The Semantics of Infinity
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Sometimes, as I lie deep in Earth's shadow, when the ghosts of spooky actions at a distance emerge from the Šther to entangle photons in their quantum snares, an especially disturbing spectre haunts my dreams. It taunts me with a question expressed through a family of interrogatives: When you carry out an integration between infinite limits, which infinity do you use? Aleph-Null or Aleph-One? When you put a lazy eight on the elongated ess as a superscript and/or a subscript, does it represent the infinity of the natural numbers or the infinity of the real numbers? And does the choice make a difference in the value of the integral?
Transfinite arithmetic, which Georg Ferdinand Ludwig Philipp Cantor (1845 Mar 03 - 1918 Jan 06) devised in the 1870's and 1880's, puts us in a bit of a quandary as to how we must answer those questions. Using set theory, which he created, Cantor put some remarkable propositions to the proof and thereby verified the existence of a grand hierarchy of ever-larger infinities. He used the Hebrew letter aleph with an appropriate subscript to represent each infinity in that hierarchy; thus, Aleph-Null represents the infinity of the counting numbers (the positive integers or natural numbers) and Aleph-One represents the infinity of the real numbers. If we take a close look at what Cantor did, we may discern the difference between those two infinities in a way that lets us answer my spectre's questions about the limits of integration.
Everything that we need to know about the concept it denotes lurks within the name itself. Infinity (Latin for without limit) denotes the mathematical concept of endlessness. From that fact we can follow Cantor's lead and deduce the properties of infinity and of infinite sets, in particular their relations to the numbers.
Consider the set of the natural numbers, the set whose elements consist simply of certain names in an ordered sequence (one, two, three, four,...and so on; ein, zwei, drei, fier,...und so weiter; odin, dva, tri, chetireh,...i tak dalyeyeh; ehad, shnayim, shalosh, arbah,...v'od; et grandly cetera). Does that set have a last element; to say in other words, does it have a largest possible number?
No, it does not. We can prove and verify that statement easily. Suppose that someone shows you a number (either the name or the place-value drawing corresponding to that name) and tells you that no larger number can possibly exist. Recalling to mind your grammar-school arithmetic, you simply add another number to that alleged largest number and thereby create an even larger number. Because the natural numbers consist of nothing more than names in an ordered sequence, we know that they possess no magic powers that enable them to negate the rules of arithmetic at any point, so you know that you can keep adding numbers to any other number endlessly. That fact, combined with the definition of infinity, makes the natural numbers the elements of an infinite set. Cantor gave the infinity of that set the name Aleph-Null, the name of the smallest infinity in his hierarchy.
Next Cantor discovered that he could do some truly bizarre things with infinite sets. Imagine that a hyperactive demon has set out to draw the natural numbers, in their proper sequence, on a straight line extending endlessly to the right and imagine that a second hyperactive demon has set out to draw under each natural number on that line its double. Again drawing on your knowledge of simple arithmetic, you know that the first demon cannot draw a number that the second demon cannot double, so now you know, as Cantor claimed, that the demons have produced and continue to produce endlessly a unique, bijective (working both ways), one-to-one matching (ubijoto matching, for short) between the set of the natural numbers and the set of the even positive integers. But the even positive integers comprise a proper subset of the set of the natural numbers; we expect it to contain only half as many elements as does the full set of the natural numbers, but Cantor has proven and verified that it doesn't, that it contains the same quantity of elements.
Thus Cantor discovered another criterion by which we can discern infinite sets. If we have a set that violates Euclid's fifth common notion, if we can devise an ubijoto match between the set and at least one of its proper subsets, then we have an infinite set.
Cantor had to do something a little different when he considered the set of the decimal fractions, the fractions that fill the number line between zero and one, and he got an amazingly different result. Imagine that we have before us two columns. In the left column, starting at the top, a hyperactive demon draws all of the natural numbers in their proper sequence and in the right column a second hyperactive demon draws all of the decimal fractions in random order, matching each fraction with a natural number. Lest we come to believe that the demons have created an ubijoto match between their respective sets, Cantor showed us how to create a decimal fraction that the second demon hasn't drawn on its list. Take from the first fraction on the list the digit in the first place to the right of the decimal point, change it, and put it into the first place of the new fraction; take from the second fraction on the list the digit in its second place, change it, and put it into the second place of the new fraction; take the third digit from the third fraction on the list, change it, and put it into the third place of the new fraction; and so continue on. Because of the way in which we create the new fraction (and we can create a vast set of alternative versions), we know that it differs in at least one place from every fraction on the demon's list, so we know that we cannot match it with a natural number, since those already have matches. Thus the set of the decimal fractions must give us an infinity greater than the infinity of the natural numbers. We call that non-denumerable infinity Aleph-One.
See how smoothly and easily that works? With that simple diagonalization procedure Cantor proved and verified the proposition that the set consisting of all possible permutations of ten symbols drawn on a line of spaces extending endlessly to the right of the decimal point contains vastly more elements than does the set that consists of all possible permutations of ten symbols drawn on a line of spaces extending endlessly to the left of the decimal point.
Let's try that again, shall we? Forget Cantor's list: let's try something different. Take a decimal fraction and reflect it through the decimal point; more specifically, take from the fraction the first digit on the right of the decimal point and put it into the first place on the left of the decimal point, take the second digit on the right and put it into the second place on the left, and so on. If, for example, you chose 0.6589, then that reflection process gave you 9856. That reflection process uniquely associates any decimal fraction with a natural number, because the reflection yields one and only one natural number and one and only one decimal fraction can yield that particular natural number. Likewise, we can reflect any natural number through the decimal point to obtain a decimal fraction. For example, reflecting 4825 gives us 0.5284000.... Again that process creates a perfectly unique association between the two numbers. Further, there exists no decimal fraction and no natural number to which we cannot apply the reflection process to yield an appropriate association. Those facts necessitate the existence, with respect to reflection through the decimal point, of an inherent ubijoto match between the set of the natural numbers and the set of the decimal fractions. That fact, in turn, necessitates that the set of the decimal fractions have a cardinality of Aleph-Null, not Aleph-One.
So how did Cantor go wrong? To find out, look again at his diagonalization proof, but do it with finite decimals. For example, if you list all of the three-digit fractions, your list will have one thousand entries, but you can only apply the diagonalization to the first three (or, more properly, to only three at a time). Thus, the diagonalization can only tell you that your list has more than three entries. If you increase the number of digits in your fractions, the number of fractions that you can diagonalize grows arithmetically while the total number of fractions on your list grows geometrically. If you extend the digits endlessly, as Cantor did, that fact does not change and you achieve nothing more than taking an infinite (Aleph-Null) set and selecting out of it an infinite subset, an achievement not essentially different from separating the infinite set of the even positive integers out of the infinite set of the natural numbers.
Ah, but we have yet to consider the set of the real numbers, the set that consists of every possible permutation of one natural number united with one decimal fraction. If the variables in an integral represent real numbers (and they usually do), then surely I should use the infinity of the set of the real numbers as the limit on my infinite integrations and that infinity should bear the name Aleph-One. After all, we create the set of the real numbers by combining two infinite sets in a multiplicative way, so that new set should have a cardinality of infinity squared. And we know that, except for one, multiplying any number by itself always yields a bigger number, so we expect that squaring infinity will yield an even bigger infinity: Aleph-Null squared equals Aleph-One.
Of course, we must test that proposition, put it to a proper mathematical proof. But for every natural number we have an infinite set of real numbers based on appending decimal fractions to that same natural number, so surely we cannot think that we can create an ubijoto match between the set of the natural numbers and the set of the real numbers! And that statement just bled all of the jaw-dropping astonishment out of what comes next (in comedy boxing they call it telegraphing your punch-lines).
Take some real number and interleave the digits of the fractional part with the digits of the integer part; more specifically, so stretch out the integer part of the number that its digits come to occupy the odd-indexed places on the left side of the decimal point, then put the first digit on the right into the second place on the left, put the second digit on the right into the fourth place on the left, and so continue on. If you choose, for example, the real number 939.573, the interleaving process turns it into 397359. We see clearly that the interleaving process creates a bijectively unique relationship between a real number and a natural number: one and only one real number will produce a given natural number and any given real number will yield one and only one natural number. There exists no real number to which we cannot apply the interleaving process. We can also reverse that process and convert any natural number into a real number: that reverse interleaving also creates a bijectively unique relationship between the two numbers. And, of course, there exists no natural number to which we cannot apply that reverse interleaving. Those facts necessitate that applying the interleaving process to the complete set of the real numbers creates an ubijoto match between that set and the set of the natural numbers, which necessarily means that the set of the real numbers has a cardinality of Aleph-Null.
So what happened to infinity squared? Though it sounds intensely mathematical, that phrase actually has no referent in the realm of mathematics, in much the same way that"faster than light" doesn't refer to anything real in physics. Because infinity denotes endlessness, it cannot represent an actual number and, thus, cannot legitimately participate in any arithmetic process, such as squaring. But in relatively technical expositions of the Cantorian theory of infinity we see equations in which Cantor's alephs mingle with actual numbers (or letters that represent actual numbers). That mingling implies that infinity denotes an actual number of some kind and thus misleads our reasoning about infinity (somewhat like trying to bring phlogiston back into modern thermodynamics; yeah, and how do you put that into a partition function?) We really need those equations to go away and eliminating Cantor's alephs from mathematics should achieve that ban automatically.
We can pursue the geometric analogy implied above, nonetheless, and convince ourselves once and for all time that no set can ever have a magnitude that goes beyond the infinity of the natural numbers. Start with the unit interval, a straight line segment whose endpoints we label zero and one. Each point on that interval represents the decimal fraction that denotes the point's location on the line as a fraction of the line's length. Thus, the unit interval represents geometrically the complete set of the decimal fractions.
Construct a unit square by so bringing two unit intervals together, with a right angle between them, that their zero endpoints coincide, then draw two more unit intervals to complete the boundary enclosing the square. Next to each of the first two unit intervals draw the complete set of the decimal fractions, then reflect the elements of one of those sets through the decimal point to convert them into the elements of the complete set of the natural numbers. Now we know that the set of points inside the square corresponds to the complete set of the real numbers, each point having a natural number and a decimal fraction as its coordinates in the square. But we know that we have an ubijoto match between the set of the real numbers and the set of the natural numbers; and we know that we have an ubijoto match between the set of the natural numbers and the set of the decimal fractions; so now we know that we have an ubijoto match between the set of the real numbers and the set of the decimal fractions (which we could have obtained directly by interleaving the digits of each real number's integer part within the digits of its decimal part). That fact tells us that the unit square contains exactly as many points as does the unit interval.
Consider the unit cube. It consists of an infinite set of unit squares set face to face. We can replace each of those squares with a unit interval without changing the number of points in the figure. That replacement gives us a unit square consisting of an infinite set of unit intervals set side by side. Then we replace that unit square with a unit interval. So now we know that the unit cube contains exactly as many points as does the unit interval.
Consider the unit tesseract, the four-dimensional unit right prism. It consists of an infinite set of unit cubes set body to body in the fourth dimension. If we condense each of those cubes into a unit interval, as we did above, and then condense the resulting unit square into a unit interval, we show that the unit tesseract contains exactly as many points as does the unit interval.
Proceeding to ever higher dimensions, we find that we can always condense the unit right prism into a unit interval. At no dimension will we encounter any mathematical phenomenon that would change that fact. So imagine a number vastly greater than any number that you can name or draw and know that the unit right prism of that many dimensions contains exactly as many points as does the unit interval. If contemplating that fact doesn't boggle your mind, nothing will.
So, except for Aleph-Null, the infinity of the natural numbers, Cantor's alephs constitute an empty set. And I can't shake off the feeling that someone could have and should have discovered that fact much earlier than I did. One hundred and thirty years seems a long time for an error to persist in the foundations of mathematics, especially since it has such a simple correction. If we conceive infinity as denoting endlessness, then Cantor's higher alephs must represent endlessnesses more endless than endlessness. That little piece of like-wow-man, hippie-dippy mysticism should have raised in mathematicians' minds more red flags than you can see in Tienanman Square on May Day. But it didn't. Indeed, I only discovered Cantor's error myself through pure serendipity (see "uh-oh" above).
I believed that I could work out a description of a set that has a cardinality of Aleph-Three. To that end I set out to write a tutorial on Cantor's work that would make his proofs of the existence of non-denumerable sets as clear as possible. I wrote the most concise description that I could of his diagonalization proof and watched in horror as my dream shattered and crumbled into fairy dust.
I don't make these comments to disparage mathematicians, but to point out the fact that the human brain did not evolve to work with abstract mathematics. In my more waggish moments I claim that Homo sapiens denotes a joke, not a species. Homo ludicrous does not possess intelligence as a born-in trait. In much the way that they invented civilization, our ancestors had to invent intelligence, subroutine by subroutine, as if cobbling together some weird computer program (and we certainly conceive the human brain as a suitably weird computer). Thus we obtain our intelligence more from our culture than from our DNA. With suitable conditioning and training we can fake intelligence, but, like the dancing bear, we don't do it often and we don't do it well. That we do it at all should astound us.
And we have done it. And we have done it with remarkable consistency. Our ancestors invented mathematics and then tried to figure out the consequences of what they had created, vastly expanding a purely imaginary realm that possesses such an exquisite inner consistency that many of the Ancients believed mathematics possesses more reality than does the world that we apprehend through our senses. As part of that process, especially as it applies to plane geometry, the Ancient Greeks invented logic in order to ensure the validity and truth of their reasoning, but it doesn't come to us easily. We don't speak in Logic and we never will, so we must, as our mothers used to caution when we used the wrong words, watch our language.
How we define the denotation of a word, however clearly and unambiguously we do so, does not guarantee that we will not misuse the word. As Benjamin Lee Whorf pointed out in his famous discourse on empty gasoline drums, the connotations of a word can work mischief in our unconscious minds and distort the meaning of a word in ways that lead us into disaster. Infinity denotes endlessness, but when we talk about the infinity of the numbers or of space, describing infinity as something those entities can never reach, infinity gains the connotation of magnitude. We usually use numbers to describe magnitude, so that connotation subtly leads us to think of infinity as a kind of number, even though the denotation contradicts any such idea. Thus, the connotation of"infinite number" may have misled Cantor in his interpretation of the results of his diagonalization procedure.
We need to apply a version of Cobbett's Rule: I describe this thing thus not only so that you will understand me, but also so that you will not misunderstand me. We need to ensure that we understand clearly the concepts that we use in our reasoning and ensure that their connotations do not sabotage the clarity of that understanding. And we need to keep our minds open to the possibility of finding yet other errors lurking in the structure of mathematics.
As for me, when Earth swings me into its shadow, I no longer dread the spectre that comes to me and questions my choice of integration limits; for I have mooted the spectre's questions. Only one infinity exists in the realm of mathematics, so the spectre's proffered choice does not. I can carry out infinite-limit integrations with a clear conscience. However, I also feel confident that the elapse of time will eventually conjure out of the cobbly worlds of Humanity's collective unconscious another spectre to trouble my dreams with impudent questions. Thus human knowledge evolves.
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