## Author Archive

## What is Wrong with Cantor’s Diagonal Argument?

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.**1**0111011000 …

2 0.1**1**010111000 …

3 0.11**0**10001110 …

4 0.011**0**1110001 …

5 0.1000**0**100011 …

6 0.00111**1**11111 …

7 0.001011**1**0001 …

8 0.1100011**0**100 …

9 0.11101100**1**01 …

10 0.011110111**0**0 …

11 0.0101010101**0** …

** .**

** .**

** .**

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 *i *’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.**1**0111011000 …

2 0.1**1**010111000 …

3 0.11**0**10001110 …

4 0.011**0**1110001 …

5 0.1000**0**100011 …

6 0.00111**1**11111 …

7 0.001011**1**0001 …

8 0.1100011**?**100 …

9 0.11101100**1**01 …

10 0.011110111**0**0 …

11 0.0101010101**0** …

** .**

** .**

** .**

*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 *d *’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.**1**0111011000 …

2 0.1**1**010111000 …

3 0.11**0**10001110 …

4 0.011**0**1110001 …

5 0.1000**0**100011 …

6 0.10111**1**11111 …

7 0.001011**1**0001 …

8 0.1100011**0**100 …

9 0.01110011**1**00 …

10 0.001110011 **!**1 …

11 0.0101000101**0 **…

** .**

** .**

** .**

*d* = 0.110001101**!**0 …

*i* = 0.001110010**!**1 …

Table 3

If the diagonal’s 10th digit (marked with ‘**!**‘ on line 10) is 1, *i *’s 10th digit has to be 0, and if the diagonal’s 10th digit is 0, *i *’s 10th digit has to be 1. In other words, since the diagonal’s and *i *’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 0.10**1**110110001 …

2 0.110**1**01110000 …

3 0.1101**0**0011101 …

4 0.01101**1**100010 …

5 0.100001**0**00111 …

6 0.1011010**?**00 !0 … = *d******

7 0.00101110**0**010 …

8 0.110001101**0**00 …

9 0.0100101?11 **!**1 … = *i******

10 0.01111011100**0 **…

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_

## I have a dream! But I can’t remember it…

In my previous post, “Is Truth Beneficial and/or Socially Constructed?,” I mentioned as a counterexample to the pragmatist theory of truth a nightmare a person had which she did not tell anyone about and kept as a secret for the rest of her life. The nightmare was so horrible and embarrassing that every time she remembered her nightmare, she was disturbed. Her life became a nightmare of sorts because of that nightmare.

Actually this kind of scenario is very rare in real life. The fact is that we tend to forget our dreams and nightmares soon after waking up. Even before we get up from bed, most of the content of our dream has already evaporated from our memory. We remember only very few, if any, of our dreams and nightmares in the rest of our lives. The ones we remember for a while are the ones which were extremely interesting or shocking for us, or those we had the chance to tell other people about on many occasions, which kept our memory of them alive. Ask yourself how many of your dreams and nightmares you still remember. I bet very few, if any.

The interesting thing is that we forget even the most vivid of our dreams and most frightful of our nightmares in the twinkling of an eye (unless our memory of them is reinforced by telling other people about them or by intentional recalling, for example). We forget our dreams even though some of them are *more vibrant* than certain waking experiences which we remember for much longer time.

Psychologists and brain physiologists tell us that dreams serve a useful function for our brain. So we have to have them. But it seems we also have to forget them fast after having them. I think there is a simple evolutionary explanation of this phenomenon. If we were to remember our dreams long after we woke up, we would be disposed to confuse the memories of our dreams with the memories of our waking experiences. Suppose I have a dream in which a friend of mine does something evil to me or an enemy of mine does a big favor for me. If my brain were to retain as lively a memory of that dream as the memories of my real life experiences, I might mistakenly think the contents of my dream correspond to some real experiences of mine that occurred in the past, and my attitude towards my friend or towards my enemy would unnecessarily be affected by that mistake. Such disorientations clearly would have negative survival value and therefore would be blocked by the mechanisms of human evolution. Hence the elusiveness of our dream contents.*

## Is truth beneficial and/or socially constructed?

There are several varieties of the pragmatist theory of truth. Since, according to the pragmatist theories, what we take to be truth is dependent on our pragmatic interests, rather than being “representations of reality,” the pragmatist theories can be regarded as anti-realist theories of truth. According to C.S. Peirce, who is one of the important figures in the pragmatic tradition, truth is, briefly, beliefs socially agreed upon in the long run. Hence “reality” is something socially constructed and is based on consensus. Another important figure, William James, thought that truth is something that has instrumental value. For James “facts” are our mental constructs which prove beneficial in the long run. The popular versions of especially the instrumentalist variety of pragmatism can be found in slogans like, “The truth is what works,” “Truth is what is convenient to believe,” “A proposition is true if believing it has advantageous results.” For the purposes of what follows, I will take the pragmatist theory to be claiming the following:

(PT) Proposition S is true IFF believing that S yields beneficial results in the long run.

I assume that (PT) is shared fully or partly by all pragmatist theories. (One could substitute “has pragmatic value” for “yields beneficial results” in (PT) to stay closer to the letter of the title “pragmatic theory of truth.”)

First off, a counterexample. Suppose I had a terrible nightmare. Every time I remember it, I get the creeps.* I don’t tell anyone about it, because my nightmare is also kind of embarrassing and I am afraid people will make fun of me or will insist that I go see a shrink to get it analyzed—which I’d hate to do. So I keep silent about it for the rest of my life. Thus my belief that I had that nightmare produces no ostensible benefits whatsoever in my life—if anything, every time my belief is enlivened by my recollection of the nightmare, this does nothing but disturb me. As I tell no one about it, my belief yields no useful results for anyone else either. It might even make me edgy in my dealings with some other people at least for a while, and this is not going to be beneficial for any of the parties. So, no useful outcome ensues from my belief either for myself or for any portion of humanity. I have no idea what caused my nightmare, so I don’t have a clue how I can prevent my or someone else’s having a similar nightmare in the future. And since I refuse to consult a shrink about it, she is not getting any monetary or academic benefits out of it either.

Yet, it is *true* that I had the nightmare, even though no one reaps any benefits out of my belief that it happened.

## A more devastating version of the Raven Paradox

C.G. Hempel’s “Raven Paradox” involves derivation of the intuitively unpalatable conclusion that observation of things like a white shoe or a rainbow confirms the raven hypothesis, “All ravens are black.” Here’s how it goes. An earlier author Jean Nicod had put forward the following criteria for confirmation of hypotheses of the form “All A’s are B’s”:

Observation of an object which has the property of being an A and also the property of being a B *confirms* “All A’s are B’s.”

Observation of an object which has the property of being an A but not the property of being a B *disconfirms* “All A’s are B’s.”

Observation of an object which does not have the property of being an A neither *confirms nor disconfirms* “All A’s are B’s.”

Add to these criteria the following highly plausible claim, which Hempel called “the equivalence condition”:

If an hypothesis H1 is logically equivalent to another hypothesis H2, then, if an observation O confirms H1, then O also confirms H2.

The equivalence condition sounds perfectly true, because to say that H1 and H2 are logically equivalent is to say that H1 and H2 make exactly the same claims about the world. Thus if a piece of evidence confirms one of the hypotheses, it must equally confirm the other one.