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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 Cantor’s 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
.
.
.
d = 0.11000110100 …
i = 0.00111001011 …

           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. 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. He asserts that this constructed number 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 this example the reals in [0, 1)) to be constructed 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. His conclusion is that the size, or cardinality, of the set of reals is greater than that of the 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.[1]

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 one on Table 2.[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.

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 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  0.01010001010
.
.
.
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 cannot consistently be either 0 or 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.

It is obvious that these two flaws arise from the fact that Cantor lists real numbers in the form of a table.  Such a table turns out to be a deceptively erroneous way of displaying the totality of real numbers.[3]

——————————————–

[1] If i were not to be in the list although its inverse is, we could not assume the list was complete to begin with.

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

[3] My much more detailed criticisms of Cantor’s arguments and my way of showing that reals cannot be more numerous than naturals can be found at: https://www.academia.edu/26887641/CONTRA_CANTOR_HOW_TO_COUNT_THE_UNCOUNTABLY_INFINITE_

Written by Erdinç Sayan

July 31, 2016 at 1:55 pm

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

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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.*

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Written by Erdinç Sayan

April 1, 2012 at 9:52 pm

Is truth beneficial and/or socially constructed?

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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.

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Written by Erdinç Sayan

March 19, 2012 at 4:11 am

Posted in Epistemology

On causes of regularities

with 21 comments

According to a standard construal, Hume proposed the following analysis of causation:

Event C caused event E iff (1) C was temporally prior to E, (2) C and E were contiguous in space and in time, and (3) events of type C are always followed by events of type E.

This is the prototype of the regularity or constant-conjunction theories of causation. Causation is linked to regularity of occurrence of events similar to C with events similar to E.

In every corner of the universe scratched matches light (when there is presence of oxygen and absence of water sprinklers around, etc.). Let me now ask a childish question: Why is this uniformity? How come scratched matches behave the same way everywhere? Do matches have telepathic communication, saying to each other, “Let’s light whenever we are scratched”? What “coordinates” or “oversees” them, so that they can display similar or repeated patterns of behavior all over the universe?

This is the same question as the question of what ensures the sameness of a law of nature in the entire universe. If one wants to say that something’s being a “law of nature” just means that it applies uniformly all over the universe, OK, then I am asking, “What sustains those laws to be effective everywhere?”. Two electrons repel each other, and an electron and a positron attract each other everywhere in the universe (or so we believe). In virtue of what is the uniformity of the behavior of the electrons and positrons and other things guaranteed? In other words, what causes regularities to hold everywhere? What is the causal infrastructure underlying regularities in nature?

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Written by Erdinç Sayan

March 5, 2012 at 1:04 am

Posted in Metaphysics

A new twist on Zeno’s Arrow Paradox

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Zeno of Elea’s arrow paradox, like some of his other extant paradoxes, aims to show that our observations of change and becoming in the world are illusions. Our senses suggest that there are all kinds of change and motion of things around us, but our reason concludes otherwise. As good philosophers, we should listen to the voice of our reason, rather than the evidence of our senses, and reject the reality of motion and change in the world.

Here’s how the Arrow Paradox is supposed to help show the unreality of motion. (What follows is a common reconstruction of Zeno’s argument.) Consider an object like an arrow which our visual experience describes as moving in its trajectory in the air. Zeno claims that at every instant of its supposed flight, the arrow occupies a region of space exactly coinciding the size and shape of the arrow. But if an object occupies a region of space coinciding with the size and shape of the object, then the object must be at rest. The arrow at every instant during its supposed flight, therefore, is at rest; it is at no moment in that time interval in motion. So, contrary to the judgment of our senses, motion is impossible.

A popular solution to Zeno’s Arrow Paradox is Russell’s “at-at theory of motion.” According to Russell, an object cannot be in motion (nor can it be at rest) at an instant. To be in motion is to be at different locations at different times. (And to be at rest during an interval of time is to occupy the same location at every instant of that time interval.) Location of an object at a single instant does not tell us anything about its kinematic status.

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Written by Erdinç Sayan

February 25, 2012 at 12:36 am

Posted in Metaphysics

A more devastating version of the Raven Paradox

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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.

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Written by Erdinç Sayan

February 22, 2012 at 11:42 pm