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How far do the natural numbers reach? How far up the number line do they go? To refine the question, In the ordered sequence of names that we call numbers can we ever come to a last name; can we come to a largest of all numbers, beyond which we cannot count? Can we find a biggest of all numbers, a number than which we can find nothing bigger? In asking that question, we seek to know the magnitude of the set of all of the natural numbers.
We can devise an answer to that question through a straightforward, almost trivial, reductio ad absurdum. Assume that I claim to have found the biggest of all numbers and that I show it to you. But you know how to add one to any number and to make up a name for any new number, so you can refute my claim by the simple expedient of adding one to my "biggest of all numbers". You know that you can continue that procedure endlessly and thus you invalidate any claim I might make that we can reach an end to the sequence of the natural numbers. There exists no number to which you cannot add another number; therefore, there exists no largest number and no last element of the set of the natural numbers. We must infer that the magnitude of the set of the natural numbers has no fixed value in the sense of something that we can count or measure. However far we go up the number line, we will find the natural numbers proceeding even further, beyond any limit that we might set on the natural numbers, without any possibility of reaching an end.
Using the Latin word for "without limit", we say that the natural numbers comprise an infinite set, one that has no last (or largest) element. That use of the adjective infinite to describe an endless sequence of ordered elements of a set then obliges us to use the associated noun, infinity, to denote the mathematical concept of endlessness. Infinity thus does not represent a place "way out there" where the number line ends; rather, it represents the fact that no such place exists, that no such place could possibly exist. As infinity denotes the mathematical concept of endlessness, so it cannot denote, or even connote, anything that would mark the end of a sequence. We need to keep that fact firmly in mind, because sometimes we use infinity as if it represented an actual number, a convenient fiction that can lead our reasoning astray.
For example, we use infinity as an endpoint in many calculations of definite integrals. When we draw the familiar lazy eight next to the elongated ess that represents the Riemannian sum of an integration, we don't mean that we intend infinity to represent an actual number, something that we know does not exist. No, that particular use of infinity stands for our use of a number, call it alpha, so vastly large relative to the other numbers in the calculation that dividing any of those other numbers by alpha yields a quotient negligibly different from zero. We don't give alpha an actual value, but merely state that it has a value large enough to ensure that the relevant parts of the calculation shrink so close to zero that we can simply ignore them and drop them from the calculation.
In such applications we use infinity and pseudo-algebraic manipulations of it in much the way that we use"sunrise" in place of the astronomically correct but clumsier "Earth's rotation bringing the sun into view over the eastern horizon". Such convenient fictions act in our minds as simplifying aids to the imagination, making abstract thought easier and thereby enabling us to achieve more with it. In mathematics, infinity takes us far out of the mental realm in which intuition guides us reliably to truth and into the realm in which we must rely on reason. Our ancestors devised reason precisely to extend their thinking into that realm. Unfortunately, they did not devise a clear delineation of the boundary between the two realms. So sometimes we forget the fictional nature of these things, come to think of them as somehow true to mathematics, and they sometimes lead our reasoning to conclusions false to mathematics. That happened to Georg Cantor in his study of infinity, as I will show in the essay on Advanced Infinity.
In the period spanning the years 1871 to 1884 Georg Ferdinand Ludwig Philipp Cantor (1845 Mar 03 - 1918 Jan 06) used set theory, which he devised, to refine our understanding of the nature of infinity by exploring the properties of infinite sets. He started with the set of the natural numbers and found that he could put the elements of that set (the counting numbers) into an ubijoto (Unique, BIJective, One-To-One) match with the elements of each of several of its subsets. That fact pretty much defines an infinite set.
Consider Cantor's best known example. Write down all of the natural numbers in a line extending to the right. Below each of those numbers write its double. Now we have written the set of the even natural numbers below the set of all of the natural numbers, which includes the even numbers as a well-ordered subset. But we have written the set and subset in a way that clearly indicates our assertion that every natural number has a matching even natural number that we can uniquely associate with it and (the bijective part) every even natural number has a matching natural number that we can uniquely associate with it. Do you believe that we will run out of even numbers before we come near running out of natural numbers (which, of course, we cannot do)? If we had a finite set, we certainly would. But then recall that we know how to multiply by two: if we think that we have come to the end of our matching, we need only take the first putative unmatched natural number and double it to generate an even number to match with it. And that process, like our adding process above, will never end.
In that one-to-one matching we have found a set and a subset of it that negate Euclid's fifth common notion, "The whole is greater than the part." We have proven and verified that a well-defined subset of the natural numbers contains exactly as many elements as does the full set of the natural numbers itself contains. Just as denial of the truth of Euclid's fifth postulate (the parallel postulate of plane geometry) led to the development of the weird non-Euclidean geometries that have found application in describing the warped spacetimes of General Relativity, so too the denial of the truth of Euclid's fifth common notion leads to non-Euclidean arithmetic, which has its own applications to modern physics.
But if we have made an ubijoto match between the positive even integers and the complete set of the natural numbers (the positive integers), how can we possibly accommodate the other subset, the set of the positive odd integers?
We answer that question by visiting Hilbert's Hotel. In this little scenario David Hilbert (1862 Jan 23 - 1943 Feb 14; the German mathematician who, on 1900 Aug 08 at the Second International Congress of Mathematicians in Paris, France, gave the mathematicians of the Twentieth Century their 23-problem homework assignment) runs a hotel with an infinite number of rooms, which he has filled with guests. (Again, I note that there exists no such thing as an "infinite number" of anything; here I use it as a convenient shorthand term for the clumsier phrase "an indeterminate number whose value lies beyond all possibility of counting, a number in pursuit of infinity".) Each guest bears a number equal to twice the number of the room they occupy, so, for example, Guest-54 occupies Room-27. In this scenario the rooms represent the set of the natural numbers and the guests represent the set of the positive even integers.
Next an infinite number of new guests, bearing the positive odd integers, arrive at the hotel and they all have reservations. What can Dr. Hilbert do? He apparently has no accommodation for them. However, the desk clerk, Mr. Cantor, knows just what to do. He tells each current guest to move into the room that shares their number, so that Guest-54 moves into Room-54, and then he tells the new guests to move into the freed-up odd-numbered rooms, each new guest moving into the room that shares their number, so that Guest-27 moves into the room that Guest-54 vacated. Thus, even with the hotel fully occupied, Mr. Cantor can still accommodate an infinite set of new guests.
That seems like a paradox. People often talk of paradox when they talk of infinity. Richard Phillips Feynman (1918 May 11 - 1988 Feb 15) offered a rather nice description of a paradox that will shed some light here: "A paradox is a situation which gives one answer when analyzed one way, and a different answer when analyzed another way, so that we are left in somewhat of a quandary as to actually what should happen." In writing down the natural numbers and putting them into an ubijoto match with the positive even integers, we have created a Feynmanian paradox. On the one hand, we know that the set of the positive even integers must contain fewer elements than does the full set of the natural numbers. And on the other hand, we have proven and verified the fact that the set of the positive even integers contains the same number of elements as does the set of the natural numbers. We can't dissolve that paradox by identifying and correcting bad assumptions in the reasoning that led to it, as we do in dissolving the Twin Paradox of Relativity. We can only get away with this absurdity by putting the magical transformation that makes it possible (whatever form that might take) "way out there" in the fantastic realm of infinity, whither no one can ever go. We thus create the ultimate example of "sweeping the dirt under the rug", but it works nonetheless as long as we take proper care with it.
We can make an ubijoto match between the elements of any well-ordered subset of the natural numbers and the elements of the set of the natural numbers. By well-ordered we mean that the subset has a first element and a criterion, usually magnitude, by which each element precedes one other element. Just as we did with the positive even integers, we can make an ubijoto match between the elements of the set of the positive odd integers and the elements of the set of the natural numbers. We can also make an ubijoto match between the elements of the set of the square numbers and the set of the natural numbers, matching each counting number with its square (and that works for the other powers as well). Can we do the same with the set of the prime numbers?
Does the well-ordered set of the prime numbers have a last element? If not, then the prime numbers comprise an infinite set and we can make an ubijoto match between its elements and the natural numbers. Euclid devised an elegant proof and verification of the proposition that no last prime number exists. Assume that such a last prime number does exist, a largest prime number that we designate P. Take all of the prime numbers, beginning with two, up to and including P and multiply them together to create the number N. By the fundamental theorem of arithmetic, the unique factorization theorem, no other product of primes will produce that number. Consider N+1. If it represents a prime number, then we know that P cannot represent the last element of the set of the prime numbers. If N+1 represents a composite number, we know that none of the prime numbers equal to or less than P can divide it evenly, so it must represent a product of prime numbers larger than P, which also necessitates that P not represent the last element of the set of the prime numbers.
We can repeat that process over and over again, each new larger prime number becoming a new value for P. We have no criterion that would tell us when we must stop carrying out that analysis, so we assume that we can carry it out endlessly. Thus, through strong induction, we infer that the prime numbers comprise an infinite set that we can put into an ubijoto match with the full set of the natural numbers.
That process of matching subsets of the natural numbers with the full set of the natural numbers led naturally and unfortunately to transfinite arithmetic, an arithmetic that merges infinity with finite numbers. The rule of addition, for example, comes to us when we match each of the natural numbers with itself plus or minus some number K. We obtain a rule that tells us that infinity plus K equals infinity. In a similar manner we can devise rules for subtraction, multiplication, division, and exponentiation with infinity. But, as Wolfgang Pauli once said of someone's theory, that's not even wrong. We cannot properly use infinity in any kind of arithmetic because it does not represent a number. Cantor neglected that fact and went seriously wrong as a result.
As something of a prelude to his more technical work on infinity, Cantor identified three types of infinity; 1) theological, 2) physical, and 3) mathematical. Briefly he described those types as:
1) the Absolute Infinite (aka God = Ω), "realized in its most complete form, in a fully independent otherworldly being...."When the average person talks about infinity as inconceivable, this is it.
2) "when it occurs in the contingent, created world...." For example, points in space comprise an infinite set.
3) "when the mind grasps it in abstracto as a mathematical magnitude, number or order type." For example, the size of the set of all positive integers.
Note that Cantor described mathematical infinity as a kind of number. Because we have used the natural numbers to define infinity, we have certainly made infinity a numerical concept, but nonetheless infinity does not denote an actual number. Referring to the geometric metaphor, we must say that infinity does not denote a point on the number line. Thus, we cannot legitimately refer to something "approaching infinity" or "at infinity". We might also note some difficulty in using "without limit" as a limit in integration.
Cantor also noted that an infinite set A is larger than an infinite set B if B can be put into a one-to-one correspondence with a subset of A but A cannot be put into a one-to-one correspondence with B or a subset of B. He used the Hebrew letter aleph to designate an infinity and applied subscripts to distinguish the size classes of the various infinities in the hierarchy. Aleph-Null, for example, designated the infinity of the set of the natural number. Then Cantor began the process of finding those sets of mathematical objects that have infinite size and determined which infinities, which alephs, correctly denote their size. And in that endeavor he found endlessnesses more endless than endlessness, something that should have alerted him to the fact that he had made a serious error in his reasoning. I will address that issue in the essay on Advanced Infinity.
Now we have proof and verification of the statement that the set of natural numbers satisfies the property of closure with respect to the processes of addition and subtraction. We offer that proof by noting that we thus have enough numbers to solve the equation X+A=B or X-A=B for any numbers A and B that we might choose. That statement remains true to mathematics when we include the fractions and the mixed numbers with the natural numbers and the negative integers, but that gets us into the topic of advanced infinity, which I describe in another essay.
Prior to the 1870's mathematicians believed in potential infinity but not in actual infinity. They used the idea of infinity in limits, but it was not well defined. It was vaguely described much as modern laypeople describe itB a number bigger than any conceivable finite number. Cantor gave it a proper description by stating that an infinite set is one that can be put into a one-to-one correspondence with a proper subset of itself. Thus Cantor effectively defined infinity as a mathematical concept by denying the validity of Euclid's fifth common notion, just as other Nineteenth Century mathematicians created non-Euclidean geometry by denying the validity of Eucli's fifth postulate.
For centuries mathematicians had accepted the idea of an incomplete infinity, a potential infinity. But they would not accept a completed or actual infinity. Galileo even introduced the statement that the even numbers have the same cardinality as the natural numbers as a paradox that precludes an actual infinity. But Cantor says,"In introducing new numbers mathematics is only obliged to give definitions of them, by which...they can definitely be distinguished from one another. As soon as a number satisfies all these conditions it can and must be regarded as existent and real in mathematics." This view contrasts with that of Leopold Kronecker, who regarded only the integers and numbers directly derived from them as real. In that controversy I take Cantor's side: one of the tasks facing the mathematician is that of parleying current knowledge into further knowledge and accepting what logic gives us, however bizarre it seems to us.
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