Seker ramat horaa '15-'16: http://u.math.biu.ac.il/~katzmik/sekerramathoraa1516.pdf

Keisler's textbook using Abraham Robinson's infinitesimals
*Elementary calculus*:
https://www.math.wisc.edu/~keisler/calc.html

Keisler's textbook in searchable format:
http://u.math.biu.ac.il/~katzmik/keislercalcsearchable.pdf

Keisler's *Foundations of infinitesimal calculus* (companion
volume)
http://u.math.biu.ac.il/~katzmik/keislerfoundations07.pdf
(searchable)

Lecture notes in Hebrew by Luie Polev Jennings:
part 1 (13
lectures),
part 2 (13
lectures).

Lecture notes in English:
lecture 1,
lecture
2,
lecture
3,
lecture 4
lecture 5,
lecture 6,
lecture 7,
lecture 8,
lecture 9,
lecture 10,
lecture on
sequences: Lecture on inf, sup, sequences, etc.

Cauchy's
definition of continuity

Cauchy's
definition of continuity in BW

Final exams and solutions:
'15 moed
A,
solutions
for final '15 moed A,
'15 moed
B,
solutions
for final '15 moed B,
'16 moed
A,
solutions
for final '16 moed A,
'16 moed
B,
solutions
for final '16 moed B,
'16
moed C,
'17 moed
A,
solutions
for final '17 moed A,
'17 moed
B,
solutions
to final '17 moed B,
'18
moed A,
solutions
to final '18 moed A,
'18 moed
B,
solutions
to final '18 moed B,
'19 moed
A,
solutions
to final '19 moed A,
'19 moed
A,
solutions
to final '19 moed A,
'19 moed
B,
solutions
to final '19 moed B,
'19 moed
C,

Old homework from '14: targil 1,
targil 2,
targil 3,
targil 4,
targil 5,
targil 6,
targil 7,
targil 8,
targil 9,
targil 10,
targil 11,
targil 12,
targil 13

Bochan from '14:
bochan

Answers to some Frequently Asked Questions (FAQ).

**Question** 1. *Does infinity belong to the hyperreals?*

Answer. The infinity symbol ∞ is often added to the reals in
calculus and analysis courses.
The resulting number system is sometimes called the extended reals.
This extended number system is of course not a field, and is not related to
the hyperreals.
The hyperreals form a field that contains many infinite elements, but the
infinity symbol is not one of them.
One can also adjoin the infinity symbol to the hyperreal line, resulting in an
extended hyperreal line.
I am not sure we will need this in the course but if we do it will be signaled
appropriately.

**Question** 2. *Which elements are added when one passes from the reals to
the hyperreals?*

Answer. What is added is all infinitesimals, all infinite numbers,
but also combinations like 1+ε where ε is
infinitesimal.

**Question** 3. *What is a finite number which is not real?*

Answer. An example already appeared above, namely 1+ε where
ε is infinitesimal.

**Question** 4. *In Cantorian set theory that all the students are familiar
with to one extent or another, there is the notion of cardinality of a
set. How is this related to the hyperreals?*

Answer. Cantor developed a theory of infinite cardinalities including
the fact that the cardinality of the reals is greater than the
cardinality of the natural numbers, etc.
This is of course a different notion of infinity than that of an
infinite hyperreal.
Cantor by the way was hostile to infinitesimals and at some point
claimed to have proved that they are inconsistent.
Another point, by the way, is that the famous philosopher Russell
accepted Cantor's claim of inconsistency as fact and reproduced it in
his books, including his famous "Principles of Mathematics".
This kind of tidbit is strictly speaking not necessary but it might
spice up class or tirgul presentation if you notice that the students
are drifting off to sleep :-)

**Question** 5. *Is there an infinitesimal number greater than
epsilon?*

Answer. The student probably answered himself, "2ε".
Another example given in class is square root of ε.

**Question** 6. *Why doesn't our lecturer let us use the concept of
"tends to zero" when speaking of infinitely small numbers?*

Answer. What I personally told them in the lecture is that the idea
of a sequence (1/n) tending to zero is an excellent intuitive point of
approach to infinitesimals.
So they can certainly use the concept provided they understand that
this is a preliminary, intuitive stage toward grasping the concept of
an infinitesimal number.
However, in the end a positive infinitesimal is a fixed number that's
smaller than every positive real number.
A student asked me in class if it is possible to think of an
infinitesimal as 0.0000...1 with a lot of zeros.
I told him that the answer is affirmative, provided that there are
infinitely many zeros there before a final 1.

**Question** 7. *Why can't one define finite numbers as numbers between
-H and H ?*

Answer. All finite numbers are indeed between -H and H if H is
infinite.
However, there are some infinite numbers that are also there.
For example, H/2 is also between H and -H.
So the property only works in one direction and cannot serve as a definition.

**Question** 8. *They asked for an example of an infinitesimal. How
does one respond?*

Answer. For advanced students, such an example can be given with
respect to a construction of the hypereals in terms of sequences of
real numbers.
Here the *equivalence class* of the sequence (1/n) provides
such an example.
This connects well with question 6 above on tending to zero.
Namely, the sequence tends to zero.
However, it is not the sequence itself but rather its equivalence class that
defines a hyperreal infinitesimal.
One can also mention that students already have natural intuitions of
such numbers, when they think about what they feel is a very small
discrepancy between 1 and 0.999... Over the hyperreals one can
formalize such intuitions if one thinks of a number 0.999...9 with a
specific infinite number of digits 9.

**Question** 9. *Is zero an infinitesimal?*

Answer. The convention following Keisler's book is to define the
number zero to be infinitesimal. This may seem contrary to intuition
but turns out to be convenient technically. Unlike every other
infinitesimal, zero is not invertible.

**Question** 10. *Why does Keisler all of a sudden goes back to the
old approach with limits after he has defined things using standard
part?*

Answer. In Keisler's approach, limits themselves are defined via
standard part. Therefore when limits start appearing in the course
this is not to be interpreted as a throw-back to the old method, but
rather as an application of the standard part approach.

**Question** 11. *How does one define continuity of a function on an
arbitrary domain?*

Answer. The notion of continuity on a closed interval is defined via
*one-sided* continuity at both endpoints. Fist we define
continuity at a point, then on an open interval (in the natural
way). But then, if we want to extend it to arbitrary intervals
(closed, half closed) we need one-sided continuity. Thus we don't
bother students with notions of continuity on a general domain where
for example any function defined on Z turns out to be continuous.
Dealing with general domains only confuses the students at this stage
in their learning.

**Question** 12. *What kind of weird definition of inverse function is
that?*

Answer. Keisler's definition of the inverse function is the
following. If *y=f(x)* is a function then *x=g(y)* is
its inverse provided *f* and *g* have the same graph in
the (x,y) plane. Thus inverse fuctions can be defined *before*
the composition of functions is defined. Keisler's definition in
terms of the graphs is a very nice one and is different from the
traditional one using composition of functions.

**Question** 13. *Wow, I am impressed. Are there ANY mistakes in
Keisler's book?*

Answer. In december '15 Meny Shlossberg pointed out that there is a
gap in Keisler's proof of the direct test on page 138. Here the
existence of a maximum is used in the proof even though such existence
is not proved until a later section. Keisler corrected it in the
online edition of the book.

**Question** 14. *How does one prove that Thomae's function is
continuous at irrational points?*

Answer. See
this.

**Question** 15. *What do the enemies of Robinson's infinitesimals say
about them?*

Answer. In the fall of '18, one of the students in 88-132 engaged
some PhDs on the internet in a discussion of Robinson's framework for
analysis/calculus with infinitesimals. Some of their comments are
reproduced below, together with a rebuttal.

>(15a) Well the most obvious thoughts include lack of definition and

>missing important parts like ultrafilters. Give me a proper

>definition. Why should there even exist such a thing? Why do they

>have the properties they do? Etc. At this point I have no idea what

>object you are talking about.

Is this fellow one of the PhD's you mentioned earlier? The construction of the hyperreals was presented in the wednesday seminar. Some of the 88132 students are still attending the wednesday seminar. The construction is rather accessible and is essentially a standard algebraic technique. The technique involves quotienting a ring by a suitable maximal ideal.

This construction was of course not presented in the freshman calculus course. Similarly, the construction of the real number system was not presented, and is almost never presented, in introductory calculus courses. Instead, both the real numbers and the hyperreal numbers are introduced axiomatically. The techniques for their set-theoretic construction construction are beyond the level of first-semester calculus.

Actually I find it quite revealing that this PhD wrote "At this point I have no idea what object you are talking about." He is frank enough to admit that he does not know what he is talking about, but apparently not intelligent enough to realize that if he does not know what he is talking about, he should naturally keep his mouth shut :-) By the way, two of the september 11th terrorist plotters apparently had PhD's; see this link. So having a PhD is not a guarantee of good character traits :-)

>(15b) simplifying material which is already basic is useless. you lost

>the topological arguments. which you are going to need anyway. you

>gain useless stuff and you lose important stuff.

What we *gain* in this approach is about 80% of the students in
freshman calculus, who according to education studies never pick up
the epsilon-delta technique properly, because they don't have proper
preparation for it at this point in their studies. Unlike many other
courses, we try to provide such preparation, by explaining the
fundamental notions of the calculus like continuity and derivative
using the intuitive notion of infinitesimal.
By the way, the remark "you lost the topological arguments"
illustrates the ignorance of this particular PhD. Since Robinson's
system takes place within the classical set theory ZFC, by definition
you don't lose anything! All the traditional mathematics is still
there. What you gain is a new technique, useful both pedagogically
and at the research level.

>(15c) It's just a matter of how most working analysts don't think about

>it, so if you give a proof of a theorem using non-standard analysis

>in a paper, analysts will either ignore it or immediately translate

>it to regular analysis.

This comment is true for Paul Halmos, who performed such a translation of Robinson's solution of Halmos's invariant subspace conjecture in 1966, based on insights gained from the infinitesimal viewpoint (Halmos' own despicable behavior is documented in this article). However, the comment is not true in general. For instance, the Fields medalist Terry Tao routinely uses Robinson's framework and ultraproducts in his publications, both articles and books.

>(15d) Hyperreal analysis wasn't created for its usefulness, more as an

>intellectual exercise to see if maybe old-school mathematicians

>weren't just talking nonsense.

This seems to be merely an ignorant remark not worth responding to. When a PhD makes such statements it is actually a good sign, since it shows that he ran out of better arguments; namely he has none :-)

>(15e) There are more examples but these were the main ones, I do hope you

>don't actually lose anything expanding the real numbers into the

>hyperreals (such as completeness).

As I mentioned in class, in Robinson's approach we work with
*both* fields R and R*, where R is embedded in R*. In other
words, we don't replace R by R* but rather work with the pair R,R*.
Thus the extremely important ordered complete Archimedean field R is
still there, so don't worry.
As I also mentioned, when formulated in the language of first-order
theory, the property does carry over to R*, since R* is an elementary
extension of R.

>(15f) Say I want to introduce the concept of numbering to students or to

>people who find this interesting, obviously you start by asking them

>to name a few numbers, you introduce the Natural numbers, you mention

>that this isn't a number line yet and there are jumps in between, you

>ask if they could name numbers that don't appear on the list, until

>you get to negative numbers, zero, and eventually rationals which

>finally make a continues line, mention that there are irrational

>numbers that are somehow not on that infinite line even though

>rationals can be found between any two numbers... But then there's

>the question: are there more numbers on that number line which we

>didn't cover?

This question certainly makes sense, in the context of the Cantor-Dedekind postulate that identifies the line in physical space with what we refer to as the "real line" in mathematical analysis. However, Keisler already pointed out that we have no way of knowning whether the line in physical space is like the "real line", like the "hyperreal line", or neither of them. The question that is more relevant is "which number system is most useful in applications and teaching?" Many people today would argue that R, like R*, is merely an idealisation that does not have a referent. One shouldn't confuse two different meanings of the adjective "real": (1) truly existing, and (2) equivalence class of Cauchy sequences of rationals.

**Question** 16. *Isn't the epsilon-delta technique
essential to calculus and analysis?*

Answer. Consider the following parable. For a number of decades, the
American Marines used to be trained to storm beaches. Class after
class of fresh recruits used to re-enact D-Day as a staple of their
basic training. Now I heard that recently the higher-ups in the US
realized that the conditions of modern warfare have changed. They
therefore reduced the emphasis on this portion of the basic training
of the Marines. Presumably they are currently trained to parachute
into enemy territory instead, etc.

*Unlike* beach-storming, epsilon-delta is still a feature of
central importance in modern analysis. What did change is that we can
now train the students in such a way as to *prepare* them for
epsilon-delta, rather than throwing it at them before they know any
calculus at all.

We have to relate seriously to the fact that 80% of the students never acquire the epsilon-delta techniques properly, yet they do pass all the math courses and graduate with a toar rishon degree from our department.

In infi1 taught via infinitesimals, we give enough background in epsilon-delta so that the top 20% of the students (those that have the option of going to graduate school in math) get an adequate grasp of it. The idea that professional-level proficiency in epsilon-delta at freshman level is indispensable can no longer be taken for granted.

Shimon Brooks had a bit of a difficulty in the first year he was teaching infi2 following a semester taught via infinitesimals in infi1. However, in the second year he learned the trick. The trick is to formulate most statements he needs in terms of limits instead of explicit epsilon-delta formulations. This seems to work in almost all cases and he reported better experience in infi2 in the years following.

Erez Sheiner got 4.87 out of 5 in student evaluations for infi2 for mathematics students during the 2017-2018 academic year (after they were taught infi1 using infinitesimals). This was the top score among all lecturers in the department. He usually gets very high scores so that's not out of the ordinary, but surely it shows that at least the math students thought the experience wasn't all that bad. His high score indicates that students trained via infinitesimals in infi1 must not have been unprepared for the material of infi2.

Even though the different method in infi1 may have required some adjustments, the experience of both Brooks and Sheiner shows that whatever difficulties do arise are manageable. And the Amir twin "experiment" confirms that the students have a more pleasant experience in infi1 at Bar Ilan nowadays, which is surely not a circumstance to be discounted.

More
on infinitesimals

Bar Ilan University "mathwiki" site for the course 88-132

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