Outer Limits of Reason by Noson S. Yanofsky Read Free Book Online

such a subjective property.
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    A more serious and related paradox is called the Berry paradox . The key to understanding this paradox is that in general the more words one uses in a phrase, the larger the number one can describe. The largest number that can be described with one word is 90. 91 would demand more than one word. Two words can describe ninety trillion. Ninety trillion + 1 is the first number that demands more than two words. Three words can describe ninety trillion trillion. The next number (ninety trillion trillion + 1) would demand more than three words. Similarly, the more letters in a word, the larger the number you can describe. With three letters, you can describe the number 10 but not 11.
    Let us stick to number of words. Call a phrase that describes numbers and has fewer than eleven words a Berry phrase . Now consider the following phrase:
    the least number not expressible in fewer than eleven words.
    This phrase has ten words and expresses a number, so it should be a Berry phrase. However, look at the number it purports to describe. The number is not supposed to be expressible in fewer than eleven words. Is this number expressible in eleven words or less? This is a real contradiction.
    We may also talk about other measures of how complicated an expression is. Consider
    the least number not expressible in fewer than fifty syllables.
    This phrase has fewer than fifty syllables. Another phrase,
    the least number not expressible in fewer than sixty letters,
    has fifty-nine letters. Do these descriptions describe numbers or not? And if they do describe numbers, which ones? They describe a certain number if and only if they do not describe that number. But why not? Each certainly seems like a nice descriptive phrase.
    Yet another interesting paradox about describing numbers is Richard’s paradox . Certain English phrases describe real numbers between 0 and 1. For example,
    â€¢ “pi minus 3” = 0.14159
    â€¢ “the chance of getting a 3 when a die is thrown” = 1/6
    â€¢ “pi divided by 4” = 0.785
    â€¢ “the real number between 0 and 1 whose decimal expansion is 0.55555” = 0.55555
    Call all such phrases Richard phrases . We are going to describe a paradoxical sentence. Rather than just stating the long sentence, let us work our way toward it. Consider the phrase
    the real number between 0 and 1 that is different from any Richard phrase.
    If this described a number, it would be paradoxical since the phrase would describe a number and yet it would not be a Richard phrase. However, there are many real numbers that are different from all Richard phrases. Which one is it? The problem is that this phrase does not really describe an exact number. Let us try to be more exact. The set of Richard phrases are a subset of all English phrases, and as such, they can be ordered like names in a telephone book. We can first order all Richard phrases of one word, then the phrases of two words, and so on. With such an ordered list we can talk about the n th Richard sentence. Now consider
    the real number between 0 and 1 whose n th digit is different from the n th digit of the n th Richard phrase.
    This is just showing how the number described is different from all the Richard phrases, but it still does not describe an exact number. The number described by the forty-second Richard number might have an 8 as the forty-second digit. From this, we know that our phrase cannot have an 8 in the forty-second position. But should our number have a 9 or 6 in that position? Let us be exact:
    the real number between 0 and 1 defined by its n th digit being 9 minus the n th digit of the n th Richard phrase.
    That is, if the digit is a 5, this phrase will describe a 4. If the digit is an 8, this phrase will describe a 1. And if the digit is a 9, this phrase will describe a 0. This phrase is a legitimate English phrase that precisely describes a number between 0 and 1, yet it is different from