Outer Limits of Reason (6 page)

K
_m
K
_i
is false for all
i
>
m

K
_m
+
1
K
_i
is false for all
i
>
m
+
1

 

K
_n
K
_i
is false for all
I >
n

 

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
.

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
.

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 such a subjective property.

 

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,

• “pi minus 3” = 0.14159

• “the chance of getting a 3 when a die is thrown” = 1/6

• “pi divided by 4” = 0.785

• “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: