Metamathematical Investigation of Intuitionistic Arithmetic and Analysis

488 Pages·1973·21.39 MB·english

by  A. S. Troelstra (eds.)

Lecture Notes ni
Mathematics
detidE yb .A Dold, Heidelberg dna .B ,nnamkcE hcirJ(Z
344
I IIIIIIIIIIIIII
A. S. Troelstra
(Editor)
Universiteit nav Amsterdam, Amsterdam/Nederland
Metamathematical Investigation
of intuitionistic Arithmetic
and Analysis
galreV-regnirpS
Berlin-Heidelberg- New York 1973
AMS Subject Classifications (1970): 02C15, 02D05, 02D99, 02H10
ISBN 3-540-06491-5 Springer-Verlag Berlin • Heidelberg - New York
ISBN 0-387-06491-5 Springer-Verlag New York - Heidelberg • Berlin
This work is subject to copyright. All rights are reserved, whether the whole or part of the material si concerned,
specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction yb photocopying machine
or similar means, and storage in data banks.
Under § 45 of the German Copyright Law where copies are made for other than private use, is payable a fee to the publisher,
the amount of the fee to be determined by agreement with the publisher.
© by Springer-Verlag Berlin - Heidelberg .3791 Library of Congress Catalog Card Number .83241-37 Printed in Germany.
Offsetdruck: Julius Beltz, Hemsbach/Bergstr.
Dedicated to
GEORG KREISEL
who has contributed so much to the
subject of this volume
Preface
The present volume found its origin in a course on functional and realiz-
ability interpretations on intuitionistic formal systems, presented at the
Rijksuniversiteit Utrecht (Netherlands) in the spring of 1970, and a course
on the metamathematics of intuitionlstic formal systems at the University of
Amsterdam in 1971 - 1972. The literature on the subject was widely scattered,
the connection between certain rules was often not made explicit in the
literathre, and some obvious questions were not answered there.
Therefore I thought it would be useful to give a coherent presentation of
the principal methods for metamathematical investigation of intuitionistic
formal systems and the results obtained by these methods, connecting results
in the literature, filling gaps and adding some new material. A first attempt
(for realizability and functional interpretations) was made in Troelstra
1971, which , however, because of a rather terse style, was not readily assim-
ilated by readers new to the field. (It still provides a useful survey of
the applications to first-order systems however.) Therefore a more elaborate
presentation, including other techniques of metamathematical research, seemed
to be called for.
Having learnt of the unpublished Ph.D. work of C.Smorynski on applications
of Kripke-models to intuitionistic arithmetic, and of Dr Zucker's thesis on
the intuitienistic theory of higher-order generalized inductive definitions,
subjects which both fitted very well into the scope of the planned volume,
I asked them to contribute a chapter each ; their contributions appear as
chapters ,V and VI respectively° The models for intuitionistic arithmetic
of finite type, functional and realizability interpretations, and normaliza-
tion for natural deduction systems, and also the general editing of the
volume I undertook myself.
Finally, N.Ao Howard contributed an Appendix supplementing discussions in
§ 2.7 and § 3.5.
The organization of the volume is primarily method-centered, i.e. the
material presented is grouped mostly around methods and techniques, and not
arranged according to the results obtained. Hence some results, obtainable
by different methods, appear at various places in the book. This will enable
the reader to compare the relative merits of the various methods.
As regards intuitionistic arithmetic and closely related systems, the
treatment is almost wholly self-contained ; some experience with classical
VI
metamathematics, and the elements of intuitionism, such as may be gleaned
from Kleene's Introduction to metamathematics and Heyting's book on
Intuiticnism suffices° The parts dealing with arithmetic can therefore be
used in a course for graduate students or a seminar.
The sections dealing with analysis are not self-contained, and serve
more or less as a running commentary on the literature, connecting and com-
paring various approaches and adding new results besides. This part was
thought of primarily as a help to the beginning researcher, to help him tc
find his way in the subject. For use in a seminar, these sections should
usually be supplemented by the reading of other papers.
In keeping with this set-up, the listing of applications for intuition-
istic arithmetic and closely related systems is rather extensive, but in
the case of analysis we have often restricted ourselves to some typical
examples ; further applications can easily be made by the reader himself
once he has understood the method, and its applications to arithmetic.
No special attention has been given to intuitionistic propositional
logic and predicate logic, because as formal systems they exhibit many
properties which do not generalize to arithmetic and analysis, and therefore
would require a separate treatment.
Speedy publication was thought more useful than final polish, so ~s not
to make the material outdated at the moment of its appearance. Hence also
the choice for publication in the "Lecture Notes in Mathematics". Even
while refraining from a completely self-contained treatment of all parts,
it was not possible to take all relevant work into account, not even on
arithmetic ; for example, N. Goodman's work on the theory of constructions
was left out altogether, since it would not easily be fitted into the
framework of the other developments and so would consume too much space°
We have no doubt that there are still many imperfections in this presen-
tation ; it hardly needs saying that the authors will be grateful for
errors, misprints, additions to the bibliography being brought to their
attention.
The contents of the present volume are primarily technical in character ;
but it is to be hoped that the material will not inspire a thought- and
mind-less multiplication of metamathematical results, without a thought
spent on their possible significance for an analysis of intuitionistic basic
notions and for foundations of mathematics in general. On the other hand,
the "philosophical interest" of the subject is not promoted by uncritical
analysis. (A single example : the interest of the well known disjunction
property ~A VB = ~A or ~B , and the explicit definability for existen-
tial statements are frequently overrated, especially as a criterion for the
VII
"constructive character" of the system considered. See e.g. the discussion
in Troelstra A.) As regards potential "philosophical interest", it seems
tb me to be more promising (but also more difficult) to look for new results
for well-known systems (possibly different in kind from the results dis-
cussed in this volume), instead of trying to extend known results to stronger
and stronger systems. Of course, to be potentially interesting, the new
results should also have a clear intuitive meaning in terms of the intended
interpretation of the systems considered.
Directions for use. In order to help the reader find his way, there is
an analytical table of contents at the beginning, a bibliography, and lists
of notions and notations at the end. Reference to the bibliography are
self-explanatory. § 3.5 refers (except in the appendix) to chapter III,
§ 5, etc.
The parts on arithmetic and closely related systems are more or less
self-contained. As such we mention especially: Chapter I, §§ 8, - I §§ 10,
11 ; chapter If, §§ 4 - I (2.4.18 excepted), § ,5 § 7 (except where results
of § 6 are used) ; chapter III,§ I (3.1.1 18), - § 2 (3.2.1 - 28 ; 3.2.33),
§ 4 (3.4.1- 14; 3.4.29), § 5 (3.5.1- 11; 3.5.16 (i), (iii)) ; § 6 (3.6.1-
3o6.16), § 7 (3.7.1-s), § s (except 3.s.7), § 9; chapter ,VI §§ I-4~
chapter V, §§ 6. - 1
Chapter I contains all generalities, and should usually be consulted
when needed only.
Acknowledgements° As regards my own contribution to this volume, I am
especially indebted to G. Kreisel, who permitted the use of unpublished
material in his course notes (apart from the general indebtedness expressed
by the dedication), to J.I. Zucker, for his patient and careful reading of
drafts of my chapters, suggesting many stylistic, expository and mathemat-
ical improvements and corrections, and to Miss Judith van Witsen, who
undertook the seemingly endless task of typing the manuscript. Some other
acknowledgements have been made in footnotes.
Amsterdam, June 1973.
A. .S Troelstra
TABLE OF CONTENTS
I° INTUITIONISTIC FOR~L SYSTE~ (A.S. Troelstra)
§I IntUitionistic logic I
Notational conventions (1,1.2) - Spector's system (1.1o3) -
- Oodel's system (1.1~) - Equivalence of Spector's and
Godel's system (1.1.5) - Equivalence of Spector's and
Kleene's formalization (I°I°6) - A natural deduction
system (1.1.7 - 1.1.9) - Deduction theorem for Spector's
system (1.1.9- 1.1.10) - Equivalence between natural
deduction and Spector's system (1.1.11)
§ 2 Conservative and definitional extensions~ expansions 14
Definition of predicate logic with equality (1.2.1) -
Definition of conservative extension (1.2.2) -
- Expansion (I~2.3) - Definitional extension (1.2.4)
Addition of symbols for definable predicates (1.2.67 -
- Addition of symbols for definable functions (1.2.7) -
- Replacement of function symbols by predicate symbols
(1.2.8) - Addition of defined sorts of variables (1.2.9-
1.2.10)
-
§ 3 Intuitionistie first-order arithmetic 18
Language of HA (1.7.2) - Axioms and rules of HA (1.3.3)
- Defining axioms for primitive recursive functions (1.3.4)
- Rule and axiom schema of induction (1.3.5) - Natural
deduction variant Of HA (1.3.6) - Eliminability of
disjunction in systems~ontaining arithmetic (1.3.7) -
- Formulation of H~ without function symbols (1.3.8) -
- Notational conventions (pairing, coding of finite
sequences, proof predicates, godelnumbers, godel- and
rossersentences, numerals) (1.3.9) - Formalization of
elementary recursion theory (1.3.10)
§ 4 Inductive definitions in HA 28
Definition of class F (I~.2) - Normal form for
elements of F (1.4.5- 1.4.4) - Explicit definability
of predicates introduced as closed under a condition
from r (1.4.5)
Partial reflection principles 33
C~delnumbering of function constants and terms (1.5.2) -
- Evaluation of closed terms (1.5.3) - Construction of
partial truth definitions (1.5.4) - Partial reflection
principles (1.5.5 - 1.5.6) - Remark on refinements (1.5.7) -
- Remark on quantifier-free systems (1.5.8) - Reflection
principle for qf-H~A (1.5.9- 1.5.10).
Intuitionistic arithmetic in all finite types 39
Type structure ~ (1.6.2) - Description of N-IqA w
(1.6o3- 1.6.7) - Definition of the k- operator ~.6.8) -
- HA as a subsystem of N-HA m (1.6.9) - Intensional
identity or equality (1.6o10) - Description of ~_~w
(1.6.11) - Description of ~_~w ~m_~H A (1.6.12) -
- Description of qf ~ c qf-~ -H~' qf
(1.6o13) - qf- A~H__-~ ~, qf-WE-H~A as equational
calculi (1.6.14) - The systems ~, qf -I~ w (1.6.15)
-
X
Simultaneous recursion and pairing ; a comparison
- of various treatments (1.6.16) - Pairing operators in
qf-W~E~-H~ ~ (1.6.17) - Historical notes, variants in
the literature (1.6.18)
§ 7 Induction and simultaneous recursion 15
Simultaneous recursion in qf_N_ y~W (1.7.2 - 1.7.7) -
- The induction lemma for qf_~_~w (1.7.8- 1.7.10) -
- Replacement of recursor by iterator (1.7.11) -
- Simultaneous recursion and the induction lemma in
qf-~ (1.7.12)
§ 8 ~ore about N-HA ~ 6o
Cartesian product types and pairing operators (1.8.2) -
- The X- operator as a primitive notion (1.8.4) -
- Reduction to pure types (1.8.5- 1.8.8) - Reduction to
numerical types in qf-WE-~ ~ (1.8.9)
§ 9 Extensions of arithmetic 66
Extensions of arithmetic expressed in ~(HA) or ~(H~A)
extended by relation constants (reflection principles,
generalized inductive definitions) (1.9.2) - Language of
~S o (1.9.3) - Comprehension principles (1.9.4) -
- Extensiona!ity (1.9. - 5 1.9.7) - HAS o + EXT + ACA is
conservative over HA (19.8) - Formulation of HAS with
X- terms (1.9.9) - Description of EL (1.9.10) - Some
notations and conventions (1.9.11) - Formalization of
elementary recursion theory in EL (1.9o12- 1.9.16) -
- Definitions of A°x, A~x, A°~,~'~ (1o9.17) - Systems
of intuiticnistic analysis based on the concept of a
lawlike sequence ; IDB (1.9.18) - Systems of intuition-
istic analysis based on a concept of choice sequence
I].9.19) - Bar induction (1.9.20) Extended bar induction
k- .9.21- 1.9.23) - Fan theorem (I.~o24) - Extensions of
N-HA ~ : IDB w (1.9.25) - Theories with bar recursion of
Jigger typ~E ~-~+ BR (1.9.26) - Girard's theory of
funetionals (1.9.27)
§ 10 Relations between classical and intuitionistic systems :
translation into the negative fragment 85
Definition of the mapping ' (1.10.2) - Definition of
Harrop formula, and strictly positive part (s.p.p.)
(1.10.5) - Definition of negative formula (1.10.6) -
- Properties of the mapping ' (1.10.9- 1.10.13)
§ 11 General discussion of various schemata and proof-
theoretic closure conditions 9o
Definition of admissible rule, and intended intuition-
istic interpretation of the logical constants (1.11.1) -
Disjunction and explicit definability property (1.11.2) -
- The schema Vx(A VBx) ~ A V VxBx (1.11.3) - The schema
Vx ~A ~ ~oVxA (1.11.4) - ~rkov's schema and rule
11.5) - Independence of premiss schemata and rules
,.11 )6 Church,s thesis and ~le (1.11;7).
XI
I. MODELS AND COMPUTABILITY (A.S. Troelstra)
§ I Definitions by induction over the type structure 97
Definition over the type structure (applicative set,
type level) (2.1.1) - Establishing properties for
applicative sets of terms (2.1.2) - Definability aspects
(2.1.3) - Sets of terms closed under ~- abstraction (2.1.4)
§ 2 Computability of terms in N-HA w 100
Definition of reduction and--standard reduction for terms
of N-HA w (2.2.2) - Comparison of standard and strict
reduCtiOn (2.2.3) - Alternative definition of ~ (2.2.4) -
- Definition of computability, strict -, standard -
(2.2.5) - All terms of N_~w are standard computable
(2.2.6- 9) - ~-HA~ W conservative over its induction-free
part for equations between closed terms (2.2.10) - Strong
computability and strong normalization (2.2.12- 19) -
- Uniqueness of normal form (2.2.20- 29) - Computability
and strong computability for k-based theories (2.2.30- 34) -
- Discussion and comparison of proofs of computability for
terms of A~H w in the literature (2.2.35)
§ 3 More about com~utabilit~ 116
Computability in ~-!~ w + IE o (2.3.1 - 5) - The equality
axioms IE I (2.3.6) - Standard computability of terms in
languages with Cartesian product type (2.3.7) - Computability
relative to assignment of functions (2.3.8- 10) -
Arithmetization of computability (2.3.11- 13)
§ 4 Models based on partial recursive function application :
HE0, HE0 125
Models : normal, extensional models (2.4oi) - Submodel,
homomorphism, embedding (2.4.3) - Construction of inner
extensional models from arbitrary models of ~-~H~ w
(2.4.5) - The set-theoretical model of E-HA w (2.4.6) -
- Description of HR0 (2.4.8) - The fermal~heori~ HR0,
HR0- (2.4.10) - Description of HE0 (2.4.11) - HE0 and h~a
inner extensional model of HR0 are different (2.4.12) -
- Provable faithfulness of HRO , uniformly in type 0
variables (2.4.13- 14) - Closed type I terms of N_~w
are ~ provably recursive (2.4.15) - Sketch of a variant
of HR0 satisfying ~- conversion (2.4.18) - Pairing in
HR0, HE0 (2.4.19)
§ 5 Term models of N-H~A w 132
Definition of CTM, CTNF, CTM', CTNF' (2.5.1-2)- Some
properties of CTM, CTNF, CTM', CTNF' ()2.5.3 - CTNF I is
isomorphic to a submodel of HRO for a suitable version
of HRO (2.5.5) - Alternative proof of uniqueness of
normal form (2.5.6) - HRO can be made into a model for
.~w + IE I (2.5.8) - Examples of versions of HR0 where
distinct normal terms are represented by the same element
(2.5.9) - IE o is weaker than IE I (2.5.10)
§ 6 Models based on continuous function application : ICF, ECF 138
Definition of ICF(Z~) (2.6.2) - In ICF a modulus-of-
continuity functional exists (2.6.3) - ICF(~I) contains
a fan-functional if ~ satisfies FAN (2.6.4)
Hereditarily continuous functionals ECF(~) (2.6.5) -
ECF(iL) contains a fan-functional if LL satisfies FAN
(2.~.6) - ECF does not contain a modulus of continuity
XII
functional (2.6.7) - A recursively well-founded, but not
well-founded tree (2.6.9) - Provable faithfulness of ICF
uniformly in type I variables (2.6.11 - 12) - The equi-
valence between ECF(~) and HRO (2.6.13- 21) - KLS
holds in HA+MpR (2.6.15 - 17) - Basis theorem (2.6.19~ -
- QF-S£~ T holds for ECF (2.6.20) - The models ECFr(U)
and ICFrlU) (2.6.22)- A variant of ICF and ECF
(2.6.23) - Pairing operators in ICF, ECF, ICF*, ECF*
§ 7 Extensionality and continuity in N-H~ w 155
Extensionality and hereditary extensionality (2.7.2-4) -
- Derived rules of extensionality (2.7.5) - Counterexample
to the rule of extensionality when variables of type level
> I are present (2.7.6) - Closed type 3 terms of N-HA w
are not extensional in every model (2.7,7) - Provable
modulus of continuity for type 2 terms of N-H~ w (2.7.8) -
Product topology (2.7.9) - "Floating product topology"
(2.7.10)
Other models of N- HA w 162
The schemata $I ~59~(2.8.2- 2.8.4) - Soarpellini's
models (2.8.5) - Compact and hereditarily majorizable
functionals (2.8.6)
§ 9 Computability and models for extensions of N-HA~ w 166
Extension of computability 'o't functionals of ~- w IDD~
and related theories (2.9.2) - Computability for bar-
recursive functionals (2.9.3) - Computability for Girard's
system of functionals (2.9.4) - Extnesions of HR0, HE0
to models for other systems (2.9.5) - Application of
K- HR0 : Computability of closed terms of ~- BB~DI w (2.9.6) -
- Extension of HE0, HEO to Girard's system of functionals
(2.9.~) - Similarly for ICF, ECF (2.9.8) - Nodels for
- ~ + BR (2.9. 9- 12),
I. REALIZABILITY AND FUNCTIONAL INTERPRETATIONS (AoS. Troelstra)
§i
A theme with variations : Kleene's FIC 175
Definition of F|C (3.1.2) - Soundness theorem (3.1.4) -
- Existence and disjunction under implication (3.1.5) -
IPR e for H~ (3.1.7) - Characterization of CIC by
deducibility conditions (3.1.8) - CIC respects logical
equivalence, and CIC holds for Harrop formulae (3.1.9) -
- CIC holds also for formulae which are not equivalent
tO a Harrop formula (3.1.10) - IP~ is not derivable in
H~ (3@Io11) - Disjunction and explicit definability
property for HA+MpR (3.1.12) - A variant of FIC (3.1.13)-
- IPR for H~ (3°1.15) - A method of dealing with
variables using partial reflection principles (3.1.16) -
- Closure under Church's rule (3.1.18) - Extension and
generalization of FIC to higher-order systems (3.1.19) -
- FIC for HAS o +PCA , with applications (3.1.20) -
- Extension to HAS (3.1.21 - 23) - Extension of
Moschovakis's methods to ID~B, ID~B I (3.1.2%)