# FOM: New book on foundations

John Mayberry j.p.mayberry at bristol.ac.uk
Thu May 17 12:33:56 EDT 2001

```The Cambridge University press has recently published a book by me
dealing with the foundations of mathematics:

The Foundations of Mathematics in the Theory of Sets
Encyclopedia of Mathematics and its Applications, Vol. 82
J.P. Mayberry
Cambridge University Press, Cambridge, 2000
ISBN 0 521 77034 3 hardback

I think it might be of interest to members of the FOM list as it
takes up many of the topics that have been discussed on the list, as
well as others which haven't been but perhaps ought to be.
I have put up the preface and the first two chapters,

www.maths.bris.ac.uk/~majpm

under "Recent Publications".
Steve Simpson has suggested that perhaps I ought to say a few
words here by way of an introduction.
The main argument of the book arises from my conviction that
there is a central fallacy - let's call it the *operationalist
fallacy* - that infects much of the thinking about the foundations of
mathematics. In its simplest form it consists in the conviction that
the natural numbers 0, 1, 2, . . . constitute the "raw data" of
mathematics, that they are simply "given" to us as a unique infinite
structure which can be characterised fully and rigorously as "the
successive images of zero under repeated applications of the
*successor operation*". On this conception, the principles of proof
by mathematical induction and definition by recursion are simply
"given" along with the natural numbers themselves, so that, in
particular, these two principles can be accepted as legitimate
without further justification: they are, in short, *self-evident*.
Another form of the operationalist fallacy is to be found in
the view of formal syntax in which the "constructions" employed
(which are really recursively defined functions) are somehow "given"
immediately as self-evidently efficacious, and so do not require
mathematical analysis or justification. The operationalist fallacy
thus underlies what I believe to be the illusion that Formalism is a
coherent foundational theory.
The first thinkers to acknowledge the operationalist fallacy
were Frege and Dedekind, Frege in his definition of "following in a
series" in the *Begriffsschrift* and the *Grundlagen*, and Dedekind
in the theory of "chains" expounded in his monograph, *Was sind und
was sollen die Zahlen?*. The clearest description of the fallacy *as*
a fallacy, however, is to be found in Dedekind's "Letter to
Kefferstein", and, moreover, Dedekind saw much more clearly than
Frege did, the central significance of the principle of definition by
recursion.
Zermelo also belonged to the anti-operationalist party, and
in his article "Sur les ensembles finis et le principe de l'induction
compl`ete" gave a purely set-theoretical derivation of mathematical
induction, a derivation, what is more, that did not rest upon any
assumption about the existence of transfinite sets. I think this work
of Zermelo's is highly significant, representing as it does a kind of
*finitary*, but nevertheless  *completely set-theoretical*, analysis
of  natural number.
In fact, the lesson that emerges from the work of all three
of these great pioneers of our subject is that the notion of "set" is
prior to the notion "natural number" and we must employ it if we are
to carry out a proper analysis of the principles of induction and
recursion in natural number arithmetic, or in formal syntax, come to
that.  But, surely, this runs counter to the widespread perception
among the general public, and, indeed, among mathematicians as well,
that set theory is the very quintessence of the "New Math", whereas
natural numbers, the familiar things we count with, constitute the
oldest part of our mathematical inheritance.
That widespread perception, however, is just a mistake. In
Chapter 2 of my book, which I have posted on my web page, I have
called attention to the fact that our notion of *natural number* is a
relatively recent invention, and that the ancient notion of "number"
(the Greek notion of *arithmos*, namely, a finite plurality composed
of units) that it has replaced is essentially our notion of "(finite)
set". What is more, the ancient notion is very much better as a
vehicle for explaining the facts that underlie the science of
arithmetic than our modern notion of "natural number".
But that is not all, for as I point out in my Preface (which
is also posted on my web page), modern set theory itself is really
best regarded as a refined and generalised version of classical
arithmetic, though, to be sure, a *non-Euclidean* arithmetic, since
it abandons Euclid's axiom (Common Notion 5) that "the whole is
greater than the part" by assuming the existence of "numbers" (i.e.
arithmoi, that is to say, *sets*) which have the same size as certain
of their proper subsets. Indeed, I am convinced that the best way to
look at the development of modern set theory is as a transition
*within classical arithmetic* from the *Euclidean* conception of
"finiteness" to the modern *Cantorian* conception. From that
standpoint, all the central axioms of Zermelo-Fraenkel set theory -
including the so - called Axiom of Infinity - are to be seen as
*finiteness assumptions* and, as such, take on that air of
self-evidence that is expected of "axioms" understood in the
old-fashioned sense of  "basic truths".
What I am calling the "operationalist fallacy" here also
arises, in a different guise, in the foundations of Cantorian set
theory as well, where it takes the form of positing "transfinite
processes" for defining functions (e.g. as in the "generation" of the
universe of sets as a cumulative hierarchy by the "transfinite
iteration" of the power set operation). In Chapter 5 I attempt to lay
down axioms for conventional Cantorian set theory that avoid taking
these supposed "processes" as fundamental.
Perhaps the most novel aspect of the book is the treatment,
in Part 4, of what I call "Euclidean set theory" which is classical
arithmetic, with its Euclidean conception of finiteness, but
reformulated as a version of Zermelo Fraenkel set theory in which
Cantor's (misnamed) "Axiom of Infinity" is replaced by an axiom to
the effect that all sets are Dedekind finite. In the theory of
"natural number arithmetic" that naturally arises in this theory
there are natural number systems (i.e. simply infinite systems) of
different lengths, for Dedekind's theorem that all simply infinite
systems are isomorphic is no longer provable. If you want an idea of
how to make sense of that, look at Part 4 of the Analytical Table of
Contents that I have posted on my web site. To see it worked out in
detail, of course, you have to turn to the book itself.

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John Mayberry
School of Mathematics
University of Bristol
j.p.mayberry at bristol.ac.uk
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