Proof Theory: An Introduction (Lecture Notes in Mathematics, 1407) 3540518428, 9783540518426

Citation preview

Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, Zurich F. Takens, Groningen

1407

Wolfram Pohlers

Proof Theory An Introduction

Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo Hong Kong Barcelona

Budapest

Author Wolfram Pohlers Institut fur Mathematische Logik und Grundlagenforschung WestfaIische Wilhelms-Universitat Einsteinstr. 62, D-48148 Munster, Gennany

1st edition 1989 2nd printing 1994

Mathematics Subject Classification (1980): 03F

ISBN 3-540-51842-8 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-51842-8 Springer-Verlag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is pennined only under the provisions of the Gennan Copyright Law of September 9, 1965, in its current version, and pennission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the Gennan Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringerScience+Business Media GmbH

© Springer-Verlag Berlin Heidelberg 1989 Printed in Gennany 46/3111-543'

- Printed on acid-free paper

This book contaiDs the somewhat extended lecture Dotes of an introductory course in proof theory ) gave duriDg the Winter term 1987/88 at the University of MUnster, FRG. The decision to publish these notes in the Springer series has grown out of the demand for an introductory text on proof theory. The booles by LSchiitte and G.Talceuti are commonly considered to be qUite advanced and J.Y.Girard·s brilliant book also. is too broad to serve as an introduction.

J tried, therefore. to write a boole: which needs no preVious knowledge of proof theory at all and only little knowledge in logic. This is of course impossible, so the boole runs on two levels - a very basic one, at which the book is self-coD~ and a more advanced one (chiefly in the exercises) With some cross-references to defiDability theory. The beginner in logic should neglect

these cross-references. In the· presentation I have tried Dot to use the 'cabal language' of proof theory but a language familiar to students in mathematical logic.

Since proof theory is a very inhomogeneous area of mathematical logic, a choice had to been made about the parts to be presented here. I have decided to opt for what I consider to be the heart of proof theory - the ordiDal aDa1ysis of axiom systems. Emphasis is given to the ordiDal analysis of the axiom system of the impredicative theory of elementary inductive definitions OD the Datura) numbers. A rough sketch of the 'constructive' consequences of ordiDal aDa1ysLs is given in the epilogue. Many people helped me to write this book. i.Columbus suggested and checked nearly all the exercises. A.WeiermalUl made a lot of valuable suggestioDS especially .in the section about alternative interpretations for Q. A.Scbliiter did the proof-reading, drew up the subject index and the index of notations and suggested many corrections especially in the part about the autonomous ordiDals of Za:,.

I am also indebted to the students of the workshop on proof theory in MiiDster who suggested many more correctioDS. Last but Dot least I want to thank all the students attending my course of lectures during the winter term 1987/88. It was their interest in the topic that encouraged me to write this book. A first version of the typescript was typed by my secretary Mrs. J.Probstbig using the Signum text system. Slie also wrote the table of contents. Many thanks to all these persons. July 19, 1989

w.

P.

MtiDster

v

TABLE OF CONTENTS

Preface CollteDta

Introduction CHAPTER I 0rdlDal aaalyaia of pure number theory The language !P of pure number theory Semantics for !fJ A formal system for pure number theory The infinitary language ~r:J:) Semantics for !fJco

§ 1. § 2. § 3. § 4. § S. § 6. Ordinals § 7. Ordinal arithmetic § 8. A notation system for a segment of the ordinals § 9. A norm function for n:-sentences § 10. The infinitary system Zoo § 11. Embedding of ZI into Zo § 12. Cut elimination for Zo § 13. Formalization of transfinite induction § 14. On the consistency of formal and semi formal systems § 15. The wellordering proof in Zl § 16. The use of Gentzen's consistency proof for Hilbert's program CHAPTER n The autonomous ordlDa1 of the iDfIDltary system Zm and the 1Jmlta of predicatMty § 17. Continuation of the theory of ordinals § 18. An upper bound for the autonomous ordinal of Zoo § 19. Autonomous ordinals of Zoo

1

7 9 1416 22 24 28 40 4S 48

50 56 59 62 67 11 75

77

78 85 90

CHAPTER. m 109 0rdlDal aDa1y81s of the formal theory for DOnlterated tnduetlve defID1tlou § 20. A summary of the theory of monotone inductive definitions over 109 the natural numbers VII

§ 21. The formal system ID. for noniterated inductive definitions § 22. Inductive definitions in !t'0 0 § 23. More about ordinals § 24. Collapsing functions § 25. Alternative interpretations for 0 § 26. The semiformal system lDoo § 27. Cut elimination for

114 116 125 139 147 160 168

§ 28. Embedding of ID, into 1D~~c:.1 § 29. The wellordering proof in ID,

179

Epilogue

187

Bibliography Subject IDdex

208

Index of

212

1D2rCa)

VIII

N0tat10DB

173

190

INTRODUCTION

The history of proof theory begins With the foundational crisis In the first decades of our century. At the turn of the century, as a reaction to the explosion of mathematical knowledge in the last two centuries, endeaYours began to provide the growing body of mathematics with a firm foundation.. Some of the notions used then seemed to be quite problematic. This was especially true of those which somehow depended upon that of lDfinlty. On the one hand there was the notion of iDfinitesimals which embodied 'infiDity in the small'. The elimination of infinitesimals by the introduction of limit processes represented a great progress ill foundational work (although one may again tlDd a justHlC8tion for inflnitesimals as it is clone today in the reid of DODStaDdard analysis). But on the other band there were also aotions which. at least implicitly. depended on 'iDfinity in the large'. G.Cantor in his research about trigonometrical series was repeatedly confroD.ted with such notions. This led him to develop a completely new mathematical theory of iDflDlty. namely set theory. The main feature of set theory is the comprebeaslOIl principle ~hich allows to form collection of possibly iDf'mitely many objects (of the mathematical UDiVerse) as a single object. Cantor called the objects of the mathematical UDiVerse 'Mengen' usually translated by 'setS'. Set theory, however. soon turned out to be a source of doubt itself. Since Cantor's comprehension principle allows the collection of all sets x sharing an arbitrary property E(x) into the set (x: E(x)} one easily runs into cODtradictiODS.1) For .instance If we form the set M := (x: X C xl, then we obtain the well-known Russelllan antinomy: ME M if and only if M f M. It is easy to CODStruet further antinomies of a J) Cantor hlmaelf was well aware of' t.he dIaUnc:t10D bet-weeD seta aDd other collect.loaa which may lead &.0 cont.radlct.1oDa. See hI.s letter to DedeklnCl from 27.7.1899 [Parkert. et al. 1987]

I

Ineroduet.Jon

simUar sort.. Another annoying fact was that the plausible looking axiom of choice (AC) For any famUy (SJc)Jc E J of non empty sets there is a choice function f: I ..... U{Sk: kEI} such that f(k)eSk for all kel bad as a consequence the apparently paradoxical possibility of wellorderiDg any set. Nobody could imagine what a wellorderiDg of the reals could look like and D.Hil~ in his famous list of mathematical problems presented in Paris in 1900, stated in his remarks concerning problem one (the Continuum Hypothesis) that it would be extremely desirable to have a direct proof of this mysterious statemenL Today we know that there is DO elementary CODStruction of a welJorderiDg of the reals. Ally wellordermg of the reals bas the same degree of construetJveDess as the choice function Jtself. The eXistence of a choice fUnctioD. however, is DOt even provable from the Zermelo FraeDkel axioms for set theory. All -these facta contributed to a feeling of uncertaiDty maoDg members of the mathematiCal society about the DOtiOD of a set that they were opposed to set theory in general. But it was' of course Dot possible to simply Jpore Cator'a discoveries. Hermann Wey} iD his paper tUber die neue GnmcIIageDkrise cIer MathematJk· [WerJ 1921] tried to CODVIDce his CODtemporaries that the fOUDdatIonaJ problems arising In set theory were DOt just exotic pheDOIIIeD8 of aD isolated branch of mathematics but also coDCenleCl analysis, the yery heart of _ . It was he who introduced the term 'foundatioDaJ CIi8Is' into the

discussloD. In his book 'Ou KontiDuum' [Weyl m8l.be had aJready suggested a development of mathematics which ayoided the use of UDreStricted set coDStrUctJons. In more modem terms one could say that he proposed a precIlcatiYe deYelopment of _ . Others, like LE.J.Brouwer. already doubted the logical basis of mathematics. Their point of attack was the law of the excluded middle. With the help of the law of the excluded middle it becomes possible to prove the existence of objects without _ them explicJtJy. Brouwer suggested developing _ on the buls of alterDatlYe JDtu1t1Ye prinCiples wbk:h excluded the law of the excluded middle. Their formaliZatioD - due to Heyting- DOW is known as intuitioDistic logic. Both approaches, WeyJts as wei) as Brouwer's. meant rigid restrictions on mathematics. D.HiJbert., then ODe of the most prominent _ , was DOt WIlling to accept any foundation of mathematics which would mutilate exiSting matllematics. To -him the foundational crisis was a nightmare baamting mathematics. In his opinion mathematics was the scieDce, the model for all scieDCeS, whose 'truths bad been proven on the basis of defJDitiODS Via infallible infereDCes' aDd therefore were 'valid overall

2

lzatroduetJon

in reality'. He felt that this position of mathematics was in danger and therefore

wanted to preserve it as it was. He was especially unwilling to give up Cotor's set theory, a paradise from which no one would expel him. In his opinion Cantor's treatment of transfinite ordinals was one of the supreme achievements of human thought. Therefore he planned a program to save mathematics in its existiDg form. He charted his program in a couple of writiDgs aDd debated it in several talks [Hilbert 1932":1935]). Therefore it would be lDadequate to try to sketch Hilbert's program In only a few seDtences.. Por a seriOUS evaluation of me staas of Hilbert's program today deeper consideratioas are necessary (cf. JSL 53 (1988». The part of Hilbert's program. however._ which was essential for the development of the kind of proof theory we want to give an introduction to in this lecture may be roughly characterized by the folloWing steps:

(cr.

1. Axiomatlze the whole of mathematics 11. Prove that the axiolll8 obtallled In step I are COIISisteIIt. Hilbert proposed that step II of his ~ the consistency proof. should be c:arriecI out Within a new mathematical theory which he called •Bewel8tbeorle', Le. Proof Theory. Ac:cordiDg to Hilbert. proof theory should use collteDtual reesoniDg in contrast to the formal iDferences of mathematics. Hilbert bimaelf was aw.-e of the fact that the reasoning of proof theory must itself Dot become the subject of critldsm. He therefore reqUIred proof theory to obtain its results by methods beyoDd the sbadow of a doubL He suggested using only fmitistic methods. By r-mitistic methods he UDderstood those methods 'without which neither reasoDiDg nor scientific action .e possible'. In my persoDal opinion, fmitistic reasoning may be interpreted as combiDatorial reasoning over finite domains. Some of Hilbert's students (e.g. AckerJDallD, J.v.NeuIllaDD, P.Bernays) soon obtained concrete results. Following Hilbert's maxim of first developing the mathematical tools necessary for the solution of a general problem by studying special cases of the problem they first tackled subsystems of elementary arithmetic. In fact they succeeded in obtainiDg consistency proofs for subsystems not containing the scheme of complete induction. It thus seemed to be just a matter of technical refinement to extend these consistency proofs to systems containing the full induction scheme. However, the systems cont:aiDing complete induction stubbornly resisted all attempts to prove their coDSistency. That this failure was neither an accident nor was due to the incompetence of the researchers, became clear after the publication of Kurt GOdel's paper 'tiber formal unentseheidbare Sitze der Principia Mathematica wad verwandter Systeme' [GOdel 1931]. In this paper GOdel proved his famous theorems which. roughly speaking, say the following:

3

lntroduetJon

1. In any formal system, satisfyingcertaiIJ lIlIturaJ reqUirements, It Is po881ble to formulate sentences which are true in the IntelJded structure but are also undecidablewlthJn the formal sptsm {Le. neither the seatlmce nor lt8 ne68tloD are provable in the formal system}. II. The colISl8tellCY proof for any formal system. If68iII IllltisfTUc CIJIIOaJcal requiremeats. IIJIJT sot be formalizedIn the system Itself ODe might think that GOdeI's theorems meant a sudden eDd to Hilbert's program. The first theorem sbows that step I in HUbert's program is IDdeed impossible. ThIs, however, might be remedied by the oberserYation that in fact it is Dot aecesaary to formalize all possible mathelll8tk:s. It would suffice just to axiomatize existing mathematics. Today we tnow that . .Iy everything til everyday's mathematics (and. except for the Continuum Hypothesis. probably all which Hilbert may have thought on is fonna1jzable ill ODe single formal system., namely Zermelo Fraentel set theory with the axiom of choice (ZPC). Most parts are eYen formaliZable In much weaker systema. GOdeI n, however. Is a lethal blow to Hilbert's program. Since the methods 'Without which Deither reasoDiDg Dor scientific action are possible' (combtDatorial reasonlDg OYer finite domains. ill our _ n ) should itself be aY8ilable In mathematics. any reasonable axiomatiZation of mathematics should allow the formaliZation of Hilbert's fiDltistle methods. Therefore there is no r-udtistic CODSisteDCJ proof for an axlomatiZatioD of stronger fragments of mathematics (Le. essentially those containing the scheme of complete tnduetlon). Luckily for the cleYelopment of proof theory, the researchers in the thirties did Dot IDterpret these resalts als having such drastic consequences. It Is bard to say why. GOdel'a results were known to the Hilbert school. For instance IJerDays mentiou them III [Bemays 19358] but although he expresses doubts about the feasibility of rmltistic consistency proofs be deniea that GOdel's results imply their impossibility. I conjecture that the true reasons were Hilbert's authority as well as the YagUeDeSS of IUs program. Since he ga'Ye no precise defiDltlon of what he meant by tmltistic methods one could hope that these methods comprtsed a tlDd of conteDtual reasoning which C8IlIlot be mathematically formalized. As a matter of fact mathematlctans did not stop searching for consistellCY proofs and in 1936 Gerhard Gentzen succeeded in pro'YiDg the consistency of elementary DUmber theory. According to GOdeI's second theorem GeDtzen's proof had to use Donfinitistic means. Gentzen succeeded in concentrating all ftonftnitistic means in one single poiDt - induction along a wellordering of transfinite ordertype. This result conrJrmed the Hilbert school's opinion that just a slight. modification

IDt.rodu~on

of the flDitistic standpoint (i.e. accepting a weak: form of transfinite induction) would suffice to make the whole program feasible. In §16 we will discuss the consequences of this 'slight modification' for Hilbert's program. There we will try to argue, in the spirit of Hilbert's program, that Gentzen's proof is of little help. This, however, does Dot mean that Gentzen's proof and his results are of no importallce. Quite on the contrary, in our opinion Gentzen's proof Is one of the deepest results in logic. To see why, we propose a r e _ o i l of his results. In poiDt of fact it is very easy to prove the consistency of pure DUmber theory. One Simply has to show that there exists a model for it. So what is the advantage of GeDtzen's consistency proof? The construction of the model Itself needs a certalD framework, e.g. set theory. Thus what is obtained" by a coDSJsteDcy proof via a model construction in set theory (or some eYeD weaker theory) is that the consistency of set theory also entails the cODaisteDCY of pure number theory. GeDtzen's proof, however, gives much more iDformatJon. It has already been mentJoDed that GeDtzeD's proof Is finitlstic apart from his use of JDductJon along a weJJordering of transfinite ordertype. ID our opJDloD this is the essential coDtribution of GeDtzen's proof. Jts consequences are twofold: 1. The Induction in Gentzen's proof need only be applied to formulas of a very restricted complexity. In addition the consistency proof Dever uses the Jaw of the excluded middle. Thus it may be formalized Within a system T based OD JDtuJtionistic logic With induction along a weJlorderiDg of transfinite ordertype where this induction scheme is restricted to formll1as of a very Jow compleXity. So the problem of the consistency of pure Dumber theory may be decided within the system T. Although the wellorderlDg is of transfiDlte order type it can easily be VisualiZed. So Jt seems to be completely plain that the system T is consistent. By GOdeJ's second theorem the proof theoretic streDgtb of the system T, as It wjJJ be dermed later in this lecture. has to exceed that of pure Dumber theory. But the subsystem To of T which is obtained from T by restrictiDg iDduetJon to iDItiaI segments of the welJorderiDg onl,. can be shown to be eqUiconsistent with elementary Dumber theory. Thus GentzeD'S proof provides a reductJon of the consistency problem for elementary number theory to that of a theory Tcr which from a conceptual point of View may be regarded as 'safer' than elementary Dumber theory Itself. This is 8ft example of reductive proof theory. In reductive proof· theory one generally tries to reduce the coasistency problem of a theory TI to that of a theory Tz- For a clever choice of T2 both systems will have the same proof

5

Inuodue:t.Ion

theoretic strength. The principles used in T z• however. may be easier to visualiZe and therefore a justlflcat10n of the system T 2 seems more plausible. This type of proof theory is of great foundational _ (d. the introduction to [BfPS] by S.Feferman). One important feature of Hilbert's program we did not mention is the 'elimination of ideal elements'. In this sease reductiYe proof theory contributes to Hilbert's procram by elimiuting complicated unperspicuous principles. Since both systems (T. and T2 in the aboYe example) are of the AIDe proof theoretical atreagdl ~1Ye proof t.heory Ia 1ft full accordance With GOdel's second theorem. 2. The fact that induction along the wellorderiDg is the oaly nonfiDltlstlc means in Gentzen's proof also suggests using this wellordering as a measure for the transrlllite content of pare number theory. PursuinC this idea one bad defined the proof t1Jeoretic ordJnaJ of a formal theory T as the ordertype of the smallest wellorderlng which is neeclecl for a CODStstency proof of T. 1'bis der-mitlon, howeYer, is somehow vague sinCe it says aotbin& about the means used besides the induction alone this wellordering (one tacitly has to assume that these at least have to be formalizable in T). To obtain a more precI8e definition one calls an ordinal CI prcmable in T If there is a primitiYe recursiVely definable wellordering N ~ A'¥ for any assigDment ., wlUc:h at most differs ill the value of !'(X) from c..

(ix) N t= 3XA· c. There is aD assigDJDeDt Y which value of Y(X) from. such that N ~ A'F.

14

at

most differs in the

§2. SeIll&l1tJc:a For

~

If FV1(t)=B we have t·=t'F for all assignments 4) and V. For closed terms t. =8, we therefore define t N := t 4t for an arbitrary assignment ~. Two closed terms s and t such that sN= iN are called equivalent. Two formulas F1 and F2 are said to be equivalent if they only differ in equivalent Le. terms t such that FV.(t)

terms.

The vaJue of t. and the relation N t= A. obviously only depend upon .tFV(t) or .tFV(A) respectively. If FV(t) ={Xl~ ....xll} or FV(A) ={xI9••• ~Xn.xl' •••9Xm}' we often write t[kt•....k.J or N t= A(ktr.. ,kD.St~••••SID] repectively JDstead of "t. or til: A. for aD assignment • such that +(Xl) = k l and .CXj ) = SJ hold for i = 1.....,D and j = l~•••.m". If F is a sentence we obviously have N F pet ~ N 1= F1Y for all assignments 4) and .,. In this case we write N t= f aDd say that the sentence F is valid In N. For nt-sentences AIX.,...•Xn ] we have N 1= VX,••.VXDA if and only if N t= A· holds fOl" any assignment •. This is the reason for calling them n:-seDtences although they prima facie are 9't-formulas. For n:-sentences we always write N 1= A instead of N F YXt•••YXnA. This notatioD sometimes will also be used for arbitrary formulas A. So, for a formula ~ N i= A meaDS 'N t= A· for all assignments .'.

2.4. ExercI8e Suppose that L is a first order language which is given by a set C of individual constants. a set F of function constants and a set P of predicate constants. We defiDe L2 and L2 analogously to 9'2 or 9'2 respectiVely. The semaDtics for L, and L2 is defined in the following way. (i) A structure Y for Lt is a quadruple (), re. 7, if) which satisfies the follOWing conditions:

*.

(a) ) is a set. (b) We ba'Ye 4'fc I such that for every CE~ there is a d'E~.

(e) 7" is a set of function on I. such that for any n-ary function symbol f E F there is a function f9': Ill-. I in :F. (d) ~ is a set of predicates on I such that there is a pVc In in ~ for every n-ary predicate symbol PEP. (il) A structure SJ for Lz is a quintuple (I,M'~97.SJ) such that (I,Y,7.~) is a structure for L, and Me Power(l) (the power set of I). (iii) If f/J .is a structure for L1 or L , then an V-assignment for S'1(i=l.2) is 2 a mapping • which assigns to any x an element 4-(x) E I and to &Df set variable X a set .(X) c I or .(X) E M respectively.

15

§3. A formal sy.stem For pure .n&UJ2ber theory

For L-tenns t and L-formulas F and an !J'-assignment • we define t· and Vt=~ analogously to 2.2 and 2.3 respectively. We write 9'i=F for an LI-formula F if 9't=~ holds for any $P-assignment • and I=F If ~I=F holds for all L1-struetures 9'. Prove the following claims: (i) 9'I=A-t F ~ cSPI=A .... VxF if x fFV(A) (ii) 9't==F ~ A ::0 cSPp:3xF -. A if x. FVlA) (iii) t=VXF -+ Fx(Y) (iv) t=FxCY) -+ 3XF

(v) (vi)

cSPf=A-.F => YI=A .... VXF if X.FV(A) 9'I=F-+A

~

9'1=3XF .... A if X.FV(A)

§3. A formal system for pure number theory Still in the spirit of Hilbert's program we are trying to establish a formal system which derives as much valid sentences of N as possible. In a f....st step we are going to deal With those sentences which are valid because of their logical structure. Every formula of !fJ carries a sentential and a quantifier structure. To clarify the sentential structure of an SI-formula which is 8iven by the logical connectives ',1\ and V we introduce the sentential subformulas of aD ~-formula.

3.1 IDdIICtIft cIefJDltIcm of the set AT(F) of sentential subformuJasof an ftJformula F (j) If F is atomic or a formula QxA or QXA respectively where Qe{V,3), then ATIF) = {F}. Cii) If F is a formula ,~ theD ATCF) = {F)u ATCA). (W) If F is a fonauJa (AJ\ B) or (A v BJ.. theD AT(P) = (F) v AT(A) v AT(B). Formulas A such that AT(A) =tAl are called sententialatollJ8. By AE we deDote the set of all sentential atoms of !fl. We derme AE(F) :=AE n AT(P).

3.2. Defbdtloll (j) A sentential assignment is a mapping B : AE -+ { 1,f I. (il) The truth value A B of a formula A under a gJYeD seDtential assJcDment B is given by the usual interpretation of the logical conaectiYea as truth functions (cf. 2.3.(iii)-(v) and 10.12. below).

16

§3. A formal sy.tem for pure .Dumber theory

One should notice that only the values of B restricted to AE(A) are needed in the computation of AB. (iii) A formula A is sententially valid if AB assignments B.

3.3. Lemma

If AEAT(F). then FVj(A)c FVJ(Fl for j

=

t holds for all sentential

=1,2..

The proof is an easy induction on the defJDition of AEAT(F).. An assjgnment • canoDicaJJy induces a sentential assignment B. by defining A·· t c:> til:: A· for all sentential atoms A. For these assignments we have the follOWing lemma.

=

3.4........ N F A· holds if and only jf AS.:: t.

Proof by induction on the length of the formula A.. 1. If A E AE. then we have N

1= A· ~ A" = t by definition.

2. If A is a formula ,B, then we have N 1= A•

c> 111=

B· ~ sS.

=f ~ A" = t.

3. If A is a formula (8 0 C). then we have N t= A if and only if N F Band/or N 1= C By the induction hypothesis this holds if and only if BBcb = t and!or C" t. But this is equivalent to (B ~ ~)~ = 1.

=

As a corollary to 3.4. we obtain the follOWing theorem.

3.5.. Theorem If F is &elJtenUalJy valid.. then N

1= F..

Concerning the quantifier structure of an

~-formula we

just need the follOWing

observation.. 3.6 lemma If F is a formula ,Ax(t)v 3xA or ,VxAv Ax(t)., then N

t= F.

17

§3. A

Formal sy.sc:em For pure number t.heory

Proof From N FAx Ct)4t we have to conclude N F 3xA·. By induction on the length of A we easily obtain N FAx(t)~ c> N F Ax~·)·. Hence N 1= 3xA·. In the second case we have to show that N t= VxA· implies N t= Ax(t)·. But N t= YxA· implies N F= Ax(~·)· and therefore also tl t= Ax(t)·. In the proof of 3 ..6. the careful reader wdl have ftot.lced Uuat. the proof fteeda t.he eddltlonal hypot.hesla t.hat. t.he term t. is subst.itutable for x In A. Le.. none of the free variables occ:urrlDa in t muat. be bouDd in A. Here aad 1ft future we ~1l1 tacltly . . .UDle t:.bat t.b1a prereqUlalt. ••"Ways IS aatlafled. Thla means no reatrlCt.lOn since by renamln8 the bound variables In A. we may alwaya obtain t.hat. t. I.s subst.ltutable in A.

Now we are prepared to formulate the axioms and inference ruJes of the formal system Z. of pure Dumber theory. The language of Z. is the "..st order lcmcuage !l't· 3.7. Lop:aI uIoma of the formal system (j) ejj)

Zs.

3.8. Lop:aI (V) (3)

~.

Every formula of the form ,YxA v Ax(t) ad ,Ax(t) v 3 xA is a Jogical

axiom of

(mp)

Zt

Every sententJally valkl formula is a logical axiom of

mr.eace. of the

formal system

A"A v SI- B ,A v 81- ,Av VxB } ,Bv AJ- ,3xBv A

Zt

if xfFVI (A)

(modus ponens) (V-rule) (3-rule)

The variable x of a quantifier inference is called its eigenvariable.. 3.9.. Equality axloma of the formal system ~ Among the constants for the primitive recursive relations we have a constant = for the equality relation. Although we could derive the properties of the equality relation from its der-ming axioms (contained in 3.10.) we prefer to formulate them explicitly as a separate group of axioms. This is in coincidence with the usual treatment of formal systems where the equality symbol often is regarded as a logical symbol. (i) Vx(x = x) (ii) VxYy (x

=y ..... y = x)

(iii) VXI YX2 VX3 (Xl (iv) VxVy( X

=

X2!\ X2

=X3 ..... XJ

=y ..... t = tx(y»

(v) VxVy( x = y --+ (F ..... Fx(y»).

18

= Xa)

§3. A £onaal ~ys~m £or pure number d!eory

3.10. MetMmettcal aloma of the formal system (0 The successor axioms Vx('.Q=~x) VxVy(~x =!y -+

x

Zt

=y)

Sn = All for all nEN. (jj) The derwng axioms For prillJjtiverecursive Functions are given by the universal closures of the following formulas

(~Xt,···.,Xft) =1 (f.; Xl~··· .,x!) = Xk (Sub(g"h.,... ,!lm)x.···Xn) = (glbtxs .. ·Xn) .•. (hmx.···Xn» (x = 0 -+ «Rgh) x ••••xnx) = gXt ••• xn) 1\

« Rgh)xl••. xnx) = hx••.. xny(CRgh) x•.•• Xny» RXt···xn'" XR.Xt···XIl = 1

(x = jy --. (ill)

The induction axiom is given by the scheme (IND) Ax(O)

1\

Yy(Ax(y) .... Ax(~) .... YxA

3.1t.1Dduc:t.lft defbaWoD of Zt ~ F We are going to define the formal derivation predicate for ~. be read as 'F is formally derivable in Z;.'. (i)

If A is one of the axioms 3.7.,3.9. or 3.10, then

Z. I-

Z. I-

F should

A.

If Za ~ Ai (i =1 or j = 1.2) holds for the premise(s) of an interference according to 3.8. whose conclusion is A., then we also have %,.1- A. (ii)

3.12. Remark The system Zt is an extension by definitions of the better known system PA of Peano arithmetic. PA is formulated in a first order logic with equality. The only nonlogical symbols of PA are the binary function symbols '+' for addition and ' · , for multiplication~ the unary function symbol ! for the sucessor function and a constant ~ for the natural number o. (The equality symbol is counted among the logical symbols). The axioms of the group 3.10.(JJ) are then replaced by the defining axioms for ,D.., .§.. '+' and I.'. i.e. by the universal closure of the folloWing formulas x+,Q= xandx·Q=~ x+~Y

=..§(x+y) x·.§y = (x·Y)+x

Here as usual we have written (x+y) instead of (+xy) and (x-y) iDstead of (.xy).

ApparentJy PA is a subsystem of

z.., i.e. we have

19

§3. A formal system For pure number theory

(1)

PA ~ F ::::> Z. t- F for every formula F in the language of PA.

We also have the opposite direction of (1) which means that extension of PA. But we may even prove:

Zt

is a conservative

(il) For every ~ -formula F there is a formula Fpin the language of PA such that Z. F ~ Fp •

r-

This means ~t every symbol of Zt can be defined in PA. For this reason Zt is called an extension of PA by dermitioDS. The proofs of () and (Ii). however. reqUire methods from the theory of recursive functions aDd will Dot be given here. 3.13. SoImdDeu theorem for l£~ ~F,

Zt

then N F F.

Proof By induction on the definition of

we show that Zt t- F Implies N 1= ~ for any assignment •. If F is a logical axJom then we obtain N 1= pet by 3.5. or 3.6 respectively. The claim is obvious for the equality axioms and the mathematical axioms. We only should 'check the induction scheme. Here we have to show that N ~ A(.Q)· and N t= Yy(A(y) --. A(jy))· imply N t= AlnJ· for all DEN. But N 1= Vy(A(y) -+ A(§y)· and N 1= A(~)· imply N t= A(Sn)·. Since we have N 1= A(Q)· and every natural DUmber is obtained from 0 by finite applications of ,the successor function we easily obtain by metaillduetion on n that N t= A(!!..). for all DEN. In a last step we show that the valklity in N is conserved by the inference rules. From N t= A· aDd N 1= (A .... B). we immediately obtain N t= Bc.. If F is the conclusion of an instance of the V-rule then F must be a formula A -+ VxB and we obtain by the iDduction hypothesis N t= (A --. B). for all assignments •. It remains to show that N t= A· implies tJ t= Bx(!!.). for all DeN. For an arbitrary neN we obtain an assignment 1f' by defining Y(x) := n, 'F(y) := .(y) for all y*x and ,,(X) := . (j) Every number constant is a term. (ii) If tl •... ,tn are terms and if f is a constant for an n-ary primitive recursive functio~ then (1t1, .... ,tn ) is a term.

4.3. 1DcI1lc:tbe defbdtlon of the formulas of !fJfXJ (i) If t l •...•tn are terms and .R. is an n-ary relation constant, then (R,tt...tn) is a formula. (ii)

(ill)

1\ {Ai

If t is a term and X a set variable. then

(t EX)

and

(t f

X) are formulas.

If I is a nonvoid mdex set and (Ai)1 e I a sequence of formulas. then E I J and V t Ai : j E I} are formulas.

: i

We often write 1\ Al and V Al instead of /\ {AI: iE I

1£ I

j E J}

or V {A 1 :

j EI

.

) respectively.

As usual we write AJ /\ ••• AAn or A1v ... v ~ instead of !\{A1,...A,.} or V{AJ....-An} respectively.

Formulas built according to one of the clauses (il or (n) are called atomic.

The language !i'n is the sublanguage of !l'00 which is obtained by restricting the index set ) in clause (ill) to countable sets only. For technical reasons we do not count the negation symbol , among the basic symbols of the language.. This, however, does Dot mean any restriction since we may define it in the follOWing way. 4.4. IDda'd1ft cIefbdtIcm of , A (j) ,

Rtt ...t o

is the formula Ets .••tft where

R means

the primive recursive

relation complementary to R

X) == (tt Xl, A A- == V , A- ,

(il) , (tE (iii) ,

iE)

1

lEI

I

, (tf X) , V Aj lEI

== (t E Xl,

==

A , A-. iEl J

4.5. Lemme "A= A

The proof is an easy induction on the definition of

,A.

4.6. ltemark Sometimes we will be forced to extend the language !t'~ by number variables. We usually will only need fmitely many number variables Xl' _. - ,xn- We denote this extended language by !PUJ(x., ... ,xn ). The terms of the extended language

23

are then defined by adding the clause (0) Every number variable is a term to definition 4.2.

§ 5. Semantics for 9'Q ) In!/'aJ we do not have any number variables and therefore do not need aD assignment for them. An assignment for !t'co is a mapping • from the set of variables into the power set of N. We derIDe t· for S'aJ-terms t as in 2.2. Since there are no free number variables in t we always have t. =tN. S.l IDdactbe cIefbdtIoll of N (i)

N

(iii) (iv)

t=

(.ftt ...tn)· ~

t= (t e X). N F (t. X)· N 1= 1\ AI· IE I N t= V AI· lE I

(ii) N

t=

~

~

XE.(1tN, ••• ,~) t N e c.(X)

~

~ ••(X)

~

tl

~

N

t= As· t= AI·

=1

for all iE I for some jel

As in the semantics for!t' we denote by N F A that N assignments •.

t=

A. holds for all

Our first goal is to obtain a more S1Jltaetical description of the validity relation for the language S'o. For this reason we are going to introduce a CODCept of iDf"JDitary derivations which completely characterizes the validity of !fJa-formulas in N. Again for technical reasons we will not solely derive smgle formulas but rather finite sets of formulas. These finite formula sets are to be interpreted as the disjuDCtion of their members. As syntactical variables for finite sets of formulas we use capital greek letters' such as 4,r~.,.... We always wllJ write 4.F instead of 4v{F}. 5.2. IDducttte defmltlcm of

t:a

A

(Axl) If lp(tjN,.....t nN ) =1 and (~tl ••. tn) e4, then ti 4 (Ax2) If t N a: ~ then Ii 4, tE X. Sf X (1\) If ti /1, Ai for all i E I, then ~ A, /\ {Ai: iE I} (V) If ti fl., Aj for some IE 1, then Ii 6., V {Aj : Ie I}

24

§s. SemaDtJcs for .f'00

5.3. Soundneu theorem for

If ii ~. then

t:g

Nt=V {F : FE~}.

Proof

Ii A. (Ax1) Then A contains an atomic formula (~t,. ... tn> such that xp (ttN,•••,tr> = 1 and by S.l.(i) we obtain NI=(V{F: FE6})· for every assignment ct. (Ax2) For any assignment 4» and term s and t such that SN = t N we either have N F(tE X)· or N F(S. X)·. Hence N I=(V A v (tE Xl v (sc Xl)•. (1\1 By the induction hypothesis we have N I= EBA and ~« » := A (ii) If GE BA and 0(0) is not reducible or an axiom according to (Axi) or (Ax2). then it is a .. BA for all j< fa) (i.e. 0 is a top node of the tree). (ill)

If oeB6 and

&(0)

reducible with distinguished redexl~A{. then O-EBA 25

§s. Sem.ntlcs for 91CD

for all iE I and S (o.

Vx< Ih(n)-1 (0< x => (n)x+l ~ (n)x).

and

*

= 0" D2 0) C3x< min{lh(nt),lh(D2)}«Dl)X< (n2)x)J\ VY< x (O (Dt)y = (D2)Y» v Clh(nt)< Ih(D!) 1\ Vx< Ih(nt)«nt)x = (n2)x»].

Dl ~ A, then we have AJ.At for all iE I by the induction hypothesis. By the same inference (8) we thus obtain ~ 6~ Al for all iE I.

I?

S. If 1\ Ai is the main formula of the last inference, then this inference is a 1\iE} CIt inference whose premises are ~ 1 Ao, Ai for all i E J. By the structural rule we obtain ~1 A.Al'l~At and by the induction hypothesis eiA,A1 for all iel. This implies A.Ai for all iEI by 10.5.

t:

10.8. V - _ ad V -exportation (I)

~ A. AI

(ii) ~

A,As v

,An implies

'P"'

ft

A,A. v ... v An.

v AD implies ~ A. A1,.••,An •

Proof (i)

By iterated application of

~

rp-A,A2-•...An,A1 v ••• v An => (il)

V -inferences we obtain

~

A,A.....,A,. =>

rp A, A3 ••••,An,A. v.•• v An =>•••=> ICItp + D ~"'2

The proof is by induction on

A,~v

A ••• v'-;-sl

CI.

1. If ~ v ... v An is not the main formula of the last inference, theft either 4 is an axiom and so is A,~,••• .An or we have the premises ~ AJ • ~ v •.. v~. But then we have ~ AJ'~'••• '~ by the induction hypothesis and obtain ~A,~,•••.An by the same inference.

2. If Al v ...v An is the main formula of the last inference. then it is an Vinference whose premise is ~o A, AlAs v •..v An. By the induction hypothesis It follows ~6,A1'.•.An and by 10.5. ~ A,AI'•• .An 10.9. Tautolo8J 1emma Suppose that F is an 9'co(x.,••. ,xn ) formula. t=(~'.-.tn) and .=(St'...•sn) are ntuples of!l'CD -terms such that Sl and t 1 are equivalent for i =1,.••,D. Now iF F1 == Fx(t), F2 == Fx (.) and ex rk(F)9 then we have ~cx A.F1, ,F2 for all finite formula sets A.

=

Proof by induction on rk(Fs) 1. If F• .is an atomic formula RtJ ... t ll • then we have Fz :: Rs••• .sn • If Rt•...t D is valid • we obtain by (Axl) ~ A,F1,1Fz. Otherwise ,Rs•...s D is valid and we 52

§ 10. TIJe Jnilmtary 8y.s~m ~

again by (Axl) obtain Ii- ~,FI' ,F2• 2. If Ft == t l E X, then ,F2 :: Sl • X. But then I;- A,Ft , ,F2 holds by (Ax2). 3. If F1 == A A~, then ,F2 :: V 'A~ and by the induction hypothesis we have ax t ~~\ . .IE I . . .axl+l 1 FirA, A 1 "A1 for all lEI. Usmg an V-mference we obtain t-;;-A '~1,'F2 for all iE I. Since (I =sup{cxl +l:ie I} we have 2«1+1 < 2( NI=Ai[S] for some ie 1I4·NI=Ai[T] for some i E J => N t= F[T]. 13.9. _

lemma

Suppose that ~ is a transitive wellfounded!fJ,-def"mable relation on N and A is a rJDite set of X-positive !f1«)-formulas. If ~ ,Prog(-(,X), It f X,....,tn f X, 4, then it follows NJ=VIFx[-(,.]:FeA} where y = ~2cx and ~ =max{lt~I