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form of self-reference. That is, there must be some chain of reasoning that is circular and returns to where it started. This was the common belief until Stephen Yablo came up with a clever paradox called Yablo’s paradox . Consider the following infinite sequence of sentences:
    K 1 K i is false for all i > 1
    K 2 K i is false for all i > 2
    K 3 K i is false for all i > 3
    Â 
    K m K i is false for all i > m
    K m + 1 K i is false for all i > m + 1
    Â 
    K n K i is false for all I > n
    Â 
    Every statement declares all the further statements to be false. Notice that no sentence ever references itself, nor is there any long chain that has some sentence referring back to itself. Nevertheless, this is a paradox in the sense that one cannot say that any sentence is either true or false. Imagine that for some m , we have that K m is true. K m says that all of K m + 1 , K m + 2 , K m + 3 , . . . are false. Splitting this up, we have that K m +1 is false and all of K m +2 , K m +3 , . . . are false. However, K m +1 says that all of K m +2 , K m +3 , . . . are false, which makes K m +1 true. Hence, by assuming that K m is true, we get a contradiction about the status of K m +1 . This can be viewed in figure 2.4 .
    Figure 2.4
    Yablo’s paradox—assuming true
    In contrast, imagine that for any m , we assume K m is false. That means that not all K n for n > m are false and there is at least one n > m with K n is true. But we saw that if any K n is true, we get a contradiction as in figure 2.5 .
    Figure 2.5
    Yablo’s paradox—assuming false
    When we assume that any K m is either true or false, we arrive at a contradiction. This is a contradiction without any self-reference.
    2.3  Naming Numbers
    Numbers are the most exact concepts we have. There is no haziness with the idea of 42. It is not a subjective idea where every person has their own concept of what 42 really is. And yet we will see that there are even problems with the description of numerical concepts. First a short story. In the early twentieth century, the mathematician G. H. Hardy (1877–1947) went to visit his friend and collaborator, the genius Srinivasa Ramanujan (1887–1920). Hardy writes: “I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. ‘No,’ he replied, ‘it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.’” 6 In detail, 1729 is equal to 1 3 + 12 3 but it is also equal to 9 3 + 10 3 . Since 1729 is the smallest number for which this can be done, 1729 is an “interesting” number. 7
    This tale brings to light the interesting-number paradox . Let’s take a tour through some small whole numbers. 1 is interesting because it is the first number. 2 is the first prime number. 3 is the first odd prime. 4 is a number with the interesting property that 2 × 2 = 4 = 2 + 2. 5 is a prime number. 6 is a perfect number—that is, a number whose sum of its factors is equal to itself (i.e., 6 = 1 × 2 × 3 = 1 + 2 + 3, etc.). The first few numbers have interesting properties. Any number that does not have an interesting property should be called an “uninteresting number.” What is the smallest uninteresting number? The smallest uninteresting number is an interesting number. We are in a quandary.
    What went wrong here? The contradiction came about because we thought we could split all numbers into two groups: interesting numbers and uninteresting numbers. This is false. There is no way to define what an interesting number is. It is a vague term and we cannot say when a number is interesting and when it is uninteresting. 8 “Interesting” is a feeling that a person gets sometimes and hence is a subjective property. We cannot make a paradox out of