Mathematical Association of America

Seaway Section Conference

Rochester, New York

October 20, 2007

Updated: October 30, 2007

**INFINITESIMALS IN MODERN MATHEMATICS**

http://www.jonhoyle.com/MAAseaway

**Jonathan W. Hoyle**

**ABSTRACT**

This talk will tour the various modern mathematical models of
infinitesimals. After a
brief historical overview of their use by mathematicians over the
centuries, modern definitions will be introduced.
By extending the ordered field of the reals, Non-Archimedean
values appear, and depending upon the construction, standard theorems
from analysis may produce very unusual results.
An introduction of Nonstandard Analysis will be made first,
outlining the constriction of Hyperreals and an overview of
interesting results pertaining to this structure.
This will be followed by a tour of other mathematical systems
involving infinitesimals, including Surreal numbers, Superreal
numbers, Super-real fields and Smooth Infinitesimal Analysis.

**OUTLINE**

1. Introduction & History2. The Hyperreals (Nonstandard Analysis)

2.1 R

^{∞}2.2 Equivalence Relationship

2.3 Infinities Both Great and Small

2.4 Hyperreal Terminology & Results

2.5 Internal Sets & the Transfer Principle

2.6 Nonstandard Proofs

2.7 Hyperreal Incompleteness

3. Other Approaches

3.1 The Surreal Numbers

3.2 The Superreal Numbers [Tall]

3.3 The Super-Real Numbers [Dales & Woodin]

3.4 Smooth Infinitesimal Analysis

4. Conclusion

5. Further Reading

**1. INTRODUCTION
& HISTORY**

What is an infinitesimal? The
mathematical definition we will use here is:

A non-zero number ε is aninfinitesimalif |ε| <^{1}⁄n for all n ∈N.

(Some treatments define infinitesimals to include 0,
whilst others do not. For
the purposes of this paper, we will assume it does not, although we
will allow for exceptions as needed.)
It is easily demonstrated that no infinitesimals exist within
the field of the real numbers **R**.
If ε were an infinitesimal in **R**, then ^{1}⁄ε
∈ **R** as well. Without
loss of generality, assume ε were positive.
If we let k be any integer greater than ^{1}⁄ε,
then ε > ^{1}⁄k,
which is a contradiction. This
simple proof was known to the ancients and would seem to indicate
that infinitesimals simply do not exist.
Despite this, infinitesimals have had a long and somewhat
controversial involvement in mathematics.

The use of infinitesimals can be traced back thousands of years and
has been fraught with logical difficulties and paradoxes from the
beginning. Archimedes used
them for his thought experiments, although he removed them for his
final rigorous proofs. Both
the infinite and infinitesimal were considered inherently
inconsistent concepts at the time and were generally avoided.
It was during the development of Calculus that their use
became pragmatically necessary. Despite
Newton's genius in the discovery of Calculus, he was unable to find a
way around postulating their existence.
Newton could not adequately defend his inconsistent treatment
of infinitesimals (sometimes as zero, sometimes as non-zero), as
Bishop Berkeley rightly criticized.
It wasn't until 19th century mathematician Karl Weierstrass
did Calculus receive a rigorous treatment, removing the references to
infinitesimals. At that
point, infinitesimals were all but banished from the mathematical
world.

In the 1960's, mathematician and logician Abraham Robinson
investigated ways in which the concept of the infinitesimal could be
revived in a rigorous and consistent manner, avoiding the logical
paradoxes which plagued their earlier use.
His success in this led to the new branch of mathematics
called *Nonstandard Analysis* (NSA).
Over the next few decades, other consistent formulations of
infinitesimals in number systems were developed, including John
Conway's *Surreal Numbers* in the 1970's, David Tall's
*Superreal Numbers* in the 1980's, and most recently the
*Super-Real Fields* of Dales & Woodin and *Smooth
Infinitesimal Analysis* popularized by John L. Bell.

For more on the history of the infinitesimal, visit this Stanford
page and this Wikipedia
entry.

The purpose of this paper is to tour these various approaches of the
modern use of the infinitesimal. A
more detailed introduction to Nonstandard Analysis will be given
first, as this is by far the most common usage of infinitesimal
analysis. This examination
will include Hyperreal foundations, results and highlights.
Following this, we will tour four other approaches involving
infinitesimals. Since many
of the concepts of these other approaches are similar with those in
NSA, these sections are shorter overviews.

**2. THE
HYPERREALS (NONSTANDARD ANALYSIS)**

In 1965, Abraham Robinson introduced Nonstandard Analysis, a
mathematically robust construction of a Non-Archimedean extension of
the real numbers. This
extension, called the Hyperreals, denoted ***R**, includes both
infinitesimal and infinite elements.
Since Robinson's original work, there have been further
developments of Nonstandard foundations.
Today there are essentially three foundational approaches:
Robinsonian superstructures, Nelson's axioms to Internal Set Theory,
and Keisler's elementary and more intuitive axioms.
These three are nicely compared and contrasted by K. D.
Stroyan in *The Infinitesimal Rule of Three* published in
*Developments in Nonstandard Mathematics* [1995, ISBN:
0582279704]. Each
approach has its advantages, and the Stroyan article is recommended
for anyone serious about the study of NSA.

In this presentation, I will be constructing ***R** as outlined by
Robert Goldblatt's *Lecture on the Hyperreals* [1998, ISBN:
038798464X], as this is both concrete and rigorous.

**2.1 R**^{∞}

First we begin with **R ^{∞}**, the set of ordered
infinite sequences of

< 0, 0, 0, 0, … >< 1, 2, 3, 4, … >

< 1, 0, 1, 0, … >

< 2, 3, 5, 7, … >

< -1, π, 0.0001, 10

^{10}, √17, … >

Every possible sequence of real numbers, convergent or
otherwise, are members of **R ^{∞}**.
We will also use as a convention the subscript notation to
refer to specific elements of a sequence.
Thus, for x ∈

**
**Standard arithmetic operations will be extended to

a + b = < a_{0}+ b_{0}, a_{1}+ b_{1}, a_{2}+ b_{2}, … >a - b = < a

_{0}- b_{0}, a_{1}- b_{1}, a_{2}- b_{2}, … >a ⋅ b = < a

_{0}⋅ b_{0}, a_{1}⋅ b_{1}, a_{2}⋅ b_{2}, … >a / b = < a

_{0}/ b_{0}, a_{1}/ b_{1}, a_{2}/ b_{2}, … >a

^{b}= < a_{0}^{b}^{0}, a_{1}^{b}^{1}, a_{2}^{b}^{2}, a_{3}^{b}^{3}, … >

Where operations are not defined, such as division by 0
or 0^{0}, we will informally say that
the indices of the sequence in which they occur are also undefined.
Finally, we will extend functions over **R** to include
**R ^{∞}** by the following convention: If

f(a) = < f(a_{0}), f(a_{1}), f(a_{2}), … >

**2.2 EQUIVALENCE
RELATIONSHIP**

At this point in our construction, we have a fully usable **R ^{∞}**
structure, but this alone is not sufficient for our definition of

We would also like to be able to consider two sequences as
"equivalent" if they are equal at "almost all" indices.
For example, the sequences < 3, 3, 3, 3, … > and
< 3, 0, 3, 3, … > differ only at the second index and
should be considered in all other ways identical.

We wish to define an equivalence relationship ~ on **R ^{∞}**
such that < a

1. All subsets ofNare either "large" or "small", but not both.2. All finite sets are "small".

3. All cofinite sets are "large".

4. The complement of a "large" set is "small", and vice-versa.

Furthermore, we need to ensure that our definition of "largeness" supports transitivity of ~. In other words, we wish that whenever a ~ b and b ~ c, that a ~ c always holds. This implies that the intersection of two "large" sets is always "large".

This can be accomplished by using what's called a *non-principal
ultrafilter* on **N**. It
is not important for the moment to know how a non-principal
ultrafilter on **N** is constructed; it suffices to accept only
that one exists. (For
those interested in understanding more about ultrafilters, visit
this Wikipedia
page.)

Now we are in the position to compose our equivalence classes.
We write a = b whenever the agreement set **E** = { i | a_{i}
= b_{i} } is "large" and a ≠ b if
**E** is small. Likewise,
we can define a < b whenever the agreement set { i | a_{i}
< b_{i} } is large, etc.
Moreover, any predicate on real numbers which returns truth or
falsity can be extended in this fashion.

The set of equivalence classes we have defined in this quotient ring
of **R ^{∞}** is our set

Since this definition of ***R** is dependent upon our choice of
ultrafilter, there are an infinite number of possible ***R**
constructions from which to choose.
How different are these various ***R**'s?
The answer depends on some set theoretical considerations.
If one assumes the Continuum Hypothesis, then the choice of
ultrafilter is irrelevant, as all possible ***R**'s are
isomorphic. Without the
assumption of the Continuum Hypothesis, the situation remains at this
time undetermined.

**2.3 INFINITIES
BOTH GREAT AND SMALL**

We are now ready to look at some properties of ***R**.
As we have already seen, ***R** contains all of **R**,
since each r ∈ **R** is a member of a distinct equivalence
class < r, r, r, … >.
What we will now show is that ***R** is a proper superset
of **R**. Let us
define:

ω = < 1, 2, 3, 4, … >

To see that ω is infinite, note that for any natural number n, the agreement set { i | i > n } is cofinite, and thus ω > n for all n. Likewise, we can consider the inverse of ω, called ε:

ε =^{1}⁄ω = < 1,^{1}⁄2,^{1}⁄3,^{1}⁄4, … >

By similar inspection, we see that ε is smaller
than any positive real number and thus is infinitesimal.
With exponentiation operations defined in ***R**, there are
examples of increasingly small hyperreals (ε^{2},
ε^{100}, ε^{ω},
etc.) as well as ones which are increasingly large (ω^{2},
ω^{ω}, ω^{ω}^{ω},
etc.). We can even
calculate combined infinite and infinitesimal arithmetic, such
as:

ω^{ε}= <1.000…, 1.414…, 1.442…, 1.414…, 1.379…, 1.379…, 1.348… >

**2.4 HYPERREAL
TERMINOLOGY & RESULTS**

Below are some interesting results generated from NSA:

1. We sayxisinfinitely closetoy, ifx - yis infinitesimal, writtenx ≈ y. (For example, ω^{ε}≈ 1).2. The set of hyperreals infinitely close to

xis called thehaloofx, denotedhal(x).3. The set of infinitesimals is simply

hal(0).4. If

x∈*R,r∈Rand x ≈ r, then we callrtheshadowofx, denotedr = shd(x).5. The set of all hyperreals a finite distance from x is called the

galaxyofx, denotedgal(x).6. The set of finite hyperreals is simply

gal(0).7. Every finite hyperreal is infinitely close to

exactly onereal number.8. For

r∈R,x∈*R,x ≈ r, we callrthestandard partofx.9. For

r∈R,x∈*R,x ≈ r, we callx - rthenonstandard partofx.10. All finite hyperreals can be uniquely expressed as

r + iwhereris a real number andiis an infinitesimal.

**2.5 INTERNAL
SETS & THE TRANSFER PRINCIPLE**

Sets within Nonstandard Analysis are divided into two types:
*internal* sets and *external* sets.
Internal sets are those which have analogs in standard
analysis over the *-transformation.
For example, the *-transformation maps the unit interval U =
[0,1] in **R** to the internal set *U = [0,1] in
***R**. Although U and
*U may appear identical at first glance, they are quite different
sets. For one thing, *U
contains infinitesimals whereas U does not.
The set U of standard reals also exists in ***R**, but it
is an external set not an internal one.

The Transfer Principle says that statements and properties true of
standard analysis have "equivalent" statements and properties true in
Nonstandard Analysis. These
equivalents, however, involve only internal elements of NSA via the
*-transformation. A
concrete example may help elucidate this point.
Consider the Archimedian Property of real numbers:

For anyx, there existsn∈Nsuch thatn > x.

In **R**, this property true, but in ***R** it is
false as currently stated. To
get the true equivalent in NSA, we must replace the sets described
within the statement into their *-transformations.
Doing so produces the following true statement of
***R**:

For anyx, there existsn∈*Nsuch thatn > x.

**2.6 NON-STANDARD
PROOFS**

If the Transfer Principle implies that NSA contains all of the same
theorems of classical analysis, what is the benefit gained from
studying NSA? Although it
is true that the theorems of standard analysis hold true in NSA
(under the *-transformation), it is untrue that only *- transferred
theorems are theorems of NSA. Theorems
involving external sets are one without classical analogs.
One such theorem is: the cardinality of ***N** is equal to
the cardinality of **R** (the size of the continuum).
Such results are not demonstrable from the Transfer Principle
alone.

More importantly, however, are the pedagogical benefits of using
Nonstandard proofs. Nonstandard
proofs are typically more intuitive than standard proofs.
Consider, for example, the standard definition of
continuity:

A function f iscontinuousat a pointxif: for every_{0}ε > 0, there existsδ > 0such that |x - x|_{0}< δ→ |f(x) - f(x_{0})|< ε.

Such a definition obfuscates the intended conceptual meaning of continuity. Students must memorize a mathematical formulation, rather than understand the concept. Worse still, the standard "delta-epsilon" proofs are often exceedingly difficult for novice to follow.

Consider instead the nonstandard definition of continuity:

A function f iscontinuousat a pointxif: when_{0}x ≈ xthen_{0}f(x) ≈ f(x_{0}).

Memorization is less important, as they key concept is
made obvious: continuity essentially means that infinitely close
points map to infinitely close points.
For *f* to be continuous at *x** _{0}*,
it is sufficient to show that

Nonstandard proofs tend to be much shorter as well.
Let us compare two proofs for the continuity of the function
*f(x) = ax + b*. The
first will be a standard delta-epsilon proof, and the second will be
a nonstandard proof.

Standard Proof: Let
*x** _{0}* be arbitrary.
We note that

|f(x) - f(x_{0})|

=|ax + b - ax_{0}- b|

=|ax - ax|_{0}

=|a| ⋅|x - x|_{0}Therefore, by choosing

δ = ε/|a|, we have:|

x - x|_{0}< δ⇒ |

x-x|_{0}< ε/ |a|⇒ |

a| ⋅|x -x|_{0}< ε⇒ |

ax-ax| <_{0}ε⇒ |

ax+b-ax_{0}- b| <ε⇒ |

f(x) - f(x_{0})| <εSince

xwas arbitrary, we have this holding for all_{0}x.Q. E. D.

Nonstandard Proof: Let
*i* be an arbitrary infinitesimal.
We note that

f(x + i)

= a(x + i) + b

= (ax + b) + ai

= f(x) + ai

≈ f(x)Q. E. D.

**2.7 HYPERREAL
INCOMPLETENESS**

It is natural to suppose that by enlarging the reals to include
additional elements, we have somehow made the number line more
complete. It then may come
as a surprise to you that the opposite is the case: ***R** is, in
fact, *less* complete than **R**.

To understand this better, it is worth looking at another well known
incomplete set: **Q**, the set of rationals.
Consider the equation *f(x) = x*^{2}*
- 2*. For no *q* ∈
**Q** do we have *f(q) = 0*.
One can choose rationals that can get *f* as close to 0
as we like, but no rational number will get to 0 exactly.
In other words, **Q** has a "hole" in it where √2
should be.

Dedekind formalized the concept of completeness in the following
manner: A partially ordered set is *complete* if every subset
with an upper bound has a least upper bound.
Take for example, the set T = { 1, 1.4, 1.41, 1.414, 1.4142, …
}. It is bounded above,
but T has no least upper bound in **Q**.
However, T has a least upper bound in **R**, namely √2.
This is because **R** is a complete ordered field, whereas
**Q** is an incomplete ordered field.

It is easy to prove that ***R** is incomplete as well:

Theorem: ***R** is Dedekind Incomplete.

Proof: Let **I** = the set of infinitesimals.
Note that any positive real number is an upper bound of
**I**. Assume **I**
has a least upper bound *δ*.
Either *δ* ∈ **I **or** ***δ*
∉ **I**. If *δ*
∈ **I**, then *2δ* ∈ **I**.
But *2δ > δ*, which contradicts the
assumption that *δ* is a least upper bound.
So assume *δ* ∉ **I**.
Then *δ/2* ∉ **I**, and so *δ/2*
is an upper bound of **I** as well.
But since *δ/2 < δ, *δ cannot be the
least upper bound. Therefore,
I has no least upper bound, and hence ***R** is incomplete.

So then a question that might arise is this: Can one "complete"
***R** by continually appending missing elements to it?
Unfortunately, this has an unsatisfying answer.
It has been proven that every complete ordered field is
isomorphic to **R**. Thus,
to "complete" ***R**, the elements needing to be added are
precisely those which remove the gaps and distinctions between the
infinitesimals, the finite and the infinite.
You essentially get back **R** (or something
indistinguishable from it).

Although incomplete, ***R** does however have the property of
being *internally complete*: that is, all *internal*
subsets of ***R** with an upper bound have a least upper bound.
This is due to the Transfer Principle.
The converse of this can be used to prove the externality of
some sets. For example:
**I**, **N**, and **R** each has an upper bound but no least
upper bound; therefore, each of these sets must be external.

Interestingly, hyperreal incompleteness implies that there are
equations in ***R** which have solutions for internal sets but not
for external ones. For
example, consider the set function S(X) as:

S(X)= ∑_{n∈X}^{1}⁄2^{n}, for X a subset of*N.

*S(X)* has a solution for any internal X, eg:
*S(***N**)* = 2.
However, *S(X)* does not have a solution for every
external set. X = **N**
has no solution, as *S(**N**)* exceeds any
value in *hal(x)* for *x* < 2, but is always strictly
less than any value in *hal(2)*.
*S(**N**)* essentially "points" to a
hole in ***R**.

**3. OTHER
APPROACHES**

Discussions involving the modern use of infinitesimals usually refer
to those in the hyperreal system of Nonstandard Analysis.
However, the past few decades of mathematical research have
developed alternative approaches to extending **R** which include
non-Archimedean values. We
will tour four of them: Conway's *Surreal* numbers, Tall's
*Superreal* numbers, Dales & Woodin's *Super-real*
numbers and finally Bell's *Smooth Infinitesimal Analysis*.

**3.1 SURREAL
NUMBERS**

John Conway's construction of surreal numbers is reminiscent of
Dedekind cuts used to generate real numbers.
The surreal construction is recursive cut of two sets of other
surreal numbers.

Construction Rule: If *L* and *R* are two sets of surreal
numbers, such that no member of *R* is less than or equal to any
member of *L*, then the ordered pair denoted { *L* |
*R* } is a surreal number.

Essentially, a surreal number is two sets of surreals with the left
side *L* being smaller than the right side *R*.
We also allow *L* and/or *R* to be empty.
In the case where *L* is the empty set, we simply write {
| *R* } for { ∅ | *R* }; likewise { *L* | } for
{ *L* | ∅} and { | } for { ∅ | ∅ }.
We will define the surreal number 0 as { | }.
To define "larger" and "smaller", we need an inequality
definition:

Comparison Rule: For two surreal numbers a and b, where a = {
*a** _{L}*,

Essentially, all the elements in *L* need to be greater than the
elements in *R* for { *L* | *R* } to be a valid
surreal number (allowing for the empty set in either *L* or
*R*. Equality is
defined, as you would expect as follows: a = b if a ≤ b and b ≤
a.

As stated above, we begin by postulating our first surreal number 0 =
{ | }. With 0 in place, we
can now apply the first recursive iteration, generating these three
potential numbers: { 0 | }, { | 0 } and { 0 | 0 }.
This final one is not a valid surreal number since 0 ≤ 0,
so we are left with the first two newly generated surreal numbers,
which we name 1 and -1 respectively.

For the second iteration, all combinations of 0, 1 -1 and the empty
set are examined. Many of
them, such at { 0 | 1, -1 }, are not valid surreal numbers, since
they fail the comparison rule. Others
turn out to be equal to equivalent classes we already have, such as {
-1 | 1 } = { | } = 0. However,
four new equivalence classes are generated: { 1 | }, { | -1 }, { 0 |
1 } and { -1 | 0 }. These
we name 2, -2, ^{1}&frasl2,
-^{1}&frasl2
respectively.

The third iteration produces the surreal numbers 3, -3,
^{1}&frasl4,
^{3}&frasl4,
-^{1}&frasl4
and -^{3}&frasl4.
In general, the nth generation will produce ±n, and all
multiples of ±^{1}&frasl2^{n}.
We define a surreal number's *birthday* as the index
number corresponding to the iteration in which it is generated.
For example, the birthday of -3 is 3, and the birthday of ^{9}⁄16
is 5.

We define S_{i} as the set of all
surreal numbers whose birthday is ≤ i.
We define S_{ω} as the union of all S_{i}
for i ∈ **N**. Do
we have all the numbers we want in S_{ω}?
Unfortunately, most of the members of** R **remaining
missing, even at S_{ω}.
This is because at no finite iteration do rational numbers
like ^{1}&frasl3
appear. The solution is to
extend this process through transfinite induction, yielding the
missing numbers. In
general, we allow birthdays of any transfinite ordinal number.

Our very next birthday, ω+1, turns out to be very interesting.
As you might expect, the remainder of our rational numbers
come in at this point. What
is surprising is that ω+1 is also the birthday of all the
remaining reals as well! It is a bit unexpected that transcendental
numbers like π share the same birthday as a pedestrian rational
such as ^{1}&frasl3.
It doesn't stop there, as ω+1 is the birthday of our
first infinitesimal! The
surreal number defined as

ε = { 0 | …^{1}&frasl8,^{1}&frasl4 ,^{1}&frasl2 , 1 }

comes in iteration ω+1 and is provably an
infinitesimal, since ε < ^{1}⁄n
for all finite n. Its
inverse is also generated at this point:

ω = { S_{ω}| } = { 1, 2, 3, … | }

With additional iterations, more and more infinitesimals
and infinite numbers become generated.
It turns out that the entirety of **On** (the class of
ordinals) are eventually generated and become members of the class of
surreals. Moreover, new
and unusual numbers never defined before begin appearing in later
transfinite iterations, such as strange beasts as:

√ω = { 1, 2, 3, … | …^{ω}⁄8,^{ω}⁄4,^{ω}⁄2, ω }log ω = { 1, 2, 3, … | …

^{4}√ω,^{3}√ω, √ω, ω }

It has been speculated that the surreal numbers encompass the largest class of numbers possible.

**3.2 SUPERREAL
NUMBERS ****[Tall]**

A less ambitious but much more accessible approach to defining
infinitesimals is one by David Tall from the University of Warwick.
His motivation was to create a system which was more intuitive
for students and to make Calculus concepts easier to grasp.
The simplicity of his approach is very appealing, as it
quickly gets to the use of infinitesimals without the large
construction found ***R**'s construction.

Tall begins with **R** and appends a postulated infinitesimal ε.
Other infinitesimals are generated by closure over the basic
arithmetic operations of +, -, ⋅, /, and the result is the set
of superreals R (not to be
confused with the super-reals of Dales & Woodin, described
below). Every superreal
can be uniquely expressed as:

z = a_{n}ε^{n}+ a_{n-1}ε^{n-1}+ … + a_{1}ε + a_{0}+ a_{-1}ε^{-1}+ … + a_{-m}ε^{-m}for a_{i}∈R, n,m ∈N

Note that whenever a_{i}
= 0 for all non-zero i, z is simply a real number.
Infinitesimals are of the form:

δ = a_{n}ε^{n}+ a_{n-1}ε^{n-1}+ … + a_{1}ε

By allowing ε to have an inverse, the superreals also contain infinite values as well. As this is less useful for Calculus purposes, one can also construct R such that ε is not invertible, and thus all superreals would be finite but still contain infinitesimals.

Tall's construction entails a pragmatic approach to function
extension. Whereas any
real function can be extended into the hyperreals, only analytic
functions are considered for superreal extension.
Since analytic functions are expressible as power series,
superreal functions can be defined by extending the series into the
infinitesimals. For
example:

sin δ = δ - δ^{3}⁄3! + δ^{5}⁄5! + …e

^{δ}= 1 + δ + δ^{2}⁄2! + δ^{3}⁄3! + …

Tall's superreals are algebraic in nature and do not
carry the baggage of complicated superstructures or the need for
First Order Logic. R
is not as all-encompassing as ***R**, but it does pragmatically
provide most of the intuitive benefits of NSA without the same
cognitive overhead. The
result is an Infinitesimal Calculus which is reminiscent of
Leibniz.

For more information on David Tall and his theory of superreals,
visit his
web site.

**3.3 SUPER-REAL
NUMBERS ****[Dales & Woodin]**

Another extension of real numbers calling itself *superreal* was
introduced by Garth Dales & Hugh Woodin in their 1996 text
*Super-Real Fields: Totally Ordered Fields with Additional
Structure* [ISBN: 0198536437].
The *super-reals* described therein should not be
confused with those of the same name by David Tall described above.
Conveniently, the two can be distinguished by observing the
convention that Dales & Woodin hyphenate the term whereas Tall
does not.

The super-real field is a more abstract extension of **R**, in
fact more abstract than even ***R**.
It begins with the ring C(X) of real-valued continuous
functions on a topological space X.
X can be any Tychonoff space, not necessarily **R**.
If P is a prime ideal of C(X), and A is a factor algebra of
C(X) / P, we let *F* be a quotient field of A strictly
containing **R**. If
*F* is not order-isomorphic to **R**, then *F* is a
super-real field. When P
is the maximal ideal, it is shown that F is the hyperreal field,
showing that super-real fields are more general than the hyperreal
ones.

The material presented is one of models and does not delve into the
particulars of Calculus. It
presumably shares a great deal in common with ***R**, since
superreals are essentially generalizations of hyperreal fields.

For more information on super-real fields by Dales & Woodin, view
this
American Mathematical Society review.

**3.4 SMOOTH
INFINITESIMAL ANALYSIS**

John L. Bell describes an unusual approach to the topic with Smooth
Infinitesimal Analysis (SIA). There
is no predefined construction involving models within **R**, as
there is with the hyperreals or surreals.
Instead, he introduces the concept of the *nilpotent
infinitesimal*, that is, a nonzero number so small that its square
is 0. The set of
infinitesimals is simply defined as:

I= {x|x= 0 }^{2}

In standard analysis, **I** = { 0 }; in SIA, there
exists nonzero elements of **I**.
These smooth infinitesimals are not invertible, and so do not
form a field. Thus, there
are no infinite values in SIA. The
non-invertibility of smooth infinitesimals prevents general division
by them, but this is compensated for by the Infinitesimal
Cancellation Law, which says:

For allx,y∈R, ifx⋅ ε =y⋅ ε thenx=y.

Although unusual, this system has some interesting
properties from the perspective of Calculus.
For example, for a function *f* and infinitesimal ε,
we have:

f(x +ε) = f(x) + f'(x)⋅ ε

Geometrically, this implies that functions are linear
over an infinitesimal interval. Curves
are essentially the concatenation of infinitesimally straight lines.
This is called the property of *microstraightness*.
A surprising result from microstraightness is the proof that
*all* functions are continuous and infinitely differentiable!
This is why this branch of analysis is called
*Smooth*.

There is no Transfer Principle in SIA, so many standard theorems of
classical analysis do not hold in this approach (for example, the
Intermediate Value Theorem is false).
The single major drawback to SIA is its inconsistency with the
*Law of the Excluded Middle*.
This deeply undermines the conceptual gains made by SIA,
leaving a great deal of cognitive friction.
Despite this deficiency, Smooth Infinitesimal Analysis has
generated a fascinating new perspective on the old subject of
Calculus.

For more information, view *An
Invitation to Smooth Infinitesimal Analysis*.

**4. CONCLUSION**

Abraham Robinson introduced a rigorous system of infinitesimals into
modern mathematics, avoiding the pitfalls, contradictions and
paradoxes which plagued their use in centuries past.
In the four decades since, much more investigation and study
has been accomplished with greater understanding of Nonstandard
Analysis. New alternative
number systems involving infinitesimals are already in use, including
Conway's surreal numbers and Bell's nilpotent infinitesimals.
The 21^{st} century surely holds new and exciting
developments in this area.

**5. FURTHER
READING**

Nonstandard Analysis:Goldblatt,

Lectures on the HyperrealsRobinson,

Non-standard AnalysisDavis,

Applied Nonstandard AnalysisCutland et al,

Developments in Nonstandard MathematicsVath,

Nonstandard AnalysisStroyan & Luxemburg,

Introduction to the Theory of InfinitesimalsHermann,

Nonstandard Analysis: A Simplified ApproachKim,

Nonstandard Analysis and ApplicationsLutz & Goze,

Nonstandard AnalysisDiener & Diener,

Nonstandard Analysis in PracticeHurd & Loeb,

An Introduction to Nonstandard Real AnalysisWikipedia,

Hyperreal numberWikipedia,

Non-standard Analysis

Surreal Numbers:

Conway,

On Numbers and GamesAlling,

Foundations of Analysis Over Surreal Number FieldsGonshor,

An Introduction to the Theory of Surreal NumbersWikipedia,

Surreal Number

Superreal Numbers [Tall]:

Tall,

Intuitive Infinitesimals in CalculusTall,

Mathematical IntuitionTall,

The Notion of Infinite Measuring Number

Super-Real Numbers [Dales & Woodin]:

Dales & Woodin,

Super-Real Fields: Totally Ordered Fields with Additional StructureWikipedia,

Superreal number

Smooth Infinitesimal Analysis:

Bell,

A Primer of Infinitesimal AnalysisWikipedia,

Smooth Infinitesimal Analysis