Here we are again on #MathsMonday, looking at the questionable statements of SmartmanApps, #debunking the #disinformation and seeing what real #mathematics it accidentally uncovers.
Our now-familiar friend says that infinitesimals are a standard thing taught in schools. This is false, with the possible exception of wherever he teaches, because infinitesimals have by and large been ejected from mathematics due to being unnecessary and confusing.
But what is true is that infinitesimals *used* to be taught, and he has the very old textbooks to prove it. Letโs find one of those textbooks, and its definition of infinitesimal:
> lim a = 0 means that a is an infinitesimal quantity
(Advanced Algebra, p226 by Collins, J. 1911. American Book Company. Symbols altered for typographical necessity)
So we need to know what โlim x = aโ means, which is furnished to us on page 297 (it is no credit to this textbook that it uses undefined terms without even an apology):
> If a variable x assumes a given sequence of values such that the numerical difference between a constant a, and the variable x becomes and remains less than any assignable quantity, however small, then x is said to approach a as its limit [written] lim x = a
This is essentially the way Cauchy defined limits, and is the same as the definition of a limit I gave last time, with two changes:
1. Itโs less easily translated into the symbolic version due to not using directly equivalent language like โfor everyโ.
2. Instead of a sequence, we talk of a โvariableโ.
It is the second difference which is more consequential. In modern mathematics, in any given context, a variable only takes on a single value. If you think of x as possibly taking on multiple values, say 2 and 0 in alternation, then the following argument no longer works:
โx is not less than 1, and x is not greater than 1, therefore x is equal to 1โ.
Imagine asking โis x less than 1?โ and finding that x could be 2, so answering, โnoโ; then we may ask โis x greater than 1?โ and find that x could be 0, so answer โnoโ again. But concluding that x is equal to 1 is not valid. So, we must be smarter about such arguments. Maybe our Smart friend is that smart, but I would prefer not to depend on it.
The solution modern mathematics has taken to this is that if we want to consider a changing quantity, we encode it into a different kind of object: a function. The function accepts a parameter, and that parameter determines what value it returns. Thus the dependence of that value on the parameter is made explicit; we save ourselves from the possibility of getting confused, because โx(0) is not less than 1, and x(1) is not greater than 1, therefore x(2) is equal to 1โ is obviously rubbish!
In school it is typical still to describe functions in the old-fashioned language of variables, so you probably remember seeing a linear function described by โy = mx + cโ, where x is a variable, and y another variable which *depends* on it. Further on in school itโs typical to replace โyโ with โf(x)โ which makes this dependence explicit.
Because functions are an entirely different type of thing, we also never compare them to numbers without first describing exactly how that comparison should happen. We donโt ask, โis f equal to 1โ? Because the answer is clearly โnoโ, and similarly f is not less than or greater than 1; f is a function, not a number so these questions donโt even really make sense. This is not the case with variables; it is perfectly reasonable to ask whether x is equal to 1 - this kind of question is embedded in every single equation a school pupil is asked to solve.
Clearly, mathematicians and pupils of yesteryear were all able to work like this, but it is unnecessarily confusing and error-prone.
## Sequences
So, we will use functions instead of variables, and since it will not cause any extra difficulties, we will specifically use sequences which are functions whose inputs are natural numbers. This turns Cauchyโs definition of the limit into the modern one, and it turns his definition of an infinitesimal into โa sequence with a limit of zeroโ.
We have some questions to answer: while it is natural to compare a variable to numbers (such as when solving a system of inequalities) comparing a function (or sequence) to a number is less natural. What does โsin < ยฝโ mean? Is it true, or false? Clearly sin(x) < ยฝ for some x but not for others.
Nonetheless, if we want sequences to occupy the role of infinitesimals, we need to be able to perform such comparisons, because the fundamental property of an infinitesimal e is that 0 < e < 1/n for all natural numbers n. Thus how we set up infinitesimals is fundamentally a matter of how we set up rules for comparing the sequences which represent them.
Let us first point out that instead of comparing sequences to *numbers* we can replace any fixed number with the constant sequence all of whose elements are that fixed number. And secondly, letโs establish that the *normal* equality of sequences is simply that each of their corresponding elements must be equal, that is, their first elements must be equal to each other, their second elements must be equal to each other, and so on. This means that there are many sequences whose limit is zero but which are not equal to one another. One of those sequences is the constant sequence all of whose elements are zero, i.e. the sequence which we are using instead of the number zero, written (0, 0, 0, โฆ)
I should point out here that it is a standard fact about limits that the limit of a constant sequence is always that constant. This should be easy to see from the definition, but in a university course you would prove it rigorously. This means, also, that any sequence whose limit is zero must be different from any constant sequence, except that the zero sequence is equal to itself. This is good for our use of sequences which converge to zero (i.e. have limit zero) as infinitesimals, since they should be different from numbers larger than zero.
## Equivalence and Order
But we still have to do more work, because to be infinitesimals, these sequences need to be smaller than 1/n, for every natural number n. We know the first step to making sense of this: replacing 1/n with the sequence (1/n, 1/n, 1/n, โฆ). The next step is creating a notion of โsmallerโ and โlargerโ sequences.
In what follows, I will fix a sequence e representing an infinitesimal, e := (1, ยฝ, โ
, ยผ, โฆ), and x representing a small finite positive number, x:= (0.1, 0.1, 0.1, โฆ)
### Global Domination
One very simple way of setting up an ordering is to say that (a1, a2, a3, โฆ) < (b1, b2, b3, โฆ) if a1 < b1 and a2 < b2 and a3 < b3, and so on, for each index. This is no good for our purpose, because look at e and x. If we are doing things right, we should have e < x, but e1 = 1 is not less than x1 = 0.1. Even worse, if we go to the 11th element, x11 = 0.1 is not less than e11 = 1/11, so neither e < x nor x < e: this is not a **total** ordering, i.e. sometimes two objects just do not lie in any particular order.
### Lexicographic Ordering
The lexicographic (so called because itโs the ordering used by dictionaries - i.e. lexicons) ordering is perhaps the next most obvious way of ordering something thatโs made out of multiple things which are themselves ordered.
Using this ordering weโd say that (a1, a2, a3, โฆ) < (b1, b2, b3, โฆ) if a1 < b1, or a1 = b1 and a2 < b2, or a1 = b1 and a2 = b2 and a3 < b3, or โฆ and so on.
This *does* give us a total order; we can compare any two sequences and either one is less than the other, or they are exactly the same sequence. However, we still have a problem because we find that since 0.1 < 1, we have defined x < e, which is not what we wanted.
This ordering fails to capture the idea that it is the *eventual* behaviour of infinitesimals that we care about, and instead makes the early behaviour the most important.
### Eventual Domination
We canโt just do a โreverse lexicographic orderโ because we canโt start at the *last* element of a sequence composed of infinitely many elements.[^4] But we can do something else: we can say that (a1, a2, a3, โฆ) โค (b1, b2, b3, โฆ) if *from some point* all the aโs are smaller than all the bโs. You can hopefully see that, with this ordering, e โค x: from the 11th element onwards, all eโs are below 0.1. And indeed if y is *any* constant sequence consisting of rational numbers, eventually e will drop below that number and stay below.
It is not for no reason that I used โค in this definition instead of <. That is because this is again not a total order among all possible sequences: a sequence could oscillate above and below another sequence forever.
### Ordinary Real Numbers
What Cantor did in his development of real analysis was to notice that there is a certain kind of sequence that behaves very nicely, which nowadays we call a *Cauchy sequence*. It has the property that as the sequence continues, its values become arbitrarily close to all later values:
> A sequence a is *Cauchy* if, for every ฮต > 0, there is a number N such that for all n, m both greater than N, |a(n) - a(m)| < ฮต
The structure Cantor worked with was all of *these* sequences (of rational numbers) rather than arbitrary sequences. Then, he said that two sequences were equivalent if the difference between them becomes arbitrarily small. If not, then (one can prove) the difference between the values of a and b is either eventually positive, in which case we say a is larger, or eventually negative and we say b is larger.
We can go on to define arithmetic between such sequences and prove that everything works out wonderfully but there is one snag for us: there are *no* infinitesimals! What was the fate of our supposedly infinitesimal sequence e? Well, unfortunately the difference between it and the zero sequences becomes arbitrarily small, so they are rendered equivalent: any sequence which is โinfinitesimalโ in the Cauchy sense becomes merely zero.
[^4]: Unless the infinite order type is a *successor ordinal* which is a concept beyond the scope of this post, and is not the case for our sequences.
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