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Archive for July 2016

What is Wrong with Cantor’s Diagonal Argument?

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Cantor gave two proofs that the cardinality of the set of real numbers is greater than that of the set of natural numbers. According to a popular reconstruction of the more widely known of these proofs, his diagonal argument, Cantor randomly tabulates the real numbers in the interval [0, 1) in an array. I will use a binary version of the Cantorian table for ease of exposition and give Table 1 as an example:

1    0.10111011000 …
2    0.11010111000 …
3    0.11010001110 …
4    0.01101110001 …
5    0.10000100011 …
6    0.00111111111 …
7    0.00101110001 …
8    0.11000110100 …
9    0.11101100101 …
10  0.01111011100 …
11  0.01010101010
                 .
                 .
                 .

Table 1

Cantor asks us to assume for reductio purposes that this table is “complete” in the sense that it lists every real number in the interval [0, 1). The digits on the diagonal of the list (boldfaced) give us the infinitely long number d:

d = 0.11000110100 …

He then constructs a new number i from the diagonal of the table by substituting every 0 after the decimal point in d by 1, and every 1 in d by 0:

i = 0.00111001011 …

He asserts that this constructed number i is a real number in [0, 1) which cannot be found in the putatively complete list, because i is guaranteed to differ from each number in the list as the number’s digit that falls on the diagonal is changed in i. Since no list of real numbers in [0, 1) to be set up in the same fashion will manage to include all of the real numbers in [0, 1), it follows, according to Cantor, that no such table will have the items in it being enumerable by natural numbers. In other words, there can be no bijection between the set of natural numbers and the set of real numbers in [0, 1). His conclusion is that the size, or cardinality, of the set of reals even only in the interval [0, 1) is greater than that of the whole set of naturals.

Let me point out an immediate puzzle with Cantor’s argument. I will call two binary numbers in [0, 1) “inverses” of one another if one number has 1 in a certain location after the decimal point, the other number has 0 in the corresponding location; and if it has 0 in that location, the other number has 1. Thus the numbers on lines 3 and 7, for example, are inverses of each other. So are the numbers on lines 5 and 10. Crucially, d and i are also inverses of each other. If we are to assume this list is complete, every number in the list must have its inverse also included in the list. Cantor says i is not included in the list. But how come? The number ’s inverse, namely d, is in the list—it is on line 8. Since its inverse is in there, i must also be in there, like every other pair of inverses. But if i cannot be in the list, for the reasons Cantor points out, then there must be something wrong with the starting assumption (for reductio purposes) that  all the reals in [0, 1) are tabulated.

I see two flaws in Cantor’s diagonalization procedure. These flaws make it infeasible to construct the number i and thereby invalidate his proof.  Let me illustrate the first flaw on Table 2. It is obvious that, since d is a real number in the interval [0, 1), d must be included in the list—if we are to assume the list is complete. Suppose it is on line 8.

1    0.10111011000 …
2    0.11010111000 …
3    0.11010001110 …
4    0.01101110001 …
5    0.10000100011 …
6    0.00111111111 …
7    0.00101110001 …
8    0.1100011?100 …
9    0.11101100101 …
10  0.01111011100 …
11  0.01010101010
                  .
                  .
                  .
d = 0.1100011?100 …
i = 0.0011100?011 …

            Table 2

The n-th digit of d after the decimal point comes from the n-th digit, after the decimal point, of line n in the table. Thus the first digit of d, namely 1, comes from the first digit of line 1. The fourth digit of d, namely 0, comes from the fourth digit of line 4, and so on. It will be noticed that the 8th digit of d (marked with ‘?‘ on line 8) is indeterminate: unlike the other digits of line 8, the 8th digit of line 8 does not come from any other line than line 8 itself. But then there is nothing to determine whether the 8th digit of line 8 must be 0 or 1. [1]

Thus, if d is included in the supposedly complete table, d will fail to satisfy a necessary condition for being a real number in [0, 1), viz. having all determinate digits. This of course entails that the value of the 8th digit of ’s inverse i is also indeterminate, and hence i too fails to be a real number in [0, 1). The upshot is that Cantor’s supposition that he can unproblematically take the inverse of the diagonal of the table is false.

Let us forget about the first flaw for now and assume, contrary to fact, that there is a determinate diagonal of the Cantorian table. (Thus I assigned 0 to the 8th digit of line 8 in Table 3 below, for illustration’s sake.) The second flaw arises from the following phenomenon: the diagonal and its inverse clash at some digit. Suppose the inverse i of the diagonal were on line 10 in Table 3:

1    0.10111011000 …
2    0.11010111000 …
3    0.11010001110 …
4    0.01101110001 …
5    0.10000100011 …
6    0.10111111111 …
7    0.00101110001 …
8    0.11000110100 …
9    0.01110011100 …
10  0.001110011 !1 …
11  0.0101000101
                  .
                  .
                  .
d = 0.110001101!0 …
i = 0.001110010!1 …

            Table 3

If the diagonal’s 10th digit (marked with ‘!‘ on line 10) is 1, ’s 10th digit has to be 0, and if the diagonal’s 10th digit is 0, ’s 10th digit has to be 1. In other words, since the diagonal’s and ’s 10th digits coincide, we get the result that their shared 10th digit can be neither 0 nor 1. Hence inclusion of i in Cantor’s table leads to paradox! But as we have argued earlier, i must be included (assuming d could) in a table that claims to be complete—complete even if only for reductio purposes.

Actually not only the diagonal, but some other lines in a Cantorian table also give rise to problems. In the example in Table 4 one of the lines parallel to the diagonal, call it a “quasi-diagonal line,” is indicated with boldface. Let the number d* consist of the first two digits of the first line appended to the digits of the quasi-diagonal line. If the table is to be assumed complete, the number d* will have to be a member of the table. Suppose it is on line 6. For the reasons explained above, the 8th digit of d* after the decimal point is indeterminate, and similarly for the 8th digit of i*, which we assumed to be on line 9. Moreover, the digit of i* marked with ‘!‘ is a seat of contradiction—if it is 1 then it must be 0, and if it is 0 then it must be 1—and similarly for the corresponding digit of d*.

1    0.101110110001 …
2    0.110101110000 …
3    0.110100011101 …
4    0.011011100010 …
5    0.100001000111 …
6    0.1011010?00 !0 …    = d*
7    0.001011100010 …
8    0.110001101000 …
9    0.0100101?11 !1 …    = i*
10  0.01111011100
11  0.010101010100 
                  .
                  .
                  .

            Table 4

So, Cantor’s table involves an indeterminacy not only on the path of its diagonal. Since there are infinitely many quasi-diagonal lines in the table (only one of which is illustrated in Table 4), in actual fact the Cantorian table is replete with indeterminacies and contradictions. In other words, there are infinitely many loci of the first and the second flaws in Cantor’s table.

These two flaws arise from the fact that Cantor lists real numbers in the form of a table. Such a table appears to be an innocuous way of displaying the totality of real numbers, but this appearance is deceptive. For, in a putatively exhaustive table of reals, there would have to be numbers—the diagonal number d and numbers like d*—that must cross each of the other numbers, including themselves and their inverses, at some digit, and this gives rise to the flaws we have pointed out. [2]

——————————————–

[1] I concealed the first flaw when setting up Table 1 earlier, for convenience of exposition.

[2] My much more detailed criticisms of Cantor’s arguments and my way of showing that reals cannot have a higher cardinality than naturals can be found at [inaccessible temporarily–sorry]: https://www.academia.edu/26887641/CONTRA_CANTOR_HOW_TO_COUNT_THE_UNCOUNTABLY_INFINITE_

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Written by Erdinç Sayan

July 31, 2016 at 1:55 pm