One hundred years of russells paradox mathematics, logic, philosophy

de Gruyter Series in Logic and Its Applications 6
Editors: W. A. Hodges (London) · R. Jensen (Berlin)
S. Lempp (Madison) · M. Magidor (Jerusalem)


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One Hundred Years of Russell’s Paradox
Mathematics, Logic, Philosophy
Editor

Godehard Link

≥

Walter de Gruyter
Berlin · New York

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Editor
Godehard Link
Philosophie-Department
Ludwig-Maximilians-Universität München
Ludwigstr. 31/I
80539 München
Germany
Series Editors
Wilfrid A. Hodges

School of Mathematical Sciences
Queen Mary and Westfield College
University of London
Mile End Road
London E1 4NS
United Kingdom

Ronald Jensen
Institut für Mathematik
Humboldt-Universität
Unter den Linden 6
10099 Berlin
Germany

Steffen Lempp
Department of Mathematics
University of Wisconsin
480 Lincoln Drive
Madison, WI 53706-1388
USA

Menachem Magidor
Institute of Mathematics
The Hebrew University
Givat Ram
91904 Jerusalem
Israel

Mathematics Subject Classification 2000: 03A05; 00A30, 03-06

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Library of Congress Ϫ Cataloging-in-Publication Data
One hundred years of Russell’s paradox : mathematics, logic, philosophy / edited by Godehard Link.
p.
cm. Ϫ (De Gruyter series in logic and its applications; 6)
Includes bibliographical references and indexes.
ISBN 3-11-017438-3 (acid-free paper)
1. Paradox.
2. Liar paradox.
I. Title: 100 years of Russell’s
paradox. II. Link, Godehard. III. Series.
BC199.P2O54 2004
165Ϫdc22
2004006962

ISBN 3-11-017438-3
ISSN 1438-1893
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Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie;
detailed bibliographic data is available in the Internet at ϽϾ.
Ą Copyright 2004 by Walter de Gruyter GmbH & Co. KG, 10785 Berlin, Germany.
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Preface

In June 2001 an international conference was held at the University of Munich, Germany, to commemorate the centenary of the discovery of Russell’s paradox. It brought
together leading scholars from the following fields:
• Russell Studies
• Mathematical Logic
• Set Theory
• Philosophy of Mathematics
The aims of the event were twofold, historical and systematic. One focus was
on Russell’s logic and logical philosophy, which were shaped so much by his own
and related paradoxes, in particular since new material has become available in recent
years from the rich sources of the Russell Archives through the publication of the
Collected Papers of Bertrand Russell. But the second aim was of equal importance:
to present original research in the broad range of foundational studies that draws on
both current conceptions and recent technical advances in the above-mentioned fields.
It was hoped to contribute this way to the well-established body of mathematical
philosophy initiated to a large extent by Russell a hundred years ago.
The conference featured plenary sessions with distinguished invited speakers, section meetings with both invited and contributed papers, and as special events two panel
discussions, “Russell in Context” and “The Meaning of Set Theory”, as well as two
symposia, one on “Propositional Functions” and the other on the “Finite Mathematics
of Set Theory”. The Russell panel, chaired by A. Irvine, was composed of N. Griffin,
P. Hylton, D. Kaplan, and A. Urquhart. Y. Moschovakis chaired the panel on Set Theory, with S. Feferman, H. Friedman, D. C. McCarty, and W. H. Woodin as panelists.
The symposiasts of the first symposium, chaired by R. Wahl, were G. Landini and
B. Linsky, those of the second S. Lavine and K.-G. Niebergall, chaired by the editor.
The papers collected in this volume represent the main body of research emanating
from the proceedings of the conference. All authors delivered a talk at the conference

except J. Mycielski, who was unable to attend but sent his invited paper, and H. Field,
who attended the conference as a discussant and was invited by the editor to contribute
his present paper. G. Jäger and H. Schwichtenberg were each joined by a co-author
for their paper included here. A number of invited speakers delivered a talk, but
were unable to contribute a paper to the volume because of other commitments; these
are W. Buchholz, P. Hylton, H. Kamp, P. Martin-Löf, Y. Moschovakis, C. Parsons,
G. Priest, and A. Urquhart.

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vi

Preface

The papers were all originally written for this volume, with two exceptions. The
original place of publication of H. Field’s paper “The Consistency of the Naive Theory
of Properties” is The Philosophical Quarterly, volume 54, No. 214, January 2004.
It is reprinted here by kind permission of Blackwell Publishing and the author. A
condensed version of Hazen’s paper appeared as A. P. Hazen and J. M. Davoren,
Russell’s 1925 logic, in: Australasian Journal of Philosophy, volume 78, No. 4,
pp. 534–556, December 2000. Reprint of this material as part of the present paper by
kind permission of Oxford University Press and the co-author.
The occasion to publish this volume would never have arisen without the essential financial support from a number of institutions that made the conference possible
in the first place. It is therefore fitting to express my deep appreciation here to the
Deutsche Forschungsgemeinschaft (DFG ) for covering the main bulk of the conference costs through funds made available to the interdisciplinary DFG graduate program
Graduiertenkolleg “Sprache, Information, Logik” (SIL ) at Munich University, and
special conference funds; furthermore to the following institutions and private sponsors
for filling the remaining financial gaps: the Bayerisches Staatsministerium für Wissenschaft, Forschung und Kunst, the Philosophy Department, University of Munich,
the Gesellschaft von Freunden und Förderern der Universität München (Münchener

Universitätsgesellschaft e.V. ), the University of the German Federal Armed Forces
Munich, Apple Computer Germany, and Alpina Burkard Bovensiepen GmbH & Co.
Many people helped in various ways to organize the conference and to prepare
this volume. I wish to thank all of them, in particular Solomon Feferman for his
encouragement, and Andrew Irvine for generous advice, in the early stages of the
conference project. Ulrich Albert carried the main organizational burden of the conference as “front liner” on the internet and at the conference site; Sebastian Paasch
was responsible for the technical part of the preparation of the volume. My thanks
to both of them for their efficiency and their commitment far beyond the call of duty.
Ulrich and Sebastian were assisted at different stages by Marie-Nicole Ehlers, Martin
Fischer, Roland Kastler, Uwe Lück, Michaela Paivarinta, Hannes Petermair, Marek
Polanski, Christian Tapp, Mai-Lan Thai, Julia Zink, and Mauricio Zuluaga. Daniel
Mook helped with checking the English of non-native speakers.
I owe a special debt to Karl-Georg Niebergall, who was always close at hand for
any advise on matters of content pertaining to the contributions of the volume, and
whose philosophical seriousness, enthusiasm and unfailing technical judgment I have
had the privilege to enjoy in many years of joint seminars and private discussions.
Finally, I wish to extend my thanks to Dr. Manfred Karbe of Walter de Gruyter
Publishers for supporting the decision to publish the volume, and for his sustained
encouragement, advice, and patience during the phase of its preparation.
May 2004

G. L.

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Table of Contents

Preface

v

Godehard Link
Introduction. Bertrand Russell—The Invention of Mathematical Philosophy

1

W. Hugh Woodin
Set Theory after Russell: The Journey Back to Eden

29

Harvey M. Friedman
A Way Out

49

Sy D. Friedman
Completeness and Iteration in Modern Set Theory

85

Kai Hauser
Was sind und was sollen (neue) Axiome?

93

Gerhard Jäger and Dieter Probst
Iterating Operations in Admissible Set Theory without Foundation:
A Further Aspect of Metapredicative Mahlo

119

Solomon Feferman
Typical Ambiguity: Trying to Have Your Cake and Eat It Too

135

Karl-Georg Niebergall
Is ZF Finitistically Reducible?

153

Tobias Hürter
Inconsistency in the Real World

181

Michael Rathjen
Predicativity, Circularity, and Anti-Foundation

191

John L. Bell
Russell’s Paradox and Diagonalization in a Constructive Context

221

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viii

Table of Contents

Peter Schuster and Helmut Schwichtenberg
Constructive Solutions of Continuous Equations

227

Kai F. Wehmeier
Russell’s Paradox in Consistent Fragments of Frege’s
Grundgesetze der Arithmetik

247

Andrea Cantini
On a Russellian Paradox about Propositions and Truth

259

Hartry Field
The Consistency of the Naive Theory of Properties

285

Ulrich Blau
The Significance of the Largest and Smallest Numbers
for the Oldest Paradoxes

311

Nicholas Griffin
The Prehistory of Russell’s Paradox

349

Gregory Landini
Logicism’s ‘Insolubilia’ and Their Solution by Russell’s Substitutional Theory

373

Philippe de Rouilhan
Substitution and Types: Russell’s Intermediate Theory

401

Francisco Rodríguez-Consuegra
Propositional Ontology and Logical Atomism

417

Bernard Linsky
Classes of Classes and Classes of Functions in Principia Mathematica

435

Allen P. Hazen
A “Constructive” Proper Extension of Ramified Type Theory (The Logic of
Principia Mathematica, Second Edition, Appendix B)

449

Andrew D. Irvine
Russell on Method

481

Volker Peckhaus
Paradoxes in Göttingen

501

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Table of Contents

David Charles McCarty
David Hilbert and Paul du Bois-Reymond: Limits and Ideals

ix

517

Jan Mycielski
Russell’s Paradox and Hilbert’s (much Forgotten) View of Set Theory

533

Shaughan Lavine
Objectivity: The Justification for Extrapolation

549

Geoffrey Hellman
Russell’s Absolutism vs. (?) Structuralism

561

Robert S. D. Thomas
Mathematicians and Mathematical Objects

577

Holger Sturm
Russell’s Paradox and Our Conception of Properties, or: Why Semantics
Is no Proper Guide to the Nature of Properties

591

Vann McGee
The Many Lives of Ebenezer Wilkes Smith

611

Albert Visser
What Makes Expressions Meaningful? A Reflection on Contexts and Actions

625

List of Contributors

645

Name Index

649

Subject Index

653

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Introduction. Bertrand Russell—The Invention of
Mathematical Philosophy
Godehard Link

1. The Grand Conjunction
It is a theorem of elementary logic that, given an arbitrary two-place relation R, there is
no object y such that for any x, x bears R to y just in case x does not bear R to itself. If
the domain is the universe of sets, and R is the membership relation, then the theorem
says there is no set consisting of those and only those sets that are not members of
themselves. However, the ‘naïve’, unrestricted, comprehension scheme claims the
contrary and thus produces Russell’s paradox. By the same token, if R is the relation

of predicability and the domain consists of predicates then there is no predicate that is
predicable of just those predicates that are not predicable of themselves. A collection
schema assuming the contrary yields the form of the paradox in which it was first
formulated by Russell in May 1901. In the 1901 draft of Part I of his Principles of
Mathematics he writes:
The axiom that all referents with respect to a given relation form a class
seems, however, to require some limitation, and that for the following
reason. We saw that some predicates can be predicated of themselves.
Consider now those … of which this is not the case. … [T]here is no
predicate which attaches to all of them and to no other terms. For this
predicate will either be predicable or not predicable of itself. If it is
predicable of itself, it is one of those referents by relation to which it was
defined, and therefore, in virtue of their definition, it is not predicable of
itself. Conversely, if it is not predicable of itself, then again it is one of
the said referents, of all of which (by hypotheses) it is predicable, and
therefore again it is predicable of itself. This is a contradiction. ([70]: 195)
It is of some historical interest that Russell should first write down his paradox in the
predicate version. According to his own account ([80]: 58) he had discovered it by
analyzing Cantor’s paradox of the greatest cardinal, where the context was set theory.
Given that, it seems that Russell had already isolated the precarious scheme inviting
paradox, viz., diagonalizing a relation and negating it, which is the common core of

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the Liar, Cantor’s Theorem, and Gödel’s Incompleteness Theorem. As Gödel would

put it years later:
By analyzing the paradoxes to which Cantor’s set theory had led, he
freed them from all mathematical technicalities, thus bringing to light the
amazing fact that our logical intuitions (i.e., intuitions concerning such
notions as: truth, concept, being, class, etc.) are self-contradictory. ([28]:
131)
Of course, the historical record suggests that it took Russell some time to realize
that he had discovered something that fundamental.1 In particular, it was not before
Frege’s reaction of shock, a full year later, upon receiving Russell’s famous letter
describing the paradox—now also in its class form—that Russell came to appreciate
its real impact and felt reassured enough to publish his findings. Thus the paradox
first appeared in print with the publication of the Principles in May 1903.
According to existing reports the paradox was independently discovered by Ernst
Zermelo. The first reference to this appears in a letter that Hilbert wrote to Frege in
late 1903 thanking him for a copy of the second volume of the Grundgesetze which
Frege had sent to him. “The example you give at the end of the book,” says Hilbert,
referring to the afterword in which Frege acknowledges Russell’s paradox, “has been
known to us here,” adding in a footnote: “I think it was 3-4 years ago that Dr Zermelo
found it after I had communicated my [own] examples to him” ([24]: 24). It is likely
that by his “[own] examples” Hilbert meant what has become known as “Hilbert’s
Paradox”,2 to which I return presently.
Given the small step from the proof idea of Cantor’s Theorem to the idea of the
Russell class it should come as no surprise that an astute mind like Zermelo hit upon
it by himself. What might be surprising is that Zermelo didn’t publish it. Before
arriving in Göttingen—and this is less known—he had already left a mark in the
history of science in quite a different field. As an assistant of Max Planck in Berlin
he had, drawing on a theorem of Poincaré’s, devised what is called the recurrence
paradox, which together with Loschmidt’s reversibility argument put a serious threat
to Boltzmann’s program of reducing thermodynamics to statistical mechanics.3 So
why didn’t he ‘strike’ again, this time threatening Cantor’s newly erected edifice of

set theory? The usual speculation about this is that at the time “the example” wasn’t
taken seriously by him, nor by Hilbert, for that matter.4 Most likely, Zermelo, who
had become a convinced Cantorian, viewed the problem as belonging to the same
category as Cantor’s paradox of the greatest cardinal and Burali-Forti’s problem of the
greatest ordinal. Presumably, these were all problems of a ‘regional’ kind, rather to
1 See Griffin’s contribution in the present volume.
2 See [57], and also Peckhaus’s paper in this volume.
3 The father of quantum theory seemed to be impressed by it, which might have been part of the reason

why he remained skeptical about Boltzmann’s atomism for such a long time; see, e.g., [46]: 26f.
4 For instance, assuming the paradox was discovered and communicated to him before the Paris meeting
in August 1900, Hilbert could have mentioned it there in his famous lecture on mathematical problems.

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Introduction

3

be played down than exaggerated,5 having to do with extrapolating arithmetic into the
transfinite, and living within the bounds, or even on the fringes, of Cantor’s theory; or
so it seemed.
By the time he wrote his letter to Frege, however, Hilbert at least must have
changed his attitude regarding these anomalies. For there he called his own examples “even more convincing contradictions” ([24]: 24), more convincing because he
considered them to be of a “purely mathematical nature.”6 He continues with the
explicit assessment that there is something seriously amiss in the very foundations of
logic and mathematics. So the diagnosis was there now, and it was the same that the
philosopher Russell had arrived at. However, the cure was not to come for years. As
for Hilbert’s part, the main focus of attention at the time lay elsewhere, on classical

number theory, culminating in his proof of Waring’s theorem in 1908, on integral
equations and on mathematical physics, activities intensified by his close cooperation
with H. Minkowski, up to the latter’s premature death in 1909 (see, e.g., [68]). Thus
in Göttingen, progress in the foundational work was basically left to Zermelo, who,
however, had a different, much more mathematical, perspective, being preoccupied
with the axiom of choice and its justification. While he did lay the foundations of
modern set theory in the course of it, these circumstances nonetheless gave Russell a
start of a number of years which he needed, after many abortive attempts full of blind
alleys and frustrating toil, to find his own way out of the paradoxes. By bringing all
the ‘regional’ versions of the paradoxes together, including the ‘semantical’ ones, he
was able to broaden the scope of the investigation and to contribute to the foundations
of logic proper.
Paradoxes like the Liar or those about the nature of infinity had been around as
famous puzzles since antiquity, but the tools to tackle them had not been forthcoming
in all that time. What counted as logical discourse, in Cambridge and elsewhere7 was
replete with ambiguity and obscurity. Precision and rigor were missing, and traditional
logic had no way to come to serious terms with such genuinely logical problems as
the paradoxes, old and new.
It is a curious fact about the history of knowledge that early on, at the time of the
Greeks, logic should have dissociated itself from mathematics. Perhaps this is simply
due to the contingent fact that the man who invented formal logic, Aristotle, was for
all his greatness not known for excelling in the field of mathematics. At that time
5 Those earlier problems didn’t even seem paradoxical to Cantor and Burali-Forti, as G. Moore [54]
argues.
6 See ([57]: 168), for the phrase used by Hilbert. The authors explain that Hilbert’s construction doesn’t
make use of the particular rules of ordinal and cardinal arithmetic set up by Cantor. Also, the editors of
[24] add a footnote to the passage quoted in which they refer in turn to a remark by O. Blumenthal in
his biographical essay of Hilbert [6]. There Blumenthal speaks of “the example, devised by Hilbert and
nowhere leaving the domain of purely mathematical operations, of the contradictory set of all sets formed
by union and self-mapping [Selbstbelegung]” ([6]: 422).

7 The work of the neo-Hegelian philosopher F. Bradley, for instance, marks a point of reference for the
appreciation of the tremendous change that the subject of logic was to undergo in the course of Russell’s
contribution.

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began what I like to call the Babylonian captivity of logic in the realm of metaphysics,
lasting for over 2000 years. When in the age of the scientific revolution Aristotelian
metaphysics became the target of modernizers like Galileo, logic was considered part
and parcel of metaphysics and was dismissed together with the philosophy Galileo
fought against. For him, formalization of logic was obsolete; what was needed from
logic he considered as natural and no real subject of study, certainly no precondition
for founding the new science of physics. In his polemic The Assayer [25] Galileo said
famously:
[Natural] philosophy is written in this grand book, the universe, which
stands continuously open to our gaze. But the book cannot be understood
unless one first learns to comprehend the language and read the letters in
which it is composed. It is written in the language of mathematics, and
its characters are triangles, circles, and other geometric figures without
which it is humanly impossible to understand a single word of it. ([25]:
237f.)
Thus, if even syntax was in the domain of geometry, little room was left for logic. In
fact, the subject of logic was not only ignored, it acquired an outright bad reputation
with scientists and mathematicians over the centuries, still to be felt in Poincaré’s
polemics. Even Kant, while certainly not prejudiced against logic, said nonetheless

that logic hadn’t made any progress since Aristotle. He saw this subject as a beautiful
finished edifice which could only suffer from further additions to it:
If some of the moderns have thought to enlarge it [logic] by introducing
psychological chapters on the different faculties of knowledge (imagination, wit, etc.), metaphysical chapters on the origin of knowledge or on the
different kind of certainty … or anthropological chapters on prejudices,
their causes and remedies, this could only arise from their ignorance of
the peculiar nature of logical science. ([44]: Preface, B VIII)
Here Kant should not be taken to mean that logic cannot be applied to other fields of
knowledge, but rather that the subject matter of logic proper is neither psychology nor
metaphysics. But what Kant complained about had happened all the time and was
soon to culminate prominently again in Hegel’s work and that of his followers. This
was also the situation in England when Russell arrived at the scene.
So what was Russell’s point of departure? In hindsight, it can be said that Russell
didn’t discover his paradox quite by accident. The late 1890s saw him deeply involved
in studying what was known among idealist philosophers as the ‘contradictions’ of
the infinitely large and the infinitely small. At the time, philosophy in Cambridge was
dominated by neo-Hegelians, and Russell adopted their point of view.8 Thus, in spite
of his mathematical training, he took a philosophical interest in mathematics and its
foundations. Even after he had given up his idealist convictions he kept his thoroughly
8 See, for instance, [41], [37], [54].

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Introduction

5

philosophical outlook. His ontology had become pluralistic, but the ‘theoretical terms’
of his logic consisted of the same kind of philosophical entities that traditional logicians

like Bradley would use, viz., propositions, concepts, and, with a novel emphasis on
their importance, relations. In contrast, for instance, reflection on language as a
prerequisite to modern logic played no particular role yet in Russell’s work. The
Principles, even in those parts where he introduced the innovation of the denoting
relation for the first time, were written in a rather traditional philosophical style,
certainly not apt to draw a large readership among mathematicians.9 The situation
was just like Frege described it in the Preface of his Grundgesetze [23], where he said
that he had to give up on the mathematicians who would call his work “metaphysics”
because of expressions like ‘concept’, ‘relation’, ‘judgement’ occurring in it. There
were those two separate communities, and it seemed hard to imagine how they should
take interest into one another. Mainstream mathematicians did not only care little
about philosophy but considered even Cantor, who was really one of them, as too
philosophical to be taken seriously. It is a telling historical detail that Felix Klein’s
widely read lectures on the development of mathematics in the nineteenth century [45]
mentions Cantor only in passing. Russell clearly expressed this state of affairs in My
Philosophical Development [80], where he says:
The division of universities into faculties is, I suppose, necessary, but it
has had some very unfortunate consequences. Logic, being considered to
be a branch of philosophy and having been treated by Aristotle, has been
considered to be a subject only to be treated by those who are proficient
in Greek. Mathematics, as a consequence, has only been treated by those
who knew no logic. From the time of Aristotle and Euclid to the present
century, this divorce has been disastrous. ([80]: 51)
Here is now what I consider Russell’s historic achievement. Open-minded, quick,
and polyglot, he was able to engage philosophers and mathematicians alike in an
excitingly new foundational debate. And crucially, this was not just another epistemological enterprise, important as meta-reflections are for clarifying the conceptual
and methodological premises of a given discipline. It was all that, but it was more:
it opened up new horizons for technical research of an unprecedented kind, deriving from a few basic logical premises. Russell, who was surely not equipped with
the most advanced mathematical expertise of his time,10 nevertheless almost singlehandedly produced viable solutions to the new problems that were rigorous enough to
spark interest in the mathematicians’ camp, in particular, with Poincaré and the Hilbert

school. Poincaré, highly skeptical about symbolic logic, still had a hand in uncovering
the presuppositions of quantification, while Hilbert was of course the one influential
mathematician who recognized early on that the paradoxes did pose a problem for
9 For instance, Peckhaus [this volume] cites G. Hessenberg in Göttingen for “scathing” comments on

Russell’s book.
10 Mathematics at Cambridge was not the best to be had in Europe at the time, as Russell admits ([80]:
29).

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the foundations of mathematics, and that something had to be done about it.11 Some
of the finest younger mathematicians who came to specialize in the new discipline
or at least substantially contributed to it at some point in time (Weyl, Bernays, von
Neumann, Gödel) took their starting point from the framework of Principia in one
way or the other. On the philosophers’ side, Wittgenstein, Ramsey, Carnap, and Quine
stand out; each of them adopted the Russellian outlook as well, although Wittgenstein,
of course, was soon to strike out in quite a different direction.
It is this combination of transdisciplinary philosophical query, mathematical training and logical expertise that characterize Russell’s paradigm of mathematical philosophy. Russell therefore occupied a unique role in bringing about what I like to call,
in an astronomical metaphor, the Grand Conjunction of the old and venerable fields
of philosophy, logic, and mathematics.
Let us remind ourselves of some important features of Russell’s work that are
representative for this new enterprise.

1.1. Some Russellian Themes

1.1.1. Symbolic Explicitness
Today it is hardly necessary to stress the benefits of symbolism, and in particular,
the salutary role of a good symbolism. The Peano–Russell notation of Principia was
indeed a good one, serving a generation of logicians well (Frege, for instance, was less
lucky in convincing others to use his symbolism, in spite of his clarity of insight and
meticulous exposition). At first, when Russell seized upon Peano’s notation, it was
perhaps not more than an expedient to express ideas succinctly; historically, however,
it is more than that. It was a unique way, in philosophy and even in mathematics, of
‘making it explicit,’ to use a catch phrase of today. Distinctions could be drawn that
wouldn’t be noticed otherwise, thus preparing the ground for new questions and lines of
research. The organizing power of the symbolism made Russell become definite about
his primitives, aware of what could be defined in terms of what, and what was needed
to prove an assertion. In his case in particular, it also led to productive strictures with
the conceptual apparatus he started out with. This is most clearly seen in his gradual
retreat from the received metaphysical approach towards his fundamental innovations,
as documented in his writings from the Principles leading all the way up to Principia.
In the latter work, he finally was in the position to give proofs and derivations from
first principles, more geometrico. It strikes the eye that he had thereby left behind
everything that was so far considered ‘logic’ in the philosophers’ sense.
Even so, we should not pass over the weak points that could not be covered up by
the extensive formalism; as Gödel said: “It is to be regretted that this first comprehensive and thorough going presentation of a mathematical logic and the derivation
11 See again Peckhaus’s paper in this volume with references therein, and also [50].

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Introduction

7

of Mathematics from it is so greatly lacking in formal precision in the foundations
(contained in *1–*21 of Principia), that it presents in this respect a considerable step
backwards as compared with Frege” ([28]: 126). Also, what we today take as the
hallmark of modern mathematical logic, a notion of metatheory, was barely to be
found (but see below). Related to this failure is the notorious laxity in matters of use
and mention, which Quine famously harped on and induced him to basically rewrite
Principia from scratch. Thus I think it is fair to say that Russell’s logic was still
in a pre-paradigmatic stage of development. The real ‘founding document’ of the
new discipline was Gödel’s paper on formally undecidable propositions [27]; but as
its complete title shows, the point of reference were the Principia Mathematica of
Whitehead and Russell.
1.1.2. On the Road from Metaphysics to Logical Syntax
In his unique laboratory of ideas reflected in the Principles,12 Russell worked with
traditional philosophical entities like propositions and concepts. Propositions were
certain complexes of ‘terms’, that is, objects,13 which—though clearly patterned after
their syntactic form—were no syntactic entities. Rather, they were conceived in
analogy with regular physical objects consisting of various extra-linguistic parts. The
notion of ‘constituent’ used in this connection was ambiguous, meaning both a part
of an expression and a part of complex entity. Real syntactic leverage is introduced,
however, when Russell begins to inquire into the notion of substitution, with which
he experimented a lot over the years. This investigation had the natural tendency
to acquire a syntactic character almost by itself. But for Russell the transition to
syntax was by no means automatic. Quite to the contrary, he struggled with the
formalism and its interpretation for a long time, trying to do equal justice to the
demands of metaphysics and the emerging discipline of symbolic logic. This is the
kind of tension mentioned above that led to all the endemic ambiguities that modern
readers of Russell justly complain about,14 wondering time and again whether Russell
is talking about entities or expressions. I think that the failure to distinguish clearly
between the expression and the object it signifies stems from a neglect of language
in Russell’s early work where it is not expressions but concepts that denote ([71]:

chap. V). Philosophically, his own ‘linguistic turn’ was still to come, in spite of all the
symbolic machinery he had already adopted.15 Even in his seminal essay On Denoting
12As Quine says of this book: “It is the more remarkable, then, that this prelogical logic of Russell’s, this
morass of half-formulated problems, should already contain the embryo of twentieth-century philosophy”
([62]: 5).
13 Recall that the word ‘term’ in the Principles means an entity and not an expression.
14 For instance, even in Principia he could still write: “A function, in fact, is not a definite object, … it
is a mere ambiguity awaiting determination” ([85]: 48).
15 This judgement seems to contradict what Russell himself says about that time: “I was very much
occupied, in the early days of developing the new philosophy, by questions which were largely linguistic. I
was concerned with what makes the unity of a complex, and, more especially, the unity of a sentence” ([80]:
49). However, the very example he gives—involving the philosophically charged concept of part-whole—is
an indication of the fact that it is still mainly metaphysics that he worries about.

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[72] we find him wavering between the newly introduced ‘denoting phrases’ and the
old ‘denoting complexes’. The infamously obscure Gray’s Elegy argument occurring
there bears ample witness of the fact that Russell had not quite worked out the basic
semiotic relations.16
The other principal case is point is the ambiguous status of the propositional functions, which, after Russell had abandoned classes as regular objects, had to carry the
main ontological burden of Principia’s philosophy. The exact criterion of identity of
propositional functions—or of some equivalent intensional notion of property—has
remained a matter of debate up to the present day, that is, whether they should be
conceived as equipped with outright syntactic granularity or something more coarsegrained yet short of extensionality.

When I speak of Russell’s road to syntax I don’t mean to say that he left metaphysics
behind completely; not so, as his theory of logical atomism shows. But in his hands
metaphysics or ontology became more amenable to a logical treatment in the modern
sense. The development also showed, however, how problematic the old metaphysical
conception of ontology became which still constituted the philosophical underpinning
of Principia. Russell’s intellectual crisis in 1913, produced by the events surrounding
the unfinished Theory of Knowledge manuscript [75], testifies to this; however great
the role of Wittgenstein’s criticisms was in “paralyzing” Russell, the internal problems
of traditional metaphysics still occupying his mind were such that they led the whole
program of logical atomism into an impasse.17
1.1.3. The Re-Invention of Ockham’s Razor As a Logical Tool
This is, of course, Russell’s famous method of eliminating dubious entities by the
technique of incomplete symbols in his theory of description, first laid down in On
Denoting and elaborated in *14 of Principia. It not only swept away the notorious
round square and all its lookalikes, but also expelled classes as first-class citizens from
his ontology. Now it is one thing to proclaim that entities should not be multiplied
beyond necessity, and quite another to devise an impeccable logical method of filling
this slogan with meaningful content. That this was not an altogether trivial matter is
shown by the complication arising from the presence of, first, logical operators, which
are sensitive to scope, and secondly, more than one descriptive phrase, where the order
of elimination becomes an issue. Russell provided viable solutions to these problems,
albeit refined by others later on in various directions. Thus, his theory of descriptions
is rightly hailed as a landmark of genuine logical philosophy.
The advent of this method marked a decisive point on Russell’s road to logical
syntax. The denoting concepts of the Principles evolved into denoting phrases and
dissolved altogether as an ontological substance. Incomplete symbols are plainly
16 However, Landini’s paper in this volume contains an intriguing case for the claim that in spite of all

this, Russell’s “substitutional theory” can be reconstructed rather faithfully in a way as to contain a viable
solution to the paradoxes and a surprising justification of logicism prior to Principia.

17 On this, see also Rodríguez-Consuegra’s paper in this volume.

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9

expressions, there are no ‘incomplete entities’ to be had. Incompleteness in this sense
derives from the semiotic function attached to linguistic expressions. Not every part
of a meaningful expression needs to be meaningful, some parts may require a context
to become so. This notion of context doesn’t make sense with extra-linguistic objects,
which (usually) lack a semiotic role. In the special case of mathematical objects, it
is true that Frege called functions “incomplete” or “unsaturated”; but we mustn’t take
that literally here: in fact, Frege first models logic after mathematics by assimilating
concepts to mathematical functions, and then uses the syntactic (indeed, chemical)
metaphor of saturation to characterize functions again.
In the foundations of mathematics, the main benefit Russell derived from the
technique of incomplete symbols was, of course, the prospect of doing mathematics
without a commitment to classes.
The following theory of classes, although it provides a notation to represent them, avoids the assumption that there are such things as classes.
This it does by merely defining propositions in whose expression the symbols representing classes occur, just as, in *14, we defined propositions
containing descriptions. … The incomplete symbols which take the place
of classes serve the purpose of technically providing something identical
in the case of two functions having the same extension. ([85]: 187)
In view of current discussions about anti-Platonist approaches to an understanding of
mathematical practice18 we have to remember that Russell’s no-classes theory didn’t
spring from any nominalist convictions—quite the opposite, he was a realist regarding
universals, in particular, relations. The Quinean extensionalist reform, which basically

equated all universalia with classes, was still to come, so ousting classes was not a
nominalist project for Russell. Rather, it seems that the effects the paradoxes had
on him were such that he never trusted classes again, which he took responsible for
engendering the antinomies. By reducing class talk to propositional function talk
Russell paved the way for a conceptualist approach to set theory.
1.1.4. The Nature of Quantification
As far as quantification is concerned Russell had to come a particularly long way
from where he started out. For instance, in his Principles he cannot make up his
mind about the semantic mechanism involved in the assertion of a noun phrase like
some man. He wonders “whether an ambiguous object is unambiguously denoted,
or a definite object ambiguously denoted” ([71]: 62). To save his newly discovered
denoting relation from this ambiguity he shifts it to the object denoted, with obviously
little convincing results. Metaphysical constraints about what can or cannot not be
part of the proposition expressed by an assertion involving phrases like some man
or all men make him come up with an account that is squarely at variance with the
Frege–Peano style of quantification which Russell uses elsewhere in the same book.
18 See, e.g., [21], [40].

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But there is a piece of linguistic philosophy emerging, which Russell puts to logical
use, when he draws a distinction between the genuinely universal quantifier all and
the logical meaning of free-choice any. This distinction, which appears in [74] and
made it into the first edition of Principia, can be viewed as an instance of Russell
stepping out of his single, all-encompassing formal system and moving towards some

sort of metalanguage.19 In [74], after reviewing the paradoxes, he concludes that they
all have in common “the assumption of a totality such that, if it were legitimate, it
would at once be enlarged by new members defined in terms of itself. This leads us
to the rule: ‘Whatever involves all of a collection must not be one of the collection’ ”
([74]: 63). This is one form of the Vicious Circle Principle, which however, as
Russell goes on to say, makes “fundamental principles of logic” meaningless, such
as ‘All propositions are either true or false’. Here he finds a role to play for any:
“Hence the fundamental laws of logic can be stated concerning any proposition, though
we cannot significantly say that they hold of all propositions” ([74]: 68). This is
because in Russell’s type theory, the logical tool of overt quantification is, by necessity,
parametric, i.e., relative to a given type; truly universal claims can only be made by a
quantifier-free, schematic assertion. That might be interpreted as foreshadowing the
finitary assertions of Hilbert’s metamathematics. However, Russell never made the
step to distinguish between an object language and a metalanguage, and so he didn’t
really know what to do with this observation in technical terms. Consequently, the
separate role of any was given up in the second edition of Principia. It seems to me,
though, that what Russell hinted at comes closer to what has been reconstructed in
recent years as the concept of arbitrary object,20 which is a philosophical idea.
Suffice it to say, then, that Russell did arrive at the modern notion of a quantified
sentence as a separate form, equipped with a specific range for the bound (“apparent”)
variable. But, as Goldfarb [35] rightly points out, since there was no notion of model, in
which the quantifiers and some non-logical vocabulary could be variously interpreted,
neither he nor Frege, for that matter, can be said to have developed the full-fletched
concept and machinery of current quantification.
1.1.5. The Hierarchy of Types
It was suggested above that it took Russell’s original logico-philosophical frame of
mind not only to recognize the serious nature of the problem of the paradoxes but
to sit down and work out a ‘non-regional’, comprehensive solution as well. Putting
the paradoxes in a proper philosophical perspective Russell also tried to include the
so-called semantical paradoxes into his solution. Having absorbed Cantor’s work,

and with Peano’s symbolic logic at his disposal, he now felt that he finally had in his
hands the technical means to deal with the whole array of paradoxical phenomena
besetting the foundations of logic and mathematics. The final result he came up with
was the theory of ramified types, RTT for short ([74], [85]). RTT consists of two main
19 This was noticed by Hazen [38].
20 See Kit Fine’s [22].

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components, a simple theory of finite types, STT, and the device of ramification. STT
is designed to take care of the set-theoretic paradoxes, while ramification is a way to
resolve the semantical paradoxes.
The basic idea of STT is simple enough. It derives from a platonist ontology of
individuals and properties (concepts, propositional functions), regimented into levels,
in which not only individuals fall under properties but properties in turn under other
properties, provided the subsuming property is always one level higher up. Assuming individuals to have type 0, we get properties of individuals as type 1 properties,
properties of type 1 properties as type 2 properties, etc.; thus, the above concept of
self-predicability cannot be expressed anymore. There is a syntactic ban on reflexive predication, negated or not. The same is true, derivatively, for classes, blocking
Russell’s paradox.
Virtually all of the usual mathematics (except set theory proper, of course) can
be carried out in the framework of simple type theory; indeed, only a few types will
be actually used to ascend from the natural numbers, which can be either given as
individuals or constructed as second-order properties in the Frege-Russell style,21 to
the higher systems of the rational and real numbers, and on to real-valued functions,
function spaces, etc.

However, not STT, but Zermelo’s system Z of set theory (plus the axiom of choice,
and strengthened by Fraenkel’s replacement axiom) carried the day as the standard
framework for mainstream mathematics. This was inevitable; mathematics had been
moving towards a unification of its disciplines, but Bourbaki style, not for the price
of being put in a syntactic straitjacket. In standard set theory, the types can easily be
retrieved from the notion of rank, and the axiom of foundation blocks ∈-cycles. Moreover, while Russell’s logicist conception assimilates membership to predication, with
its characteristic trait of non-transitivity, membership in set theory is relational from
the outset, allowing for, and indeed making essential use of, transitivity in modelling,
for instance, the well-ordering on the ordinals by the ∈-relation. The predicational
view precluded that, and with it the related idea of cumulativity, which was to become
the central idea of the Zermelo–Gödel hierarchy of sets, typically pictured as the ‘funnel of sets’. Iteration along transfinite types was essential for this hierarchy, whereas
in the predicational picture, Russell could see no use for transfinite types. These were
the main limitations of the non-ramified part of Russell’s theory of types.
While type theory did not become the general framework for everyday mathematics it did survive in important quarters of specialization, in particular, foundational
studies, recursion theory, and an array of applications, from computer science (see,
21 Of course, the infinitely many objects the logicist needs to construct in order to be able to interpret
arithmetic don’t come for free. Russell saw no way of avoiding an axiom of infinity (but see Landini’s paper
in this volume for an argument to the contrary, at least as far as Russell’s substitutional theory is concerned).
In contrast, the (consistent) system of second-order logic with ‘Hume’s Principle’ added, which has been
called ‘Frege Arithmetic’ [5], does produce the infinite number series; isn’t the neo-logicist thereby making
good on his claim after all that arithmetic is logic? I don’t think so. Hume’s Principle adds ‘ideology’ (in
Quine’s sense) to second-order logic, by introducing the operator ‘the number of . . .’. Even Frege himself
didn’t seem to regard Hume’s Principle as a primitive truth of logic; see ([39]: 286).

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e.g., [10], [3]) and artificial intelligence to categorial grammar theory ([2], [55]) to
higher-order intensional logic ([53], [26]). The foundational work was mainly concerned with justification; in the course of these developments, the original setup was
either simply used, as, e.g., in Robinson’s non-standard analysis [69], or modified,
either by the introduction of functional types22 or more recently through amendments
towards greater flexibility.23 There is also the influential Martin-Löf style intuitionistic type theory [51], important both for its underlying philosophy and its range of
applications.
1.1.6. Predicativism
This issue is a paradigm case of the kind of intimate connection between logic and
philosophy described above. Brought into the discussion by Poincaré, the idea of
a predicative definition was taken up by Russell and turned into a technical tool for
dealing with the semantical side of the paradoxes, that is, the ramification of the type
hierarchy.24 It was only natural for Russell, who had come to deny the existence
of classes beyond a convenient notation, to adopt a constructivist attitude towards
propositional functions, which served as non-extensional proxies for classes. But
then, classes depended on the definitional history of the propositional functions which
gave rise to them. In particular, no propositional function could be introduced by
referring to a totality to which it already, in an intuitive sense, belonged. This is the
Vicious Circle Principle; Russell’s formulation of it in [74], quoted above, is carried
over to [85], where it is paraphrased as “If, provided a certain collection had a total,
if would have members only definable in terms of that total, then the said collection
has no total” ([85]: 37). Now Gödel ([28]: 135) pointed out that these two versions
are not really synonymous, and that not ‘involvement’, but rather ‘definability only in
terms of’ is the critical notion. It is the Vicious Circle Principle in this latter sense that
found its technical expression in the predicative comprehension principles of Principia
and, later on, of the standard systems like NBG set theory or the fragment ACA0 of
second-order arithmetic. According to Gödel, the principle applies
… only if the entities involved are constructed by ourselves. … If, however, it is a question of objects that exist independently of our constructions, there is nothing in the least absurd in the existence of totalities
containing members which can be described (i.e., uniquely characterized) only by reference to this totality.
([28]: 136; two footnotes omitted)

This is Gödel’s well-known pronouncement of his platonist convictions. Thus the
issue of predicativity brought into focus two major and opposing philosophies regarding mathematical objects: platonic realism and constructivism. They have been
22 Beginning with Hilbert’s and Ackermann’s effort in the twenties to Church’s lambda calculus to Gödel’s
famous Dialectica interpretation.
23 See especially Feferman’s [12], and up to the system W of [14].
24 For an comprehensive overview of predicative logics and their philosophy, see [38].

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described at length in the literature, and need not be rehearsed in detail here. But the
interesting thing is that there are two historical developments deriving from one and
the same source, while conflicting in their philosophies: On the one hand, there is the
predicativist tradition starting with Hermann Weyl and leading up to the modern field
of predicative proof theory initiated by Feferman and Schütte; but on the other hand
there is Gödel who transformed the idea of ramification in such a way as to produce the
first model of Zermelo–Fraenkel set theory (given ZF is consistent), the constructible
universe. He achieved this by adopting an explicit non-constructivist attitude towards
the objects of set theory (or towards the ordinals anyway, working constructively from
there). Another often quoted remark of Gödel’s, in a letter to Hao Wang, makes this
clear: “However, as far as, in particular, the continuum hypothesis is concerned, there
was a special obstacle which really made it practically impossible for constructivists
to discover my consistency proof. It is the fact that the ramified hierarchy, which had
been invented expressly for constructivistic purposes, has to be used in an entirely
non-constructivistic way” ([34]: 404).
I’d like to make two points here. The first concerns the relationship, in the history of

the exact sciences, between the genesis of new results and the underlying philosophical
outlook (if any) that influences them. When new ideas possess at least some degree
of technical explicitness they can be exploited and recombined in quite productive
and unforeseen directions regardless of their original purpose or philosophy. In the
case of Russell’s ramified type theory this is one of the benefits flowing from the
effort made in Principia, quite independently of its subsequent fate as foundational
system. Secondly, the case shows the futility of constructing continuous genealogies
in the history of ideas. In my view, there is little substance in claims of the form, “It
was all in X”, where ‘X’ stands for ‘Russell’, ‘Frege’, or whoever the favorite hero
of an historiographer happens to be. For instance, in spite of the credit that Gödel
himself gave Russell regarding the sources of the constructible universe, we would
rather agree with R. Solovay, the commentator of pertinent writings of Gödel in the
Collected Works, who says: “There seems to me to be a vital distinction between the
precise notion of Gödel and the somewhat vaguer discussions of the ramified hierarchy
found in Russell’s writings. Thus Gödel’s well-known comments … to the effect that
his notion of constructibility may be regarded as a natural extension of Russell’s
ramified hierarchy into the transfinite now strikes this writer as much too generous”
([83]: 120). There was indeed both a decisive conceptual break and a new level of
technical sophistication that distinguishes Russell’s notion of ramification from what
it became in Gödel’s hands. Even so, the historical importance of Russell’s efforts is
not diminished by the fact that they were superseded by later developments; rather it
should be measured by its potential for generating fruitful lines of research.
Let me shortly mention two more such lines of research that can be traced back to
Russell’s predicativism. One is the theory of truth; comparing Russell’s approach to
the semantical paradoxes with that of Tarski, A. Church concludes: “Russell’s resolution of the semantical antinomies is not a different one than Tarski’s but is a special
case of it” ([9]: 301). Thus Tarski can be said to have extended the ramified hierarchy

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in his theory of truth, albeit introducing essential innovations in the course of it. The
other development, already mentioned above, concerns the modern research program
of predicative mathematics. It finds its crystallization in a recent work of S. Feferman’s [14] which takes Weyl’s predicative program in Das Kontinuum and moulds it
into a rigorous theory, called W, of ‘flexible’ finite types. There is a proof, jointly due
to Feferman and Jäger ([17], [18]), that W is both a conservative extension of, and
proof-theoretically reducible to, Peano Arithmetic. Thus W is drastically weaker than
any theory of the order of ZFC. Yet Feferman claims that basically all scientifically
applicable mathematics could be formalized in W.25 This is a much more specific argument than those usually given in the debate on the indispensability of mathematics
for the sciences initiated by Quine and Putnam. In fact, Quine acknowledges Feferman’s program, calling it “a momentous result”, if realizable: “It would make a clean
sweep of the indenumerable infinites and unspecifiable sets” ([64]: 230). For further
discussion of this, see [15].
1.1.7. Reductionism
We have already touched upon major reductionist issues in Russell’s work which
met with some success, like the elimination of definite descriptions, the reduction of
classes to propositional functions, and the predicative reform of logic. However, the
main logicist research program Russell embarked on, the reduction of mathematics to
logic, turned out to be a failure. The principal reason was, of course, that substantial
existence assumptions needed in mathematics (i.e., axioms of infinity) are alien to
the province of pure logic.26 But the program also foundered on technical problems,
like those related to the infamous axiom of reducibility.27 Also, the extension to
epistemology of the reductionist method, hailed by Russell as the “supreme maxim in
scientific philosophising” [76], and followed by Carnap in his Logischer Aufbau der
Welt, did not succeed; for many respectable reasons it was abandoned in the course of
the last century.28
25 See also the postscript to [14] in [16], 281–83.
26 It is a curious fact that Russell never seemed to have acknowledged this failure. As his discussion in

[78], for instance, clearly shows, he is fully aware of the fact that the Peano axioms have no finite models,
and that the injectivity of the successor function cannot be proved from pure logic. In fact, he calls the
axiom of infinity a “hypothesis”, like the multiplicative axiom (choice) and the axiom of reducibility, which
he doesn’t consider logically necessary. Yet in the same book he squarely equates mathematics with logic:
“Pure logic, and pure mathematics (which is the same thing), aims at being true, in Leibnizian phraseology,
in all possible worlds, not only in this higgledy-piggledy job-lot of a world in which chance has imprisoned
us” ([78]: 192).
27 See, e.g., [38] for the relevant background, but also [9] for the claim that, contrary to what has been
common coin about this ever since Ramsey’s criticism, repeated by Quine, the axiom by no means defeats
Russell’s purpose of ramification if only the essentially intensional character of Principia’s logic is taken
into account. As far as the last point is concerned, Linsky’s paper in this volume is also relevant here.
28 Basically, a theory of the physical world just cannot be a definitional extension of a phenomenalist
theory of sense-data. For a recent discussion of the philosophical flaws of the phenomenalist viewpoint
see, e.g., [60].

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Yet in the domain of logic and the philosophy of mathematics, the reductionist
spirit has survived and even grown stronger as more sophisticated logical tools became
available, like the concepts of relative interpretation between theories, conservativity
of one theory over another, or proof-theoretic reduction. This line of research flowed
from two main sources, Hilbert’s metamathematics and the nominalist tradition in
modern logical philosophy.
As for the former, it is true that Hilbert’s original program of finitary consistency
proofs was given up in the aftermath of Gödel’s incompleteness theorems; but modern

proof theory kept the basic methodology while relaxing finitist requirements [30], calibrating theories according to their relative consistency strength, investigating reductive
relations between them [13], and quite generally exploring the wealth of subsystems
of analysis, i.e., of second order number theory, for the development of most of everyday mathematics. In addition to the predicativist research program mentioned above
there is the important program of “Reverse Mathematics” [82] singling out various
set existence principles which are not only sufficient but also necessary for specific
portions of positive mathematics, thereby uncovering the precise resources used.
The other major source of reductionist philosophy of mathematics is the nominalist
program initiated by Goodman and Quine [36]. While Quine in his famous turn [61]
embraced classes as indispensible for scientific practice, H. Field [21] tried to justify
this practice by showing its conservativity over a nominalist ground theory. Yet another
approach is S. Lavine’s intriguing “theory of zillions” [48], which is committed to the
justification of unrestrained impredicative set theory on the basis of the resources of
“finite mathematics” present in J. Mycielski’s concept of locally finite theories [56].
The leading rationale common to all these approaches is a kind of reasoning that
is “conscious of its resources” [49]. The typical question would not only be, “Can
we prove it?”, but more specifically, “In which theory can we prove it?”, or “What is
the weakest theory for establishing the result or, more generally, for justifying usual
mathematical practice?” Note that from a foundational point of view the question of
accepting a theory is thereby ‘factorized’ into two component steps: (i) the technical
question as to what the necessary principles are which the theory rests on, and (ii)
the (mainly) philosophy-driven decision to embrace or repudiate the principles and
thereby the theory.
According to the received view, first principles (“axioms”) have to be not only
true but self-evident. But as A. Irvine’s paper in the present volume reminds us,
there is an element in Russell’s methodology, called the “regressive method” [73],
which calls for admitting principles that might suggest themselves as axioms only
after a considerable amount of technical work. Thus, the above decision (ii) may after
all not be grounded—or at least not exclusively grounded—in some philosophical
predilection or other, a situation definitely to be welcomed. It is interesting to note
that Gödel was aware of this feature in Russell’s work and supported it.29

29 Remarks that make direct reference to Russell can be found in ([28]: 127f.); see also his well-known
statement on the question of new axioms in set theory, in ([29]: 521).

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