Table Of ContentLecture Notes ni
Mathematics
Edited yb .A Dold dna .B nnamkcE
873
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Proceedings of the New Mexico State University
Conference Held ta saL Cruces, New ,ocixeM
August ,51-11 1980
Edited yb .F Richman
galreV-regnirpS
Berlin Heidelberg New York 1981
Editor
Fred Richman
Department of Mathematics, New Mexico State University
Las Cruces, NM 88003, USA
AMS Subject Classifications (1980): 03 F 50, 03 F 55, 03 F 65
ISBN 3-540-10850-5 Springer-Verlag Berlin Heidelberg NewYork
ISBN 0-387-10850-5 Springer-Verlag New York Heidelberg Berlin
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PREFACE
Constructive mathematics is distinguished by the fact that those
engaged in it maintain an awareness of their relationship to the objects
in the mathematical universe, and are concerned with manipulating those
objects rather than merely describing them. Following Hao Wang we may
say that constructive mathematics is a "mathematics of doing" while
classical mathematics is a "mathematics of being." Constructivists feel
that the mathematics they do is more meaningful, their theorems more
incisive and informative, because of this perspective.
The current resurgence of interest in constructive mathematics has
two principal sources. One is Errett Bishop's "Foundations of
constructive analysis". This book, in which Bishop expounds a
constructive philosophy of mathematics and develops several areas of
mathematics, using standard notation, from that point of view, can be
appreciated by mathematicians who are unschooled in logic and recursive
function theory. The other major influence has been the digital
computer. The notion of computation in principle has become familiar
through computer programming, making the idea that pure mathematics
should be based upon computation, should have "numerical meaning", more
attractive.
There are several identifiable schools of constructive mathematics,
all tracing their ancestry, in part, to L. E. J. Brouwer. Each takes
its own view of the relationship of the mathematician to the
mathematical universe, often based upon a particular ontological
conunitment. The school of Markov admits only mathematical objects that
are, ultimately, finite strings of symbols from a finite alphabet. This
results in a list of all partial functions of positive integers, from
which follows such nonclassical theorems as that every real valued
function on the unit interval is (pointwise) continuous. This list may
also be used to produce a counterexample to the Heine-Borel theorem.
The intuitionists, who are the direct descendants of Brouwer, view
all mathematical objects as creations of the mind and use this idea in
their proofs. The two characteristic features of this school are choice
sequences and bar induction. The former is the idea of a sequence of
integers generated, in effect, by a "black box" so that no information
is available beyond certain constraints given in advance and the
knowledge of a finite number of values. The latter has its principal
application in the fan theorem which says that a rooted tree, each of
whose branch points is of finite degree, and in which every path is
bounded, is finite. This theorem, which is proved classically by
VI
contradiction, is proved positively by the intuitionists by appealing to
the source of our knowledge that every path is bounded. The notion of
choice sequence leads immediately to the nonclassical theorem that every
real valued function on the unit interval is continuous; the fan theorem
implies that such a function is uniformly continuous. This contrasts
with the Markov school which constructs an example of a function that is
not uniformly continuous.
Those who follow the lead of Errett Bishop take a free-wheeling
finitist view with a minimum of (conscious) philosophical commitment.
The Bishop school is conservative, in contrast to the intuitionists and
the Markov school, in that no classical theorem is refuted. Any proof
in the spirit of Bishop is also a classical proof, the difference being
that the former will concern itself with the question of how to compute
the data needed for the conclusion of the theorem in terms of the data
given by the hypothesis.
The ultrafinitist school of Yessenin-Volpin views very large numbers,
such as 1030 , as inaccessible. In this way they arrive at different
notions of the natural number series.
There are also workers in recursive function theory whose results are
often germane to constructive mathematics, and whose informal techniques
are quite similar, but who work in a classical framework and may be
thought of as developing a classical model of constructive mathematics.
And there are those engaged in "predicative" mathematics, going back to
Weyl, who are uneasy with the construction of least upper bounds for
arbitrary bounded sets of real numbers, but quite content (unlike all
the aforementioned) with the construction of least upper bounds for
arbitrary bounded sequences of real numbers. Finally, there are those
interested in algorithms that work in practice, rather than just
computation in principle.
Constructive mathematics arrived in Las Cruces in the Spring of 1970
when Stanley Tennenbaum, who was visiting New Mexico State University's
department of mathematics, gave a series of seminars based on Bishop's
book. No one could remember a better attended seminar. In 1971 Henry
Cheng, a student of Bishop, accepted a position at NMSU, and in 1972
Bishop was the principal speaker at the annual NMSU Holiday Symposium.
During this time an active group of constructive mathematicians was
formed at NMSU including Mark Mandelkern, Gordon Berg, William Julian,
Ray Mines and Fred Richman. In 1973 Y. K. Chan, another of Bishop's
students, visited NMSU for a year.
In January 1979 Mines and Richman organized two special sessions in
constructive mathematics at the annual meeting of the American
Mathematical Society in Biloxi. During that meeting the Las Cruces
constructive group had the opportunity of meeting A. Seidenberg who had
been studying finitistic techniques in algebra for several years, Anil
Nerode who studied field theory from the point of view of recursive
functions, Andrej Scedrov an intuitionistic logician from SUNY at
Buffalo, Douglas S. Bridges who had just completed a book on
constructive functional analysis in the spirit of Bishop, and Allan
Calder who was writing an exposition of constructive mathematics for
Scientific American. We were also able to renew our contacts with
Newcomb Greenleaf who was working on Cantorian set theory from a
constructive viewpoint, and with Y. K. Chan.
Shortly after the Biloxi meeting it was decided that a larger meeting
in a more relaxed atmosphere was in order. It was apparent at Biloxi
that everyone would benefit by an exchange of ideas among the various
schools of thought in constructive mathematics. Thus the 1980 Las
Cruces conference on constructive mathematics was planned.
The papers in this volume are in the order of their presentation at
the conference. The following talks were given at the conference
but will be published elsewhere:
C. N. Delzell Maharishi Int. Univ. A CONSTRUCTIVE CONTINUOUS
and L.S.U. SOLUTION TO HILBERT'S-
17th PROBLEM
W. Julian New Mexico State U. z-COVERING DIMENSION
V. Lifschitz University of Texas CALCULABLE NATURAL NUMBERS
M. Mandelkern New Mexico State U. CONTINUITY OF MONOTONE
FUNCTIONS
J. Myhill SUNY at Buffalo CONSTRUCTIVE VS. NONCON-
STRUCTIVE IN A CLASSICAL
SETTING
lV
The conference would not have been possible without the support of the
Arts and Sciences Research Center and the Department of Mathematical
Sciences at New Mexico State University, nor without the efforts of the
organizers Bill Julian and Ray Mines.
Fred Richman
February 1981
Las Cruces, New Mexico
TABLE OF CONTENTS
Page
SEIDENBERG'S CONDITION P
Fred Richman . . . . . . . . . . . . . . . . . . . . . . . 1
FIELDEXTENSIONS
Wim Ruitenburg . . . . . . . . . . . . . . . . . . . . . 12
DEDEKIND DOMAINS
Ray Mines and Fred Richman . . . . . . . . . . . . . . . 16
EFFECTIVE MATHEMATICS THE COMPUTER ALGEBRA VIEWPOINT
James H. Davenport - . . . . . . . . . . . . . . . . . . 31
ON SOME OPEN PROBLEMS IN CONSTRUCTIVE PROBABILITY THEORY
Y. K. Chan . . . . . . . . . . . . . . . . . . . . . . . . 44
CONSISTENCY AND INDEPENDENCE RESULTS IN INTUITIONISTIC SET THEORY
Andrej Scedrov . . . . . . . . . . . . . . . . . . . . . . 54
COMPUTABILITY OF ORDINAL RECURSION OF TYPE LEVEL TWO
W. A. Howard . . . . . . . . . . . . . . . . . . . . . 87
A CONSTRUCTIVE APPROACH TO CLASSICAL MATHEMATICS
Jonathin P. Seldin . . . . . . . . . . . . . . . . . . . . 105
ON THE NOTION OF STANDARD NON-ISOMORPHIC NATURAL NUMBER SERIES
David Isles . . . . . . . . . . . . . . . . . . . . . . iii
REFLECTIONS ON BISHOP'S PHILOSOPHY OF MATHEMATICS
Nicolas D. Goodman . . . . . . . . . . . . . . . . . . . . 135
FORMALIZING CONSTRUCTIVE MATHEMATICS: WHY AND HOW?
Michael Beeson . . . . . . . . . . . . . . . . . . . . . . 146
INDEPENDENCE OF PREMISSES AND THE FREE TOPOS
J. Lambek and P. J. Scott . . . . . . . . . . . . . . . . . 191
AN INTUITIONISTIC INFINITESIMAL CALCULUS
Richard Vesley . . . . . . . . . . . . . . . . . . . . . . 208
LIBERAL CONSTRUCTIVE SET THEORY
Newcomb Greenleaf . . . . . . . . . . . . . . . . . . . . . 213
LOCATING METRIC COMPLEMENTS IN EUCLIDEAN SPACE
Bridges, Calder, Julian, Mines and Richman . . . . . . . . 241
A DISJUNCTIVE DECOMPOSITION THEOREM FOR CLASSICAL THEORIES
Joan Rand Moschovakis . . . . . . . . . . . . . . . . . . . 250
TOWARDS A CONSTRUCTIVE FOUNDATION FOR QUANTUM MECHANICS
Douglas S. Bridges . . . . . . . . . . . . . . . . . . . . 260
ABOUT INFINITY, FINITENESS AND FINITIZATION
A. S. Yessenin-Volpin . . . . . . . . . . . . . . . . . . . 274
A CLASS OF THEOREMS WITH VALID CONSTRUCTIVE COUNTERPARTS
Michael Gelfond . . . . . . . . . . . . . . . . . . . . . . 314
RATIONAL CONSTRUCTIVE ANALYSIS
James R. Geiser . . . . . . . . . . . . . . . . . . . . . . 321
SEIDENBERG'S CONDITION P
Fred Richman
i. ~@~9[~@! ~!~!~{" In the theory of discrete fields, that is,
fields in which equality is decidable, a central role is played by
factorial fields. A discrete field k is (~P~K~!Z) ~g~R~!~! if
every (separable) polynomial with coefficients in k is a, possibly
trivial, product of irreducible polynomials. Van der Waerden [vdWl]
gave an example of a countable discrete field which illustrates why the
notion of faetoriality has some content. A simplified version of this
example is achieved by letting {an} be an ascending sequence of zeros
and ones, and letting k be the union of the fields Q(ian), where Q
is the rational numbers and i is the imaginary unit. Then the
polynomial 2 x + 1 factors over k if and only if n a = 1 for some
2
n. Were k factorial then either x + 1 would be irreducible, in
2
which case n a = 0 for all n, or x + 1 would factor in Q(ia n)
for some n, in which case a = 1 for that n.
n
In the unexpurgated edition of van der Waerden's Modern Algebra
[vdW2] it is shown that a finite dimensional separable extension of a
factorial field is factorial. The treatment of this theorem in [vdW2]
has been criticized and modified in [MR] and [S3]. The notion of a
separably factorial field turns out to be, in some sense, more natural.
Every finite dimensional extension of a separably factorial field is
separably factorial [MR; Theorem 3.9]. Hence every finitely generated
separable extension of a separably factorial field is finite dimensional
(and separably factorial). On the other hand there are Brouwerian
counterexamples to the theorem that a finite dimensional extension of a
factorial field is factorial [S2], [MR; Example 4.3].
The property that finitely generated separable extensions are finite
dimensional characterizes separably factorial fields among countable
discrete fields as every nonconstant polynomial over a countable
discrete field has a root in some discrete extension field [JMR; Theorem
3.8]. It is an open question whether thls result holds for arbitrary
discrete fields. If we could construct root fields for polynomials over
arbitrary discrete fields the answer would be "yes". However this would
imply what might be called "the world's simplest axiom of choice":
Let S be a set such that if x E S, then x = {a,b)
where a # b. If x,y E S implies x = y, then there is
a choice function on S, that is, a function f : S - uS
such that f(x) e x for each x in S.
Let S be a set as above, and let k be the following extension of the
field of rational numbers. An element of k is a formal linear
combination of the rational number 1 and elements of us = (a : aexeS
for some x in S}. An element of k is equal to 0 if the
coefficient of 1 is 0 and the sum of the coefficients of the
elements of uS is 0. Addition is defined coordinatewise and
multiplication is determined by setting the square of each element of
uS equal to 2. If K is a discrete field containing k, and ~2 = 2
for ~ e K, then ~ provides a choice function on S by setting f(x)
equ~l to that element of x that is equal to ~ in K.
Classically this axiom of choice is trivial; constructively it
appears to be as difficult as being able to choose an element out of
each set in an arbitrary collection of two-element sets. However the
latter axiom of choice has a Brouwerian counterexample in the set of
pairs of antipodal points of a circle.
Seidenberg [Sl] introduced a condition that is an inseparable
version of separably factorial, namely that every finitely generated
purely inseparable extension is finite dimensional.
2. ~ ! @ ~ g ~!~!~- Let k be a dlscrete fleld of finite
characteristic p. Seidenberg's condition P arises from considering
th
the problem of whether a given element of k has a p root in k,
that is, whether the subfield p k of pth powers of elements of k
is detachable from k. The condition introduced by Seidenberg in IS1]
is the first conditlon in the following theorem.
THEOREM I. Let k be a discrete field of finite characteristic
p. Then the following conditions are equivalent:
i) Given aij • k for l(i(m and l(j4n, either there
exist x. E k p, not all zero, with ~ a..x = 0 for
3 13 3
all i; or, whenever E a..x. = 0 for all i, with x. • k p,
13 3 3
then x. = 0 for all j.
3
2) If K is a finite dimensional extension of k, then p K is
detachable from K.
3) If K is a finite dimensional purely inseparable extension
of k, then p K is detachable from K.
4) Every finitely generated extension field K of k with
p K c k is finite dimensional.
5) Every finitely generated kP-subspace of k is finite
dimensional.
PROOF. To derive 2 from I let Wl' '" "" Wn be a basis for K
over k, and let lP~ ~n I bi~ j. Then an arbitrary element
a ~ of K is in p K if and only if there exist x in p k such
3 ] i
that
n n x (~P n n
E3=I a3~ 3 EI=I 1 1 = Ej=I Ei=I Xlbi3o) ~
that is, aj = ~i=l n i x b i3 for all j or, since the wP 1 are
independent over k p, that the system of equations
n b = 0
x0aj ~i=l xi i3
has a nontrivial solution in k p. Obviously 3 follows from 2. To
