The infinitesimal in Calculus

[Agent Smith]

TL;DR Summary

Defining the Infinitesimal

Cobbling together a definition of the infinitesimal from bits and pieces of info gathered from books and the internet:

The infinitesimal $d$ is the positive real number greater than $0$ but less than any other positive real number.

My problem is how to express the above in logical notation. Some of my attempts follow:

1. Domain $\mathbb{R}^+$
$\{d : x \geq d\}$
In the domain of positive reals $d$ (the infinitesimal) is the positive real such that all positive reals are either equal to $d$ or greater than $d$

2. $\forall x \exists y ((x, y \in \mathbb{R}^+ \wedge x \ne y) \to y < x)$
For all x there exists a y such that if x and y are positive reals and x is not equal to y then y is less than x Here y is $d$

3. $d \in \mathbb{R}^+ \wedge \forall x ((x \in \mathbb{R}^+ \wedge x \ne d) \to d < x)$
$d$ is a positive real AND for all x such that x is a positive real not equal to $d$ then d is less than to x.

Are all the above correct definitions for the infinitesimal $d$?

 

[Hornbein]

The system that uses infinitesimals is called "nonstandard analysis". It is usually not taught and I haven't learned it. Standard analysis uses limits instead, that's what I know.

I'm fairly sure infinitesimals are not real numbers. There cannot be a smallest real number greater than zero.

 

[Agent Smith]

The system that uses infinitesimals is called "nonstandard analysis". It is usually not taught and I haven't learned it. Standard analysis uses limits instead, that's what I know.

I'm fairly sure infinitesimals are not real numbers. There cannot be a smallest real number greater than zero.

I would have to agree regarding your last sentence, and I didn't know infinitesimals are still being used in math (in nonstandaard analysis).

What I'd like to know is how to define the infinitesimal using logical notation.

What do you think about $d = 1 - \displaystyle \sum_{n = 1} ^\infty \frac{9}{10^n}$?

$1 - \displaystyle \sum_{n = 1} ^{\infty - 1} \frac{9}{10^n}$

 

[Hornbein]

I dunno. I wouldn't try to reinvent it myself.

 

[PeroK]

TL;DR Summary: Defining the Infinitesimal

Cobbling together a definition of the infinitesimal from bits and pieces of info gathered from books and the internet:

As mentioned above, there are no infinitesimals in modern calculus. Earlier naive ideas of an infinitesimal were replaced by the "epsilon-delta" methods of real analysis.

In the 1960's, the system of Hyperreal numbers was developed. This system includes the concept of infinitesimals and non-standard analysis:

https://en.wikipedia.org/wiki/Hyperreal_number

https://en.wikipedia.org/wiki/Nonstandard_calculus

Some people believe that the hyperreals are more intuitive than the real numbers, but they seem to me to add another layer of logical complexity to the idea of a number. And I'm not sure how well they extend beyond a study of the real numbers.

 

[Agent Smith]

@Hornbein & @PeroK I studied the $\varepsilon - \delta$ definition of a limit and I still can't wrap my head around it. My take on it is that given an interval that contains $f(x)$, we can find a smaller interval that contains $f(x)$ and so on ... indefinitely. Acknowledges the absence of/eliminates the need for the infinitesimal.

 

[PeroK]

@Hornbein & @PeroK I studied the $\varepsilon - \delta$ definition of a limit and I still can't wrap my head around it.

Most students struggle with real analysis. If you are learning real analysis, you should start with sequences first.

My take on it is that given an interval that contains $f(x)$, we can find a smaller interval that contains $f(x)$ and so on ... indefinitely. Acknowledges the absence of/eliminates the need for the infinitesimal.

I'm not sure that means anything.

 

[Gavran]

1. Domain $\mathbb{R}^+$
$\{d : x \geq d\}$
In the domain of positive reals $d$ (the infinitesimal) is the positive real such that all positive reals are either equal to $d$ or greater than $d$

wrong
d/2 is a positive real number and d/2 is less than d.

2. $\forall x \exists y ((x, y \in \mathbb{R}^+ \wedge x \ne y) \to y < x)$
For all x there exists a y such that if x and y are positive reals and x is not equal to y then y is less than x Here y is $d$

wrong
y/2 is a positive real number and y/2 is less than y.

3. $d \in \mathbb{R}^+ \wedge \forall x ((x \in \mathbb{R}^+ \wedge x \ne d) \to d < x)$
$d$ is a positive real AND for all x such that x is a positive real not equal to $d$ then d is less than to x.

wrong
d/2 is a positive real number and d/2 is less than d.

An infinitesimal can not be a positive real number for the same reason as an infinite can not be a positive real number.

What do you think about $d = 1 - \displaystyle \sum_{n = 1} ^\infty \frac{9}{10^n}$?

wrong

$1 - \displaystyle \sum_{n = 1} ^{\infty - 1} \frac{9}{10^n}$

wrong

 

[Mark44]

What do you think about
$d = 1 - \displaystyle \sum_{n = 1} ^\infty \frac{9}{10^n}$?

d = 0.

$1 - \displaystyle \sum_{n = 1} ^{\infty - 1} \frac{9}{10^n}$

The summation is ill-defined because $\infty - 1$ is meaningless. The symbol $\infty$ can't be used in any arithmetic expression.

 

[fresh_42]

TL;DR Summary: Defining the Infinitesimal

Cobbling together a definition of the infinitesimal from bits and pieces of info gathered from books and the internet:

The infinitesimal $d$ is the positive real number greater than $0$ but less than any other positive real number.

My problem is how to express the above in logical notation. Some of my attempts follow:

1. Domain $\mathbb{R}^+$
$\{d : x \geq d\}$
In the domain of positive reals $d$ (the infinitesimal) is the positive real such that all positive reals are either equal to $d$ or greater than $d$

2. $\forall x \exists y ((x, y \in \mathbb{R}^+ \wedge x \ne y) \to y < x)$
For all x there exists a y such that if x and y are positive reals and x is not equal to y then y is less than x Here y is $d$

3. $d \in \mathbb{R}^+ \wedge \forall x ((x \in \mathbb{R}^+ \wedge x \ne d) \to d < x)$
$d$ is a positive real AND for all x such that x is a positive real not equal to $d$ then d is less than to x.

Are all the above correct definitions for the infinitesimal $d$?

The search key "Introduction hyperreals + pdf" has brought me Hyperreals and a Brief Introduction to Non-Standard Analysis, Gianni Krakof, 2015.

I think this provides a good summary of the matter. E.g., it says on page 8 ...

Infinitesimals are defined by sequences of real numbers approaching $0$, for instance, we may take $\varepsilon= \left(\frac{1}{2},\frac{1}{4},\frac{1}{8},\ldots\right).$ This number is less than any real number in the hyperreals.

... and defines an unlimited element right after this. It also lists the arithmetic rules that can be used on page 9. This shows, that an infinitesimal represents a sequence and cannot be defined the way you tried to without caring more about the logical notation that you used, e.g. $<.$ If you try to make this rigorously then you have to proceed more along the lines of the sources I link to in this post, especially with the topological aspect of hyperreals.

I would prefer that over Wikipedia. A quick view on nLab reveals that non-standard analysis is all but trivial.

Most important if you deal with hyperreal numbers is the Transfer Principle:

​

Any appropriately formulated statement $\phi$ about $\mathbb{R}$ holds if and only if ${}^*\phi$ holds for ${}^*\mathbb{R}.$

The transfer principle justifies why hyperreals can be used in analysis instead of ordinary real numbers with limits and the epsilon framework. However, you can also read this the other way around: why use hyperreals if ordinary analysis already does the job? Students are better trained to cope with limits than they are prepared for the abstract concept of hyperreal numbers. A closer look also shows that they are far more a logical and topological subject than an analytical approach, e.g. An Introduction To Nonstandard Analysis, Isaac Davis, 2009.

 

[Agent Smith]

@Gavran and @Mark44
I had high hopes for $d = 1 - \displaystyle \sum_{k = 1} ^ \infty \frac{9}{10^k}$.
This is the definition I'm familiar with: An infinitesimal is the smallest positive real number. So does that become $\forall x(x \in \mathbb{R}^+ \to d < x)$. Thanks for pointing out how this is wrong ($d/2$), but I'd like to know if the definition is rendered correctly in logic.

@fresh_42 and @PeroK gracias for introducing nonstandard analysis to me.

 

[PeroK]

This is the definition I'm familiar with: An infinitesimal is the smallest positive real number.

There is no smallest positive real number. As far as standard pure mathematics is concerned, there is no such thing as an "infinitesimal". This is really important, since working with the Real numbers requires the property that there is no smallest number. There is not even a smallest rational number.

I had high hopes for $d = 1 - \displaystyle \sum_{k = 1} ^ \infty \frac{9}{10^k}$.

Wherever you have learned this, you'll have to unlearn it to make any progress in pure mathematics. Using the sum of an infinite geometric series, you can calculate that $d = 0$.

 

[Agent Smith]

@PeroK "there is no smallest positive real number", ok but what about the definition (as written in logic). Is that correct?

 

[PeroK]

@PeroK "there is no smallest positive real number", ok but what about the definition (as written in logic). Is that correct?

I'm not sure what you mean by that. I'm not completely familiar with the notation of formal logic.

 

[pbuk]

@PeroK "there is no smallest positive real number", ok but what about the definition (as written in logic). Is that correct?

Which definition - this one?

$\forall x(x \in \mathbb{R}^+ \to d < x)$

I would consider that to be a property of an infinitessimal rather than a definition. Another property of an infintessimal $\varepsilon$ is that $\forall n \in \mathbb N, \varepsilon < \frac 1 n$. Another perhaps surprising one (or rather pair: this pair is sometimes used as a loose definition) $\varepsilon > 0 \land \varepsilon^2 = 0$.

In maths we tend to work with definitions of sets rather than individual elements of a set, but for the hyperreals (of which the infinitessimals are a subset) the set definitions don't really help us. More useful is a way of constructing (rather than defining) hyperreals and we generally do this using sequences as explained in

Hyperreals and a Brief Introduction to Non-Standard Analysis, Gianni Krakof, 2015.

I studied the $\varepsilon - \delta$ definition of a limit and I still can't wrap my head around it.

I can understand this, many people struggle with it. However the concepts in the paper linked to above are much harder to grasp so I suggest that you concentrate on working with the reals before you look at any extension of them (and when you have improved your understanding of the reals the first extension to look at is the complex numbers which are much more useful than the infinitessimals or transfinites).

 

[fresh_42]

@PeroK "there is no smallest positive real number", ok but what about the definition (as written in logic). Is that correct?

Firstly, do not use $d.$ It is overloaded by other meanings already: a distance (metric), part of the expression $dy/dx$ which refers to a limit in ordinary analysis (Leibniz notation), a derivation (obeys the product rule), a differential form (sometimes), and possibly even more. We usually use $\varepsilon$ for infinitesimals and $\omega$ for unlimited large elements in non-standard analysis. Note, that people make a distinction between Leibniz's $d$ and $\varepsilon .$ The latter refers to the $\varepsilon>0$ in classical calculus and is a new element in the context of hyperreals, not the $d$ within the definition of a derivative.

Secondly, ..

I would consider that to be a property of an infinitesimal rather than a definition.

... this.

This shows, that an infinitesimal represents a sequence and cannot be defined the way you tried to without caring more about the logical notation that you used, e.g. <.

The paper I quoted defines infinitesimals as sequences that converge to zero, not as an arithmetic expression. Ordinary numbers are defined as constant sequences, e.g. $\pi=(\pi,\pi,\pi,\ldots).$ This shows that we are in a different environment than we were if we used ordinary notations. It automatically invokes the necessity to reconsider notation. E.g. infinitesimals are no longer unique! We have many infinitesimals instead. That's why I said that the system of hyperreals is primarily a logical challenge.

I had high hopes for $d = 1 - \displaystyle \sum_{k = 1} ^ \infty \frac{9}{10^k}$.

If at all, you will have
$$
\varepsilon= \left(1-\sum_{k=1}^n \dfrac{9}{10^k}\right)_{n\in \mathbb{N}}.
$$

@Hornbein & @PeroK I studied the $\varepsilon - \delta$ definition of a limit and I still can't wrap my head around it.

I think it would be better to work on that than to seek a bypass that uses even more abstract concepts and which leads you deep into logic and topology instead of analysis. Here is an article I wrote about epsilontics: https://www.physicsforums.com/insights/epsilontic-limits-and-continuity/ It may help you or not, but at least you can use it to ask questions without having to refer to an exotic part of mathematics. The concept is by far less complicated if you first phrase it by words and next translate it into epsilontics.

• A limit of a sequence is a number (sic!) to which the sequence members come arbitrarily close with increasing indices.
• Continuity is if you can draw a function graph in one line without cuts.
• Differentiation is when a sequence of secants becomes a tangent.

These wordy descriptions have their inconsistencies and disadvantages as do every linguistic description, and which is why we use the $\varepsilon-\delta$ and limit framework. Nevertheless, they are a good starting point to understand the why.

 

[pbuk]

Firstly, do not use $d.$ It is overloaded by other meanings already

Yes absoultely: you will see that I used $\varepsilon$ to represent an infinitessimal. In particular note that the "d" in $\frac{d}{dx} f(x)$ is sometimes referred to as an infinitessimal quantity but is not a member of the infinitessimals which will cause you a great deal of confusion if you go down this path.

My advice to you is to forget the infinitessimals

 

[Agent Smith]

We have many infinitesimals instead

?

 

[fresh_42]

If we define an infinitesimal as a sequence that converges to zero then there are many. $(2^{-n})_{n\in \mathbb{N}}$ is an infinitesimal, but $(3^{-n})_{n\in \mathbb{N}}$ is one, too. And of course many others. Krakoff defines sums and products of infinitesimals on page 9. Sums and products of zero sequences are again zero sequences. But already the question of whether they should be regarded equivalent - not equal! - leads us into the world of topology (page 5f.).

Infinitesimals and unlimiteds cannot be treated as usual numbers - they aren't. Regarding them as sequences is one possible way to deal with them. Another one would be an axiomatic approach. Both require abstract concepts beyond our common understanding of numbers. It all starts with the question: what even is a real number? Krakoff uses the approach via Cauchy sequences and sticks with them when he defines infinitesimals. This is easier than an approach via Dedekind cuts.

As I already mentioned: bypassing epsilontics and limits by using hyperreals is even more complicated than fighting your way through it. Hyperreals might be tempting. For instance, the sum or product of two infinitesimals is again an infinitesimal. That sounds intuitively obvious. However, the moment you want to actually calculate something and do analysis, you will have to find arithmetic rules, ordering, and completeness. As soon as we establish a rigorous basis for all these properties we will find ourselves in a logical and topological jungle that outplays the advantage of intuition. We have a macabre saying here: we have to die one death. Either epsilontic and limits, or logic and topology.

The modern version of calculus is not very old. Smooth functions have long been simply series expansions, epsilons have been infinitesimals, unlimiteds has been divergence, even without being treated rigorously. And physicists still use this language today without referring to hyperreals! The epsilon calculus replaced that with a rigorous framework. Of course, you can try to use infinitesimals as sloppy. But don't forget that physicists have learned epsilontic in detail before they used infinitesimals sloppy. I have found an interesting text from 1911 about Lie groups (correction: invariant transformations) that used infinitesimals a lot. Read my summary
https://www.physicsforums.com/insights/when-lie-groups-became-physics/
and figure out whether this helps you to understand the matter or makes it even more copmplicated.