Infinity

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    Infinity (Latin for without limit) simply denotes the mathematical concept of endlessness. Although it seems to have a numerical nature, infinity and its reciprocal, the infinitesimal, do not truly denote actual numbers. Nonetheless, we find the two concepts useful. In our mathematical physics and especially in our use of the calculus, we speak of something Agoing to infinity@ or Aat infinity@. And all too often we speak as if we had a definite number in mind. Over the elapse of time we have gotten careless with the term and, so, we need to review just what the word infinity and its reciprocal actually mean.

    Everything that we need to know about the concept of infinity lurks within the name itself. Infinity denotes something that has no end. From that fact we can follow Georg Cantor=s lead and deduce the properties of infinity and of infinite sets, in particular their relations to the numbers.

    Using set theory, which he created, Georg Ferdinand Ludwig Philipp Cantor (1845 Mar 03 - 1918 Jan 06) devised what he called transfinite arithmetic, arithmetic applied to infinite quantities. In the 1870's and 1880's, 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. But that raises a question: does the lazy-eight on the elongated ess of an integral represent the infinity of the natural numbers or the infinity of the real numbers? Does the choice make a difference in the value of the integral? 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 those questions about the limits of integration.

    What do we know about the infinity of the natural numbers and how do we know it? Note that infinity does not give us a number in the common sense; that is, as a specific name in an ordered set of names. We use infinity for comparison and enumeration, but not for calculation. We properly use the lazy-eight (4) purely as a limit marker, as a number-like thing that marks a numerical place where we have no possibility of a number. So we start with an enumeration and a comparison.

Aleph-Null

    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. We must infer that the cardinality of the set of the natural numbers has no fixed value in the sense of something that we can measure. It goes beyond any limit that we might set on the natural numbers; it goes beyond the realm of the finite. 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.

    The process that we use in that analysis makes infinity a number-like thing. It seems to denote a number that always eludes our grasp, a number that always lies beyond the biggest number we can name. If we conceive it as a place on the number line, that place has no specific location as the other numbers do. If we conceive it as a member of the set of ordered names, we can= t give it a specific place in that ordered sequence; it always lies beyond. We must take great care, then, in using phrases like Ainfinite number@ or Ainfinite quantity@, always bearing in mind that those phrases necessarily refer to something with an indefinite value.

    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.

    You can extend that process in a way that addresses all of the basic operations of arithmetic. Lay out the set of the natural numbers and then below each entry write the result of adding some number to it (or subtracting a number from it). In that way you demonstrate that adding to or subtracting from an infinite set does not change the number of elements in the set or, to put in the popular parlance, infinity plus (or minus) some number equals infinity. Of course, that statement does not represent an actual calculation, because it does not give us a definite value as a result. But we can use it nonetheless in devising the finite-value theorem, a kind of logical filter that we can use to deduce laws of physics.

    Likewise, we can devise ubijoto matches between the set of the natural numbers and multiples of the elements of that set to show that division of infinity by any number does not change infinity. A set that we expect to have one-third the size of the set of the natural numbers (the multiples of three) or one-fifth the size (the multiples of five) nonetheless has the same cardinality as does the set of the natural numbers. We can also create an ubijoto match between the set of the natural numbers and a set that consists of N ones, N twos, N threes, and so on, and thereby prove and verify the proposition that infinity, as a number-like entity, remains invariant under multiplication by any number N. In that way we demonstrate Cantor=s transfinite arithmetic, which tells us that infinity remains unchanged by the application to it of any of the operations of basic arithmetic: if we multiply or divide infinity by some number, raise it to some power, or extract one of its roots, we always get infinity back.

    Thus Cantor discovered another criterion by which we can discern infinite sets. If we have a set that violates Euclid= s fifth common notion, AThe whole is greater than the part@, if we can devise an ubijoto match between the set and at least one of its proper subsets, then we have an infinite set. Just as denial of the truth of Euclid=s fifth postulate (the parallel postulate of plane geometry) led to the development of the 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.

    Cantor described Aleph-Null as the smallest infinity. Can we accept that statement as true to mathematics? Assume that we have created a set of objects such that the size of the set is infinite and smaller than Aleph-Null. We can match each element of that set with one of the counting numbers, in essence assigning the counting numbers as index numbers to the elements of the set. If the number of elements in the set is less than Aleph-Null, then we must have natural numbers that we cannot match with the elements of the set because we have already matched those elements. That fact means that eventually we must come to an element of the set beyond which we have no more elements to number, a last element in the set,. But that ending point is a boundary, a mark that the set has finite size. We must thus infer that our set of objects does not have infinite size. So Aleph-Null is truly the smallest infinity.

    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. Thus we determine the relative size of infinite sets by the process of matching the set with one or more of its well-defined subsets. In that way we matched the set of the natural numbers with the set of the even natural numbers, thereby demonstrating that the set of the natural numbers does, indeed, have an infinite cardinality, that of Aleph-Null.

    Prior to the 1870's mathematicians believed in potential infinity but not in actual infinity. They used the idea of infinity in limits, but they didn=t have it well defined. It was vaguely described much as modern laypeople describe it B a number bigger than any conceivable finite number. Cantor gave it a properly technical 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 created transfinite arithmetic 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 Euclid= 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, AIn 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.

Aleph-One

    Cantor had to do something a little different from what he did with the natural numbers 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, indeed could not have, 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.

    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. That description tells us immediately that Cantor must have made an error in his analysis of the infinity of the decimal fractions. We now have the obligation to find the error and to correct it.

    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 for diagonalization, an achievement not essentially different from separating the infinite set of the even positive integers out of the infinite set of the natural numbers.

    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! But it doesn=t take us long to understand that we can do just that.

    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 Afaster 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). Those equations look legitimate because Aleph-Null looks just like any other letter that we use to represent an algebraic variable, a kind of arithmetic place marker that we will eventually replace with an actual number in order to carry out the indicated calculation. That mingling implies that infinity will eventually denote an actual number of some kind and thus misleads our reasoning about infinity. 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 that we drew 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 and thereby representing a real number. 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.

    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 and then as an actual number, even though the denotation contradicts any such idea. Thus, the connotation of Ainfinite 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 my choice of integration limits, I have mooted the question. Only one infinity exists in the realm of mathematics and thus only one infinity can serve as the limit on an integration. I can carry out infinite-limit integrations with a clear conscience.

The Infinitesimal

    Having looked at the infinite, the endlessly large, what can we say about the infinitesimal, the endlessly small? As with infinity, we must take care in talking about the infinitesimal, especially we must take care not to use the word as though it refers to a number.

    Suppose we ignored that advice. What kind of trouble will we get into? Let=s define the infinitesimal as a number smaller than any number greater than zero, or as a quantity less than any finite quantity without being equal to zero. If the infinitesimal denotes a number, then let=s represent it with H. If X represents any natural number, can we calculate H/X? If we can, then H doesn=t represent a number smaller than any number greater than zero, because we can always find a bigger X to divide it. If we can=t, then H doesn=t represent a number. So we can say, in general, that if H is infinitesimal with respect to the elements of some set, then H cannot be an element of that set because it would have to be smaller than the smallest element of the set.

    To explore the concept of the infinitesimal we describe the smallest possible fraction, which we assume (reasonably) will take the form of a unit fraction. As a ratio we describe a unit fraction as a one divided by some larger number, so our smallest possible fraction consists of a one divided by the largest possible number. But we have already seen that a largest possible number does not exist. The natural numbers grow larger endlessly, striving toward infinity with no possibility of reaching it, so the denominator in our unit fraction would grow endlessly in pursuit of the elusive infinity. Thus our fraction shrinks endlessly, coming ever closer to zero but never reaching it. Like infinity, the infinitesimal lies forever beyond our grasp and comes to no definite conclusion. We could conceivably describe the infinitesimal as one divided by infinity, but only if we keep in mind the fact that we have not thereby described a real division.

    If we try to create a smallest possible fraction as a decimal extension, the infinitesimal gets even stranger. The infinitesimal must lie between zero and the smallest possible fraction. To make a fraction infinitely small we must describe it as a string of zeros extending endlessly to the right of the decimal point with a one at the right end of that string. But an infinite string of zeros doesn=t have a right end, so we can=t actually place the one into the fraction. But neither can we dismiss the one; if we did, our putative smallest fraction would equal zero and that would leave no room for the existence of the infinitesimal. By making our fraction endlessly smaller we shove the infinitesimal right up against zero, but it won=t merge with that expression of emptiness.

    Another way of looking at infinitesimals displays them as spacers between certain elements of the set of real numbers. To see how that works calculate the decimal equivalent of 1/9. We know that we get an endless string of ones to the right of the decimal point, but look more closely at how we get them. In this infinitely long division we divide nine into one and get one tenth plus a remainder of one tenth. We divide nine into that remainder and get one hundredth plus a remainder of one hundredth. Continuing that process, we generate an infinitely long string of ones and an infinitely small remainder.

    Multiply that fraction by nine. The product of multiplying 1/9 by nine equals one. Multiplying an endless string of ones to the right of the decimal point plus a remainder by nine yields an endless string of nines to the right of the decimal point plus the same remainder. (The remainder represents a part of the original number that we have not yet divided by nine, so we also don=t multiply it by nine when we seek to undo the division.) So we say that 0.99999...plus a remainder equals one.

    An alternate way of looking at that analysis comes from the October 2006 issue of Discover Magazine in a small item titled AFuzzy Math...Almost Is Enough@ (and no proper mathematician would ever agree with that statement). Define X=0.99999..., the ellipsis indicating that the string of nines extends to the right without end. Multiply that by ten to get 10X=9.99999... and subtract X to get 9X=9.00000.... If we divide that equation by nine, we get X=1.00000....

    So on one hand we get 1.00000...=0.99999...plus a remainder and on the other hand we get simply 1.00000...=0.99999.... Which of those analyses must we accept as the correct one?

    I say that we must accept the first analysis as the correct one. The second analysis contains a very subtle error that, when corrected, leaves us with the same result we get in the first analysis. Correcting the error also illustrates the application of the infinitesimal in areas other than Aclose to zero@ .

    The error occurs in the multiplication and subtraction in the second analysis. Suppose we define X to consist of a string of N nines to the right of the decimal point. When we multiply that X by ten we get a nine on the left side of the decimal point and N-1 nines on the right side of the decimal point: the multiplication has simply shifted each nine one decimal place to the left of its original position. After we subtract X from 10X we have a nine minus a fraction that consists of N-1 zeros on the right side of the decimal point and a nine in the N-th place. We divide that number by nine and get a one minus a fraction that consists of N-1 zeros on the right side of the decimal point and a one in the N-th place. That fact remains true to Mathematics regardless of how big we allow N to become. If we let N run on toward infinity, the fraction approaches the infinitesimal, but, of course, never reaches it. Thus we must say that 0.99999...=1.00000...minus a minuscule fraction, which corresponds to what we obtained in the first analysis. In this case the infinitesimal acts as a spacer between 1.00000... and 0.99999..., one that requires an actual number in the form of a very small fraction to bridge it.

    But we don=t pursue knowledge of the infinitesimal merely in order to spoil parlor tricks. We want to have mathematics as an expression of pure logic: ultimately we want to achieve the perfect certainty that we commonly associate with arithmetic calculations. To that end we cannot allow any slippage to creep into our logic, not even an infinitesimal. In some respects mathematicians are very much like lawyers: when they speak or write formally, they do so not only to be understood, but also to ensure that they are not misunderstood. So often the formal content of mathematics differs from the content of the same subject in the popular imagination. The calculus gives us a good example of that fact.

    In the popular imagination the calculus involves the manipulation of infinitesimals. At first, even the mathematicians accepted that idea. But that acceptance went away as the formal content of Mathematics evolved.

    Consider differentiation. We have some function of x (say f(x)=x2) and we want to know the rate at which that function changes as we change the value of x. We gain that information by calculating the derivative of the function with respect to x; that is, we calculate the ratio of the amount the function changes with a given change in x to the given change in x. To make that ratio as accurate as possible we want to make the change in x as small as possible, so we use the differential dx to represent that change. We begin by calculating df=f(x+dx)-f(x)=2xdx+(dx)2 and divide that by dx to get 2x+dx. We define the derivative as the limit that function approaches as we let dx shrink without limit; in this case we get df/dx=2x.

    Of course dx cannot actually represent the infinitesimal. And we don=t need it to do so. If our function represents some astrophysical quantity that we want to study and we make x equal some number of lightyears, then making dx equal some millimeters certainly makes it small enough that dropping it from the derivative will make no discernible difference in our calculations. If that turns out not to stand true to our study, then we can always make dx smaller. The fact that we can shrink dx as much as we like makes differentiation of functions possible. But however much we shrink dx, it can never become an actual infinitesimal.

    As with infinity, we must take care never to use the infinitesimal as an actual number. It merely denotes the fact that some arithmetic processes can never end and never come to a definite conclusion. But we do have a use for it in physics. We know that an infinite set of points that we define in the classical way (as a thing with zero extent) does not yield a space of finite extent: the sum of a set of zeroes always equals zero, even if we let the set become infinite. But we also cannot define the point as having a finite, if minuscule, extent, because the set of those points has no capacity to expand. But if we define the geometric point as having infinitesimal extent, then an infinite set of them has a finite extent that can grow inherently. That fact follows from the fact that the infinitesimal gives us the reciprocal of infinity. If we add up an infinite set of infinitesimals, we must get some finite value, but not a definite one. Thus we can, indeed must, use the infinitesimal in our geometry and in the physics based upon it.

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