鲁宾逊本人对微积分的观念与思想

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注：本文十分珍贵。

袁萌 陈启清 6月26日

附件：

Numbers and Models, Standard and Nonstandard.

Dedicated to the memory of Abraham Robinson.

4 April 2010

The following is a somewhat extended manuscript for a talk at the
“Algebra Days”, May 2008, in Antalya. I talked about my personal
recollections of Abraham Robinson.

Contents

1 How I met Abraham Robinson and his innitesimals
2

2 What is Nonstandard Analysis? 4

2.1 A preliminary Axiom . . . . . . . 4

2.2 The Axiom in its nal for. . . . . . . . . 7

2.3 Some excercises . . 9

3 Robinson’s visits 12

3.1 Tu¨bingen . . . . .. 12

3.2 Heidelberg . . . . . . . . . . . . 13

4 Nonstandard Algebra 15

5 Nonstandard Arithmetic 17 References 20

1

1 How I met Abraham Robinson and his innitesimals

It was in the early months of 1963. I was visiting the California
Institute of Technology on my sabbatical. Somehow during this visit
I learned that one year ago Wim Luxemburg had given a lecture on A.
Robinson’s theory of innitesimals and innitely
large numbers. Luxemburg was on leave but I got hold of his
Lecture Notes [Lux62]. Although the topic was somewhat distant from
my own work I got interested and, after thorough reading I wished
to meet the person who had been able to put Leibniz’ innitesimals
on a solid base and build the modern analysis upon it. At that time
Abraham Robinson was at the nearby University of California at Los
Angeles, and I managed to meet him there. I remember an instructive
discussion about his theory which opened my eyes for the wide range
of possible applications; he also showed me his motivations and
main ideas about it.

Abraham Robinson 1918-74

Perhaps I am allowed to insert some personal words explaining why I
had been so excited about the new theory of inntesimals.
This goes
back to my school days in K¨onigsberg, when I was 16. At that time
the school syllabus required that we were to be instructed in
Calculus or, as it was called in German, in Dientialrechnung. Our
Math teacher at that time was an elderly lady who had been retired
already but was reactivated again for

2

school work in order to ll the vacancy of our regular teacher; the
latter had been drafted to the army. (It was war time: 1944.) I
still remember the sight of her standing in front of the
blackboard where she had drawn a wonderfully smooth parabola,
inserting a secant and telling us that y x is its slope, until
nally she convinced us that the slope of the tangent is dy dx
where dx is innitesimally small and dy accordingly.

My school in K¨onigsberg1

This, I admit, impressed me deeply. Until then our school Math had
consisted largely of Euclidean geometry, with so many problems of
constructing triangles from some given data. This was o.k. but in
the long run that stu did not strike me as to be more than boring
exercises. But now, with those innitesimals, Math seemed to have
more interesting things in stock than I had met so far. And I
decided that I would study Mathematics if I survived the dangers of
war which we knew we would be exposed to very soon. After all, I
wanted to nd out more about these wonderfully strange
innitesimals. Well, I survived. And I managed to enter University
and start with Mathematics. The rst lecture I attended to was
Calculus, with Professor Otto Haupt in Erlangen. There we were told
to my disappointment that my Math teacher had not been up to date
after all. We were warned to

1Wilhelmsgymasium. This is the same school where Hilbert in the
year 1880 obtained his Abitur.

3

beware of innitesimals since they do not exist, and in any case they
lead to contradictions. Instead, although one writes dy dx then
this does not really mean a quotient of two entities, but it should
be interpreted as a symbolic notation only, namely the limit of the
quotients y x. I survived this disappointment too. Later I
learned that dy and dx can be interpreted, not as innitesimals but
as some entities of an abstract construction called dierential
module, and if that module is one-dimensional then the quotient dy
dx would make sense and yield what we had learned anyhow.
Certainly, this sounded nice but in fact it was only an abstract
frame ignoring the natural idea of innitesimally small numbers. So
when I learned about Robinson’s innitesimals, my early school day
experiences came to my mind again and I wondered whether that lady
teacher had not been so wrong after all. The discussion with
Abraham Robinson kindled my interest and I wished to know more
about it. Some time later there arose the opportunity to invite him
to visit us in Germany where he gave lectures on his ideas, rst in
Tu¨bingen and later in Heidelberg, after I had moved there. Before
continuing with this let me briey explain what I am talking about,
i.e., Robinson’s theory of nonstandard analysis.

2 What is Nonstandard Analysis?

2.1 A preliminary Axiom

Consider the hierarchy of numbers which we present to our students
in their rst year: N⊂Z⊂Q⊂R Everything starts with the natural numbers N which, as
Kronecker allegedly has maintained, are “created by God”
(or
whatever is considered to be equivalent to Him). The rest is
constructed by mankind, i.e., by the minds of mathematicians. In
each step, the structure in question is enlarged such as to admit
greater exibility with respect to some operations dened in the
structure. In Z the operation of subtraction is dened such that Z
becomes an additive group; in fact Z is a commutative ring without
zero divisors. In Q the operation of division is dened such that Q
becomes a eld. Finally, in R every Cauchy sequence is convergent,
such that R becomes a complete ordered eld. In each step we tell
our students that the respective enlargement exists and we explain
how to construct it.

4

In order to develop what nowadays is called “analysis” the
construction usually stops with the real eld R; this is considered to
be adequate and quite sucient as a basis for (real) analysis. But
it had not always been the case that way. Since Leibniz had used
the natural idea of innitesimals to build a systematic theory with
it, many generations of mathematicians (including my lady teacher)
had been taught in the Leibniz way. Prominent people like Euler,
the Bernoullis, Lagrange and even Cauchy (to name only a few) did
not hesitate to use them.

Gottfried Wilhelm Leibniz

The Leibniz idea for analysis, as interpreted by Robinson, is to
work in a further enlargement: R⊂∗R such that the following Axiom is satised. In order to
explain the main idea I will rst state the Axiom in a preliminary
form which, however, will not yet be sucient. Later I will give the
nal, more general form.

Axiom (preliminary form). (1.) ∗R is an ordered eld extension of R. (2.)
∗R contains innitely large elements. An element ω
∈ ∗R is called “innitely large” if |ω| > n for
all n ∈N. Part (2.) says that the ordering
of ∗R does not satisfy the axiom of Archimedes.

5

Fields with the properties (1.) and (2.) were known for some time2
but the attempts to build analysis on this basis were not quite
satisfactory. Among all such elds one has to select
those which in addition have more sophisticated properties.
But for the moment let us stay with the Axiom in this preliminary
form and see what we can do with it. The elements of R are called
standard real numbers, while the elements of ∗R not in R are nonstandard. This terminology is
taken from model theory but I nd it not very suggestive in the
present context. Sometimes the elements of ∗R are called hyperreal numbers. Perhaps someone
some time will nd a more intuitive terminology. The nite elements
α in ∗R are those which are not innitely large, i.e. which
satisfy Archimedes’ axiom: |α| < n for some n
∈N (depending on α).
These nite elements form a subring E ⊂ ∗R. It contains all innitesimal elements ε which are dened
by the property that |ε| < 1 n for all n ∈ N. Itfollows from the denition
that the set of innitesimals is an ideal I ⊂ E.We have: ω innitely large ⇐⇒ ω−1 innitesimal 6= 0. It is well known that this
property characterizes E as a valuation ring in the sense of Krull
[Kru32].

Theorem: The nite elements E form a valuation ring of
∗R with the innitesimals I as its maximal ideal. The
residue class eld E/I = R.

Two nite elements α,β are said to be innitely close to each other
if α−β is innitesimal, i.e., if they belong to the same
residue
class modulo the ideal I of innitesimals. This is written as
α ≈
β . The
residue class of α ∈ E is called the “monad” of
α; this terminology
has been introduced by Robinson in reference to
Leibniz’
theory of
monads. Every monad contains exactly one standard number a
∈R; this is called the
standard part of α, and denoted by
st(α). There results the standard part
map st : E →R which in fact is nothing else
than the residue class map of E modulo its maximal ideal I. 2See,
e.g., [AS27].

6

In this situation let us consider the example of a
parabola

y = x2

which, as I have narrated above, had been used by my school teacher
to introduce us to analysis. Suppose x is a standard number. If we
add to x some innitesimal dx 6= 0 then the ordinate of the
corresponding point on the parabola will be

y + dy = (x + dx)2 = x2 + 2xdx + (dx)2

which diers from y by

dy = 2xdx + (dx)2 = (2x + dx)dx

so that the slope of the corresponding secant is

dy dx

= 2x + dx ≈ 2x since dx ≈ 0 is innitesimal. Hence: If we
step down from the hyperreal world into the real world again, by
using the standard part operator, then the secant of two innitely
close points becomes the tangent, and the slope of this tangent is
the standard part: stdy dx= 2x. I believe that such kind of
argument had been used by my school teacher as narrated above. As
we see, this is completely legitimate. It is apparent that in the
same way one can dierentiate any power xn instead of x2, and also
polynomials and quotients of polynomials, i.e., rational functions,
with coecients in R. All the well known algebraic rules for
derivations can be obtained in this way. However, analysis does not
deal with rational functions only. What can be done to include more
functions?

2.2 The Axiom in its nal form

As described by the preliminary Axiom, ∗R is an ordered eld. This can be expressed by saying
that ∗R is a model of the theory of ordered elds. The theory of
ordered elds contains in its vocabulary the function symbols
“+”
for addition, and “•” for multiplication, as well as the relation
symbol “<” for the ordering. The axioms of ordered elds are
formulated in this language. If we add to the vocabulary constants
for all real numbers r ∈R and to the

7

theory all statements which are true in R then the models of this
theory are precisely the ordered eld extensions of R. If we wish
to talk about functions and relations which are not expressible in
this language, then we have to use a language with a more extended
vocabulary. In order not to miss anything which may be of interest
let us include into our language symbols for all relations in R.3
The theory of R consists of all statements in this language which
hold in R. Thus, if we generalize the rst part of the above Axiom
as: ∗R is a model of the Theory of R, then this will allow us to
talk in ∗R about every function and relation which is dened in R.
In order to generalize the second part of the Axiom we have to
refer not only to the relation “<” of the ordering, but
to
every relation of similar kind. More precisely: Consider a 2-place
relation (x,y) dened in R. Such a relation is said to be
concurrent if, given nitely many elements a1,...,an
∈R in the left domain
of , there exists b ∈ R in its right domain such that
(ai,b) holds for i = 1,...,n.4 Such element b may be called a
“bound” for a1,...,an with respect to the relation .

Axiom (nal form). (1.) ∗R is a model of the Theory of R. (2.) Every concurrent
relation over R admits a universal bound β ∈∗R, i.e., such that (a,β) holds simultaneously for all
a ∈R which are contained in the left
domain of .

It is clear that this form of the Axiom is a generalization of its
preliminary form, and a far reaching generalization at that. It was
Abraham Robinson who had observed that Leibniz, when he
worked with innitesimals, seemed tacitly to use something which is
equivalent to that Axiom. Of course, the essential point is that
indeed there exists a structure ∗R satisfying this Axiom. This is guaranteed by general
results of model theory. The most popular construction is by
means of ultrapowers. There is some ambiguity which has to be
cleared. The Axiom refers to the “Theory of R”, and this refers to
a given language as described above, its 3Functions can be viewed
as 2-place relations and thus are included. Subsets may be dened
as the range of their characteristic functions and hence are
included too. 4The relation may not be dened on the whole of R.
The left domain of consists of those a ∈ R for which there exists a b
∈ R such that (a,b)
holds. The right domain is dened similarly. For the ordering
relation “<” the left and the right domain coincides with
R.

8

vocabulary including symbols for all relations overR. On rst sight
one would think of relations (and functions) between
individuals, i.e., elements of R. This would lead to the rst order
language (Lower Predicate Calculus), where quantication is allowed
over individuals only. But in many mathematical investigations it
is necessary (or at least convenient) to enlarge the language such
as to contain also symbols for sets of functions, relations between
sets of functions etc., and quantication should be allowed over
entities of any given type. For instance, if we wish to state the
induction axiom for the set N of natural numbers, we may say that:
“Every non-empty subset of N contains a smallest element” and this
statement contains a quantier for subsets. In order to include
such statements too Robinson works with the Higher Order Language
containing symbols also for higher entities, i.e., relations
between sets, functions of relations between sets, etc. In other
words: We interpret the above Axiom as referring to the full
structure over R and accordingly work with the corresponding higher
order language. This implies, among other things, that in
∗R we have to distinguish between internal and external
entities. Quantication ranges over internal quantities of any
given type. Here we do not wish to go into details but refer, e.g.,
to the beautiful introduction which Abraham Robinson
himself has given in his book on Nonstandard Analysis [Rob66]. See
also the rst section in [RR75]. Robinson introduced the
terminology enlargement for a structure satisfying the Axiom. As
said above, such an enlargement can be obtained by ultrapower
construction. It is not unique. In the following we choose one such
enlargement and regard it as a xed universe during our
discussion.

2.3 Some excercises

Having learned all this from Abraham Robinson, my immediate
reaction was what probably every newcomer would have done: I
wished to put this method of reasoning to a test in simple
exemplary situations. I do not have time here for a long discussion
although much could be said to convince the reader of the enormous
potential of the new way of reasoning which Robinson’s theory of
enlargements oers to us. Here let me be content with a few
examples. Let f be a standard function and consider an element
x ∈R in its domain of denition.
According to the part (1) of the Axiom, f extends uniquely to a
function on ∗R.

9

Continuity: f is continuous in x if and only if x0
≈ x =⇒ f(x0) ≈ f(x). (1) Of course, it is assumed
that x0 is contained in the domain of f, so that f(x0) is dened.
If the domain of the original f is open then f(x0) is dened for
every x0 in the monad of x. The above statement can be used as
denition of continuity of a function. Note that the usual
quantier prex ∀ε∃δ ... is missing. I have chosen this example because I
found precisely this denition in an old textbook. This was the
German “Kiepert, Dierential- und Integralrechnung” of which the
rst edition had appeared in 1863. It had been very popular, and it
got at least 12 editions, the 12th appearing in 1912 [Kie12]. The
text there reads as follows (in English translation):

If some function is given by y = f(x) then, in general, innitely
small changes of x will give rise to innitely small changes of y.
For all values of x where this is the case, the function is called
continuous.

We see that this is precisely the denition (1).

Euler’s “Analyse des inniment-petits”5

5This frontispiece of Euler’s book was used by Abraham Robbinson in
his book on Nonstandard Analysis [Rob66].

10

Derivative: Let dx be an innitesimal. Dene dy by y+dy = f(x+dx).
Then the derivative f0(x) ∈R is dened by f0(x)
≈ dy dx . (2) More
precisely: it is required that this is a valid denition, i.e., the
quotient dy dx should be nite and its monad should be independent
of the choice of the innitesimal dx. If this is the case then f is
called dierentiable at x and f0(x) is dened as the standard part
of dy dx. I have chosen this example since it is the denition
presented by my school teacher mentioned above. It is well possible
that she had been trained using Kiepert’s textbook. Integration:
Suppose the function f(x) is dened in the closed interval [a,b]
with a,b ∈ R. Let n be a natural number and
divide [a,b] into n subintervals [xi−1,xi] of equal length. We take
n innitely large; then the length dx = b−a n of each subinterval
is innitesimal. Now the integral is dened by: Z b a f(x)dx
≈ n X i=1 f(xi)dx. (3)
More precisely: It is required that this is a valid denition,
i.e., the sum on the right hand side should be a nite element
in ∗R and its monad should be independent of the choice of the
innite number n. If this is the case then f is called (Riemann)
integrable over [a,b] and the integralRb a f(x)dx is dened as the
standard part of that sum. Maybe some explanation about innite
natural numbers is in order. ∗R is an enlargement of R, and therefore every subset of R
has an interpretation in ∗R. So does N. This new subset of ∗R is denoted by ∗N. (In fact, ∗N is an enlargement of N.) Using part (2.) of the Axiom, it
follows that there exists n ∈ ∗N which is larger than every number in N, i.e., n is
innite. As to the sum on the right hand side of (3), it is to be
interpreted as follows: For every nite n ∈ N the sum sn =Pn i=1 f(xi)dx has
nitely many terms, and so sn is well dened in R. The function n
7→
sn from N
to R has an interpretation in the enlargement, i.e., it extends to
a function from ∗N to ∗R. Thus sn is dened for every n ∈ ∗N. Note that sn for innite n is not an innite series in
the usual sense. It is to be regarded as the nonstandard
interpretation of a sum whose number of terms is a natural number. The
denition (3) of the integral explains Leibniz’ idea that the
integral is essentially a sum (up to innitesimals). This idea had
led him to introduce the integral signRas a variant of the letterS
which he used for sums (instead of Σ which is used
today).

11

I have been inspired to choose example (3) because of its relation
to Archimedes’ method of measuring the area of a plane region. This
method consists of cutting the area into parallel strips of,
say, length ` and innitesimal breadth ε; then `•ε is the
(innitesimal) area of the strip and the sum of all those areas
will give the area of the whole region - up to innitesimals. The
Leibniz formula (3) does precisely this in the case of a positive
function, when the region to be measured is that between the
function graph and the x-axis. That Archimedes’ method can indeed
be interpreted in this way (contrary to what is commonly attributed
to him) is well documented by the Archimedes Codex which has been
recently discovered and deciphered; see the report [NN07] about
what is called “the world’s greatest palimpsest”.

3 Robinson’s visits

3.1 Tu¨bingen

As said at the beginning I had met Abraham Robinson in Los Angeles
in California during my sabbatical. In the summer term of 1963 I
was back at my university in Tu¨bingen. There I started a workshop
where together with some students and colleagues, we read
Robinson’s papers and his book on model theory [Rob63] which had
just appeared. We tried to understand his ideas for nonstandard
analysis and to apply them to various situations. His book on
nonstandard analysis [Rob66] had not yet been written.

University of Tu¨bingen6 6This is the Main Building where Robinson
delivered his 1966 lecture.

12

Some time later when I had heard that Robinson was in Germany, I
was able to meet him and suggested that he spend a month or so in
Tu¨bingen as visiting professor, for a course on a topic from
nonstandard analysis. He reacted favorably and so he visited
us in
Tu¨bingen in the summer of 1966.7 I had advertised his lecture
course to students and colleagues, and so he had a full auditorium.
The aim of the course, two hours weekly, was to cover the
fundamentals of model theory with particular emphasis on the
application to analysis and algebra. This job was not easy since
the students (and most colleagues) did not have a formal training
in mathematical logic; so he had to start from scratch. He was not
what may be called a brilliant lecturer who would be able to rouse
a large audience regardless of the content of his talk. His way was
quiet, with great patience when questions came up from the
students, but strong when it came to convince the students about
the impact of nonstandard applications. And this kept the attention
of the large audience throughout his lecture. In addition Robinson
was available for discussion in our workshop. Just in time his book
on nonstandard analysis [Rob66] had appeared; he presented to us
some of the more sophisticated applications. I recall my impression
that his Tu¨bingen visit could be considered as a success, and from
what is reported in Dauben’s biography it appears that Robinson
thought so too.

3.2 Heidelberg

Next year, 1967, I moved from Tu¨bingen to the University of
Heidelberg. The general academic conditions in Heidelberg in
those years were quite favorable. So it was not dicult to convince
the faculty and the rector (president) that the visit of a
distinguished scholar like Abraham Robinson would be of enormous
importance for the development of a strong mathematics group in
Heidelberg. And so in the following year, 1968, I was able to
extend a cordial invitiation to Abraham Robinson to visit us again,
this time in Heidelberg. And he came, this time not from UCLA but
from Yale where he had moved in the meantime. Again he delivered a
course on model theory and applications. To a certain extent this
job was kind of a repetition of his Tu¨bingen lecture;
again

7I am relying here on the extensive Robinson biography by
Dauben [Dau95] where this year is recorded for Robinson’s
Tu¨bingen visit. Unfortunately I did not save our letters or other
documents from that time and so I have to look at Dauben’s book for
help in the matter of dates.

13

he had a large audience. But there was a dierence. For in his
seminar, on a smaller scale, he found an audience which was highly
motivated. On the one hand, there was a group of gifted students
and postdocs who had also switched from Tu¨bigen to Heidelberg and
who had already attended Robinson’s Tu¨bingen lecture. On the other
hand, in Heidelberg there had been regular courses on Mathematical
Logic (by Gert Mu¨ller who held a position as “associate
professor”), and so there had been opportunities for the students
to acquire knowledge in this eld, in particular in model
theory.

The Mathematics Institute in Heidelberg

But the essential new feature of Robinson’s Heidelberg visit was
that he talked not only on nonstandard analysis but also on
nonstandard algebra and arithmetic; in the seminar he was able
to expound his ideas in more detail. This found a respondent
audience. His impact on the work of these young people in the
seminar was remarkable. And so it came about that he more or less
regularly visited us in Heidelberg during the following years,
continuing his seminar talks and working with those who responded
to his ideas. In the next two sections I will give some kind of
overview on the work resulting of his inuence on the Heidelberg
group, which was apparent even after his untimely death in April
1974. Robinson’s inuence was also helpful in another project. In
view of Robinson’s striking applications of model theory to
mathematics proper, I became convinced that a chair devoted to
mathematical logic could be of help to mathematicians in their
daily work, in particular if the chair was occupied by someone from
model theory. Therefore I tried to obtain help from the university
administration and the ministry of education for establishing such
a chair in the mathematics faculty. I had started this project in
Tu¨bingen already but after I moved to Heidelberg this would have
to be a chair for

14

the Heidelberg faculty. Indeed, after some time such a chair was
installed (in those times such thing was still possible). This was
in 1973. It was clear to me that Robinson’s encouragement and
judgement had been of great help in this matter. When I asked him
whether he would accept an oer to Heidelberg for this chair then
he did not say “no” but from the way he reacted it seemed to me
that he really meant “no”. After all, Heidelberg seemed to be no
match for Yale at that time. In any case, in a few months after
that the problem was not existent any more. But it should be
remembered that this chair, which is still in existence, had been
installed with the strong help of Abraham Robinson. During his
repeated visits to Heidelberg we came to know Abraham Robinson not
only as a mathematician and scholar but also as a friend. He lived
around the corner from our house and on his way to town he
regularly stepped in for a coee and conversation with us. (If I
say “we” and “us” in this context then I include my wife Erika.) He
was a man with a wide horizon and far reaching interests. If he
talked about Leibniz then one could feel not only his knowledge
about Leibniz’ life and work but also his sympathy for that
remarkable man. There was only one thing about which he strongly
disagreed with Leibniz, namely Leibniz’ insistence that “our world
is the best of all possible worlds”. Abby liked to talk to people,
and sometimes we had the impression that he knew more about our
neighbors than we did. He was keenly interested in the local
history. When we took him on tour to show him the country and its
places then it often turned out that he knew more about it than we
did, and he gave us a lecture on the history of those places. In
the course of those years there developed a friendship of rare
quality. Abby belongs to the few close friends whom I have met in
my life. I have learned much from him, not only in Mathematics but
also in questions of attitude towards the problems of
life.

4 Nonstandard Algebra

Looking at the Axiom in its nal form (in section 2.2) it is
apparent that this Axiom has little to do with the special
properties of the real number eld R. It makes sense for every
mathematical structure. And so there is not only nonstandard
analysis, but nonstandard mathematics at large. Abraham Robinson
was well aware of this; he has applied his method, partly in
collaboration with others, to various mathematical problems ranging
from topology, Hilbert spaces, Lie groups, complex analysis,
dierential algebra,

15

quantum theory to mathematical economics. There were also
investigations in the direction of algebra and number theory. As
said above, Abraham Robinson reported on this in his Heidelberg
seminar lectures. One of his rst topics was his nonstandard
interpretation of Hilbert’s irreducibility theorem (jointly with
Gilmore in [GR55]). This paper of Robinson has been said to mark a
“watershed” in the development of model theory (in the same line
with another paper of Robinson’s, of the same year 1955, on
Artin’s solutiom of Hilbert’s 17th
problem [Rob55]). Hilbert had published his irreducibility theorem
in 1892 [Hil92]. Suppose that f(X,Y ) is an irreducible polynomial
in 2 (or more) variables then, Hilbert showed, there are innitely
many specializations X 7→ t such that f(t,Y ) remains
irreducible. The coecients of f are taken from the rational eld Q
and the specialized variable t is also assumed to be in Q. Since
then there had been numerous proofs of this theorem, also over
other base elds K, e.g., number elds. Hasse had the idea to study
arbitrary elds over which Hilbert’s irreducibility theorem may
hold, and his Ph.D. student Wolfgang Franz started the theory of
such elds which today are called Hilbert elds [Fra31]. This was
the point where Abraham Robinson stepped in. He stated a
nonstandard characterization of Hilbert elds. As a follow-up of
our discussions with Robinson we were able to amend his result of
[GR55] by presenting a new, “metamathematical” proof of Hilbert’s
irreducibility theorem in the number eld and the function eld
cases. It turned out that Hilbert’s irreducibility is, in fact,
equivalent to the well known theorem of Bertini in algebraic
geometry [Roq75]. Further investigations by R.Weissauer showed that
every eld with a set of valuations satisfying the product formula
is Hilbertian. This covered all classical elds which were known to
be Hilbertian. Moreover, Weissauer found quite a number of new and
interesting Hilbertian elds, e.g., formal power series elds in
more than one variable [Wei82]. Weissauer’s paper is a good example
of the usefulness of Robinson’s enlargements. On the one hand, it
can be shown that any result which has been proved using the notion
and the properties of enlargements can also be obtained without
this. On the other hand, the use of enlargements provides the
mathematician with new methods and it opens up new analogies to
other problems which sometimes help to understand the situation.
Abraham Robinson used to say that his method may reduce a
“dynamical” to a “statical” situation. For instance, an innite
sequence t1,t2,t3 ... which preserves the irreducibility of the
polynomial f(X,Y ) under the specialization X 7→ ti leads to a nonstandard t which
renders f(t,Y ) irreducible.

16

For another topic of algebra, remember that group theory had been
started by Galois in order to study the roots of algebraic
equations. Today the notion of Galois group belongs to the basics of
algebra. But there arose the need to study simultaneously innitely
many algebraic equations; this led Krull in 1928 to the discovery
of the topological structure of innite Galois groups [Kru28], and
this developed into the theory of pronite groups. Robinson has
pointed out that pronite Galois groups can be naturally understood
within the enlargement, connected to the “nite” groups in the
sense that their order is an innite large natural number n
∈ ∗N. The corresponding pronite groups in the standard
world are obtained from these nonstandard “nite” groups by a
similar process as the derivative f0(x) is obtained from the
nonstandard dierential quotient dy dx in the manner as explained
above. Hence again: If we step down from the nonstandard world into
the standard world again, then Krull’s Galois theory of innite
algebraic extensions appears as an immediate consequence of the
Galois-Steinitz theory for nite algebraic eld extensions. There
arises the interesting question which elds K are uniquely
determined (up to elementary equivalence) by their full pronite
Galois groups GK. See [Pop88], [Koe95]. The description of the
structure of GQ as pronite group is at the focus of current
arithmetical research.

5 Nonstandard Arithmetic

Remember Hensel’s p-adic number elds which Hensel had conceived at
around 1900 and which today have become standard tools in algebraic
number theory and beyond. In the course of time it became necessary
to consider all p-adic completions at once; this has led to the
introduction of adeles and ideles in the sense of Chevalley which
play a fundamental role, e.g., in class eld theory. Now, Abraham
Robinson has pointed out that his notion of enlargement comprises
all those constructions at the same time. His enlargements are
indeed the most universal “completions” in as much as every
concurrent relation admits a bound. The classical notions of
p-adics, adeles and ideles, pro-nite groups etc. are obtained from
his enlargement by a universal transfer principle, similar to
obtaining the derivative f0(x) as the standard part of the
dierential quotient dy dx as explained above. In the ensuing
discussions with Abraham Robinson we wished to test his

17

method in some more situations of fundamental importance. The
SiegelMahler theorem seemed to us a good example to begin with.
Finally in November 1973 he invited me to Yale with the aim of
discussing in more detail the possibility of a nonstandard proof of
this theorem. Let Γ : f(x,y) = 0 be an irreducible curve dened
over a number eld K of nite degree. If Γ is of genus g > 0
then Siegel’s theorem says that Γ admits only nitely many points
whose coordinates are integers in K. Mahler had generalized this by
proving that for any nite set S of primes of K there are only
nitely many points in Γ whose coordinates are S-integers in K. The
S-integers are those numbers in K whose denominator consists of
primes in S only. Nonstandard methods seem to be useful to
distinguish between nite and innite. We work in a xed
enlargement ∗K of K, with the properties as statetd in the Axiom. If Γ
would admit innitely many S-integral points in K then it would
also admit a nonstandard S-integral point in ∗K. Such a point (x,y) is a generic point of Γ over
K and hence F = K(x,y) is the function eld of Γ over K. By
construction F is embedded into ∗K: F ⊂∗K. Now, both these elds carry a natural arithmetic
structure: F as an algebraic function eld over K and
∗K as a nonstandard model of the number eld K. What is the
relation between the arithmetic in F and in K ? In our joint
paper [RR75] we were able to prove the following

Theorem 1: If F is of genus g > 0 then every functional prime
divisor P of F is induced by some nonstandard prime divisor p
of ∗K.

18

From here it is only a small step to deduce the validity of the
SiegelMahler theorem. Abby agreed to work out the proof of the
theorem for elliptic curves, and I was to deal with curves of
higher genus. Two weeks after I had left Yale he sent me his
manuscript for the elliptic part. But he could not see any more my
part for higher genus. Actually, there is a famous conjecture of
Mordell to the eect that a curve Γ of genus g > 1 over a number
eld K of nite degree has only nitely many K-rational points,
even without specifying that they are S-integers. This conjecture
has been proved by Faltings in 1983. In nonstandard terms it can be
formulated as follows:

19

Theorem 2: A function eld F|K of genus g > 1 cannot be embedded
into the enlargement ∗K.

Clearly, this contains Theorem 1 in the case g > 1, which was my
own contribution in the joint work with Robinson. But in 1973
Mordell’s conjecture had not yet been proved and hence, at that
time, the proof of Theorem 1 was necessary also for the case g >
1. In 1973 I discussed with Robinson also a possible nonstandard
proof of Mordell’s conjecture. We planned rst to develop the tools
which we believed to be necessary for this project. However, due to
Robinson’s sudden death our plan could not be realized. In later
years Kani [Kan80b, Kan80a, Kan82] has studied systematically
function elds which are embedded into the enlargement
∗K. In my opinion, the tools and the results which have been
obtained in his work are well capable to give a nonstandard proof
of Mordell’sconjecture (together with Roth’s theorem on
rational approximation of algebraic numbers [Rot55] which is a
standard tool to deal with questions of this kind, including
theorems 1 and 2). But this has not yet been worked out. It remains
an open challenge.

References

[AS27] E. Artin and O. Schreier. Algebraische Konstruktion reeller
Ko¨rper. Abh. Math. Semin. Univ. Hamb., 5:85–99, 1927. 6

[Dau95] J. W. Dauben. Abraham Robinson. The creation of nonstandard
analysis. A personal and mathematical odyssey. Princeton University
Press, Princeton, NY, 1995. XIX, 559 p. 13

[Fra31] W. Franz. Untersuchungen zum Hilbertschen
Irreduzibilita¨tssatz. Math. Zeitschr., 33:275–293, 1931.
16

[GR55] P.C. Gilmore and A. Robinson. Metamathematical
considerations on the relative irreducibility of polynomials. Can.
J. Math., 7:483–489, 1955. 16 [Hil92] D. Hilbert. ¨Uber die
Irreducibilita¨t ganzer rationaler Functionen mit ganzzahligen
Coecienten. J. Reine Angew. Math., 110:104– 129, 1892.
16

[Kan80a] E. Kani. Eine Verallgemeinerung des Satzes von
CastelnuovoSeveri. J. Reine Angew. Math., 318:178–220, 1980.
20

20

[Kan80b] E. Kani. Nonstandard diophantine geometry. In Proc. Queen’s
Number Theory Conf. 1979., volume 54 of Queen’s Pap. Pure Appl.
Math., pages 129–172, Kingston, 1980. 20

[Kan82] E. Kani. Nonstandard methods in diophantine geometry. In
Journees arithmetiques, Exeter 1980, volume 56 of Lond.
Math. Soc.
Lect. Note Ser., pages 322–342, London, 1982. 20

[Kie12] L. Kiepert. Grundriß der Dierential- und Integralrechnung.
I. Teil: Dierentialrechnung. Helwingsche Verlagsbuchhandlung,
Hannover, 12 edition, 1912. XX, 863 S. 10

[Koe95] J. Koenigsmann. From p-rigid elements to valuations (with a
Galois-characterization of p-adic elds). J. Reine Angew. Math.,
465:165–182, 1995. 17

[Kru28] W. Krull. Galoissche Theorie der unendlichen algebraischen
Erweiterungen. Math. Ann., 100:687–698, 1928. 17

[Kru32] W. Krull. Allgemeine Bewertungstheorie. J. Reine Angew.
Math., 167:160–196, 1932. 6

[Lux62] W. A. J. Luxemburg. Non-standard analysis. A.Robinson’s
theory of innitesimals and innitely large numbers. Math. Dept.
California Institute of Technology, Pasadena, CA, 1962. 150 p.
2

[NN07] R. Netz and W. Noel. The Archimedes Codex. Revealing the
secrets of the world’s greatest palimpsest. Weidenfels &
Nicolson, The Orion Publishing Group., London, 2007. 12

[Pop88] F Pop. Galoissche Kennzeichnung p-adisch abgeschlossener
Ko¨rper. J. Reine Angew. Math., 392:145–175, 1988. 17

[Rob55] A. Robinson. On ordered elds and denite functions. Math.
Annalen, 130:287–271, 1955. 16

[Rob63] A. Robinson. Introduction to model theory and to the
metamathematics of algebra. Studies in Logic. North-Holland,
Amsterdam, 1963. ix, 284 p. 12

[Rob66] A. Robinson. Non-standard analysis. Studies in Logic.
NorthHolland, Amsterdam, 1966. xi, 293 p. 9, 10, 12, 13

21

[Roq75] P. Roquette. Nonstandard aspects of hilbert’s
irreducibility theorem. In D. H. Saracino and B.
Weispfenning, editors, Model Theor. Algebra, Mem. Tribute Abraham
Robinson, volume 498 of Lect. Notes Math., pages 231–275. Springer,
Heidelberg, 1975. 16

[Rot55] Klaus F. Roth. Rational approximations to
algebraic
numbers. Mathematika, 2:1–20, 1955. 20

[RR75] A. Robinson and P. Roquette. On the niteness theorem of
Siegel and Mahler concerning diophantine equations. J. Number
Theory, 7:121–176, 1975. 9, 18

[Wei82] R. Weissauer. Der Hilbertsche Irreduzibilitaetssatz. J.
Reine Angew. Math., 334:203–220, 1982. 16

[YKKR76] A.D. Young, S. Kochen, S. Koerner, and Peter Roquette.
Abraham Robinson. Bull. Lond. Math. Soc., 8:307–323,
1976.

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2019-06-26
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