters (the 26 letters of the alphabet and the space character), show that there are only finitely many numbers contained in S . (In fact, perhaps you can show that there can be no more than 27 50 elements in S .) Now, let the set T be all those natural numbers not in S . Show that there are infinitely many elements in T . Next, since T is a collection of natural numbers, show that it must contain a smallest number. Finally, consider the smallest number contained in T . Prove that this number must simultaneously be an element of S and not an element of S – a paradox! (B+S 3.4.20) Solution. There are only 27 50 strings consisting of fifty characters or less; furthermore most of these aren’t actually legitimate strings of English, and sometimes two describe the same number (for example, “eight” and “two cubed”). Thus there are only finitely many numbers contained in S . So there are infinitely many elements in T , since T consists of all the natural numbers except for a finite set, and removing a finite set from a countable set gives a countable set. A collection of natural numbers contains a smallest number, as we can see by checking if it contains 1, then 2, then 3, and so on. The smallest number contained in T is not contained in S , since it is contained in T . But it can be described as “the smallest number contained in T ”, which is a phrase containing less than fifty characters, so it is contained in S . If you object that “the smallest number contained in T ” isn’t really a description of 2
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that number in less than fifty characters, because we’d have to define T , then we could put a definition of T in the string. Unfortunately it’s hard to describe T in less than fifty characters! So we can describe T
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