G. Kreisel - Mathematical Logic - A Survey of Proof Theory

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A Survey of Proof TheoryG. KreiselJournal ofSymboZic L ogic, Volume 33, Issue 3 (Sep., 1968), 321-388.

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T H L OUXNAL OF SYMBOLICooicVolume 33, Number 3. Sept. 1968

A SURVEY OF PROOF THEORYG. KREISEL

81. Introduction. One might fairly say that the very meaning of our subjecthas changed since Hilbert introduced it under the name Bewektheorie (it wasmean t to be the principal tool fo r formulating Hilberts general conception of howto analyze mathematical reasoning). Specifically, the roles of the two principalelements of proof theory, namely the intuitive proofs accepted and the formalproofs (or derivations) studied, have turned out to be quite different from whatHilbert thought. In his view the hard work had been done in the discovery offormalization, and what remained was the study of certain given formal systems.But, as knowledge accumulated, it turned out that the analysis of the intuitiveproofs considered and the choice of formal systems needed most attention. Inshort, for Hilbert Beweis (in Beweistheorie) referred to formal derivations; forproof theory to be viable at the present time, it has to concern itself with the intuitiveproofs too.I think we shall get a clear a nd co rrect perspective, both on what has been donein proof theory and on what to do next, if we first recall how Hilbert himselfregarded the matter.Separating the foundations of mathematics from philosophy (epistemology).Hilbert wanted to accept only that part go f mathematical reasoning, of whichschool mathematics is typical, an d without which there could be no science at all;for more detail, see Note Ic(iii). He called Po finitist because he believed thatfiniteness was essential to its elementary character; bu t this is in itself a problematicassumption because p e works with variables (other names for goeflecting a dif-ferent philosophical analysis of what is essential to Po re (1) combinatorial or(2) concrete). O ne knows fairly well what go s, but g iving a precise definition is theobject, not the sta rting point of research here.

Received February 14, 1967, and, in revised form , May 31, 1967.This paper is based on an invited survey talk given by the author at the 1967 Annual Meet-ing of the Association for Symbolic Logic in Houston, Texas.

Th e present pap er, which deals mainly with subsystems of classical analysis, is complemen-tary t o thre e other p ublications: (i) the survey article [31], the contributions by (ii) Feferm an an d( i i i ) myself to the Congress for Logic, etc., at Amsterdam, August, 1967. The bnsic theoremson which 81-12 depend, are also contained in [31]; the present exposition is meant for readersnot familiar with non-classical systems, while [31] considers proof theory for intuitionisticformal systems, for the technical reasons given in [ 3 1 , p. 156, 3.2421. However, I give hereseveral new formulations, both of systems and results; F efermans p aper (ii) describes the use ofautonomous progressions for the characterization of intuitive conceptions 9 ts history andits problems; this important part of proof theory is neglected here altogether. The paper (iii)brings [ 3 1 ] up to d ate as far as specifically intuitionistic questions a re concerned . Several resultsfirst stated in [31] appear with full proofs and, often, technical refinements in [36].

I-J.S.L. 321


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322 G. KREISELHilbert had two discoveries available: First, the empirical discovery of formaliza-tion of (existing) mathematics in formal systems S Principia Mathematica); that

means, ifone accepts the abstract notions of mathematical practice then mathe-matical reasoning is correctly expressed by S. Since most of these notions havenothing to do with Po,here has t o be a link with Po. his was provided by H ilbertsown discovery of how to formulate in Po itself suitable adequacy conditions on aformalization.We have a certain classd of assertions in Po, translation T which maps asser-tions A E d into formulae AT of S. Recall that every assertion of Pois decidable.Th e first requirement on a formalization of intuitive d in S s this: a function Rfrom d into form al derivations of S and a proof in Poofwhere Prov, is the proof predicate for S.Second, again for A E d nd variablep,is to be established in Po. t is a familiar technical matter (e.g. [14, p. 3041) that,under quite general conditions on S, (ii) is equivalent to the consistency problem,and further that (i) and (ii) are proper adequacy conditions; cf., e.g. [35,p. 2091.Thus he had established the remarkable fact, due to formalization, that ade-quacy conditions, i.e., (i) and (ii), can be formulated in Po;more precisely, Provsbeing rudimntsry in the sense of Smullyan, the onZy assumption on Po eeded isthat rudimentary relations E Yo.The so-called separation of foundations from philosophy may be expressed asfollows (i) and (ii) should be ertablished by use of oery simple assumptions on Po butpossibly by means of complicated mathematical constructions); in other words,Hilbert thought that iio detailed philosophical analysis of Po as necessary (as,sometimes, one succeeds in answering a physical problem purely mathematically).This is Hilberts programme, stripped of some unfortunate formulations : seeNote I(c).What made this programme plausible was the tacit conviction that assertions aselementary as i ) and (ii) could be decided in Po.Strictly, it would be sufficient toassume this decidability for those S which are accepted in mathematical practice.But then there is no convincing reason why (ii) should be provable in Po or thenext system th at presents itself in practice. An d if one had to make a fresh star t onewould not have the definitive separation which Hilbert wanted. (For even strongerassumptions, cf. 94, Remark p. 326.)Consequences of Godels Theorems fo r proof theory. The following two conse-quences of the second incompleteness theorem are well known.1. There will be a need for a philosophical analysis of conceptions B, otherthan Po,nd the interest will depend on the intrinsic significance of 9. hus wedo not have the separation which Hilbert wanted.2. For a given S we cannot in general expect to establish (i) and (ii) in somegiven9 ut we shall have to search for an appropriate 9.Note that if B is changed,also the appropriate classd will in general change. Further (this is not used in thepresent paper), it should not be assumed that S is necessarily a formal system, if Bis not com binatorial, since nonrecursive descriptions may be accepted in such a 9.

(i) A -+ Prov, (nA, AT), for A Ed(ii) Prov, (p, AT)+ A


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A SURVEY OF PROOF THEORY 323But also the first incompleteness theorem affects Hilberts scheme.3. Th e choice of S becomes a central problem. With Hilberts assumptions (aslisted above), it did not really matter what systems S among those accepted in

practice, were chosen, because Po as suppo sed t o establish (i) and (ii) for all suchS. Mo reover, and independently of their relation to go,he systems S for arithmeticand analysis, which Hilbert had introduced, were thought to be complete, i.e., therules were believed to decide every proposition formulated in the language of S.So, naturally, on the one hand he gave no attention to subsystems of his formalsystems (e.g., arithmetic with induction restricted to free variable formulae in $4);they would have been regarded as artificial On he other hand, he did not loo k forextensions of his systems S ; for, by completeness, no new rules ar e required whenformulae in the language of S are considered, and, by the empirical discovery(p. 322), the concepts of m athematical p ractice were know n to be expressible in thislanguage. (As is well known, the discovery of extensions of the usual formalsystems is one of the major problems of foundations; but not for Hilberts ownconception of proof theory since the usual systems gave enough trouble )In view of G odels incompleteness theorems, there are 2 altxnatives:1. Since all formal systems are incomplete, i.e., are subsystems, they are arti-ficial: that, roughly, is the end of the kind of proof theory here considered.2. The choice of system becomes fundamental, e.g., if we are able to establish,in a given 9 i) and (ii) for a subsystem S of S, but not for S itself. Clearly, if areduction to 9 as any intrinsic significance at all, the question: Can existingmathematics by chance be formalized in S ? is just as significant as whetherit can be formalized in the obv iou s system S ; and, in general, more signifi-cant than whether existing mathematics can be formalized in some (consistent)system.This conclusion c o d ic ts with a widespread opinion dating from the beginningof this century a nd before; naturally, since the conclusion is derived from Godels(later) discoveries. It is likely that the progress of foundational research was heldup by the mathematicians habit of concentrating on the new methods introducedin a proof, such as Godels, instead of re-examining th e significance of old problemsin the light of the new result.

52. Summary @hilosophicalside). The conceptions B involved in the presentlecture are combinatorial, nonconstructive predicative (relative to the notion ofnatural number) and, to some extent, intuitionistic proofs, It is often useful, ifnot necessary, to formulate explicitly which properties of these conceptions areneeded in any particular argument, just as formal axioms of set theory formulatesome of the mathematically most useful properties of the notion of set. There isone im portant difference: since proofs are involved in 9 , ot only axioms, but alsorules of inference express significant properties of these conceptions 9: n inter-pretation of a formalism will not only assign a universe an d objects to relation an dfunction symbols, but will assign intuitive proofs to formal derivations. In otherwords, not only the set of theorems, but the set of formal derivations is im-portant.Broadly speaking, the properties of 9 iscussed will CoIicerri: principles o f p roof


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324 G. KREISELby induction and definition by recursion; definition principles for functions of typeVN -f N ; an d definition principles for functions of all finite types.

The philosophical interest of these principles is explained in $6(d), 10, NotesIV and VII. But the novice, coming from mathematics, will feel ill at ease unlessthe following objection is answered:Th e principles B considered a re more o r less traditional. Why should we restrictourselves to th em ? Would it not be better t o appr oac h the consistency problem bythe light of nature? and search for a consistency proof which is mathematically asinformative as possible? W hen we have found it we ca n then analyze its philosophi-cal significance. In other words, once again, we try to separate the foundations ofmathematics from epistemology, only in a somewhat weaker sense tha n originallyintended by Hilbert.Note I(b) is devoted to this matter.$3. Summary contd. (mathematical side). $96-12 contain mathematicalresults involving various forma systems fo r branches of .urrent (nonconstructive)mathematics or for the conceptions B mentioned in t1.e last section.Since the analysis of intuitive conceptions B has acquired such importance, prooftheory has the general character of applied mathematics. T o isolate what is mathe-matically essential it is natural to look at the methods used, and to see how theygeneralize, a s e.g., a proof concerning 3-dimensional space may generalize. Onewould expect such generalizations of proof theory to have similar functions. Thus,technically, the general problem may force us to simplify the argument, and SO

enable us to solve more complicated problems a bou t 9 itself. Also distinctions thatare subtle in the particular case, may become quite easy in the general case.Finally, the notions developed in such a generalization may be used in analyzinga new situation (here: new conception) for which the existing theory is inadequate(e.g., by use of hierarchies of systems instead of a single one, cf., footnote 1).Both these functions seem to me well illustrated by the extension of proof theoryto infinite formu lae , or, more precisely, to suitably chosen infinite formulae. I shallgive a short progress report in $13. The technical use will be evident. Personally Ibelieve the subject to be philosophically important, roughly because thoughts seemto me much better represented by infinite objects than by the words we use tocommunicate them. The idea is, of course, not new, but one would have less con-fidence in it if one did n ot have a well working theory of infinitely long f~ r m u l a e . ~a Naturally, one will try to decide a specific question by using properties of the concep tion 9which are evident o n quite a superficial understanding of 9. pecifically, one needs approxi-mations to 9 rom below for positive results, and from above for negative results; based on suchwork, the approximation, one hopes, can then be improved. In other words, we do not startwith a definirion of 9:his side of foundational research is often strange fo r the pure mathe-matician. People familiar with model theory may find a comparison with the so-called mathe-matical o r algebraic characterization of syntactically defined classes of models helpful; nobodyhas ever explained, except by a few examples, what is here meant by mathematical. But onerecognizes a good solution when o ne sees it. More imp ortant: it is, I think , an interesting ques-tion to explain th e prog ram me precisely, but this definition m ay only be possible at quite anadvanced stage of the programme.3 A more elaborate discussion of the representation of thoughts by infinite configurations isin [32], written specifically fo r Bertrand Russell: the present paper, it is hoped , is suitable for awider audience.


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A SURVEY OF PROOF THEORY 325Theory of formal systems and proof theory; a distinction. Evidently there aremany questions about formal systems which do not come under the heading of

proof theory in our sense at all, e.g., a completeness theorem with respect to anintuitive notion which does not involve any notion of proof. Thus for closedformulae A of first order predicate calculus, we have a set theoretic formula V(A)expressing that A holds in all structures, and a completeness theorem of the formVA[V(A) --f 3p Prov, (p, A)], where Provl is the proof relation for on e of the usualformulations of predicate logic. Th e very statement of completeness does not makesense in 8, f V is not defined in 8, .g., for Po.Another important example are independence pro ofs, i.e., assertions that for aspecific Ao, and for all n, iPr ov (n, A& when one simply does not know the answ er;e.g., for Quines set theory and Ao:0 = 1. It is then not only worthwhile to usemodel theoretic methods, but ridiculous n ot t o d o so. (Incidentally, quite often it israther easy and not specially interesting to convert such arguments into relativeindependence proofs which use only combinatorial methods, cf., Note 11.)Occasionally it may happe n tha t proof theory in the present sense provides theeasiest solution of a technical problem, i.e., one which is formulated in ordinarymathematical terms; in particular which does not refer to any conception 9 .Aninstance are questions ab ou t finite axiomatizability and related matters. We shallcall such technical uses of proof theory: applications, to distinguish them fromphilosophical implications.$4. Choice of subsystems. Fo r Hilberts programme a formal system S merely

functions as a compact description of m athematical practice. B ut one certainly doesnot discover such S by making a statistical analysis of mathem atical texts an d, ifthey had been so found , they wouldnt have been convincing (equally the rulesof predicate logic wouldnt have been found if one had not looked a t the intuitivenotion of logical consequence either in the now usual set theoretic sem antic senseor in the sense of intuitive logical consequence as understood by Frege; for detailson the distinction see, e.g., 42 of [33].)The general principle, which distinguishes the discovery of the basic axiomaticsystems here considered from technical ones like the axioms of group theory, isthis:We have a second order (categorical) axiomatization of a mathematical structure,and then pa ss to a first order system either (i) by using a schema or (ii) by using amany sorted calculus and giving explicit closure conditionsfor the set^".^Consider Peanos axioms for arithm etic, i.e., the structure (N, 0, S)of the set of natural numbers with a distinguished first element 0 and the successorfunction; besides the first order axioms for 0 and S, we have the second orderinduction principle

where X s a second order variable.

EXAMPLE.

VX(vx[X(x)+.X(Sx)] --f Vx[X(O) --f X(x)])The structures here considered concern arithmetic and the continuum, and, implicitly, thecumulative hierarchy of types up to the first inaccessible cardinal. All these stru ctures possesssecond order axiomatizations.


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326 G. KREISELCme (i). On e replaces this by a schema, leading to the fam iliar system of classi-cal first order arithmetic (cf., Note I11 for some formal details). N ote th at this step

is ambiguous: thus, Peano's axioms mention explicitly only the function symbols0 and S (of zero, resp. o ne argum ent), but the familiar system contains + and xtoo In technical language: the existence of additio n and multiplication are secondorder consequences of Peano's axioms, and the choice of first order language takesinto a ccount a t least some such consequences.6 It is well known that the correspond -ing schema in the first order language of (0, ) is quite inadequate for the formula-tion of arithmetic reasoning.REMARK.The formal system (with the induction schema) restricted to thelanguage (0, ) is complete: did H ilbert think tha t, for any primitive recursive func-tion f defined by use of auxiliary functions fl y . . ., k, the recursion equations fo r(f, fl,. . , J together with th e successor axioms and the schema for th e language(0,S, , fly . , J was similarly complete ?Case (ii). One considers a two sorted predicate calculus, adds membership,an d writes dow n existential axioms for the X corresponding to the formation rulesof first ord er formulae, roughly a s in the familiar finite axiomatization of set theoryby means of classes (for a general description, cf. [33, App . A]), and writes induc-tion in the formHere, we have sn evident ambiguity in the choice of existential axioms (among allthose that hold in the given structure).REMARK. Philosophically, the second form is more attractive because oneunderstands the infinite set of axioms in (i) merely because they are instances of (ii)But for a good formalization of analysis it will be necessary to separate inductionfro m set existence axioms.Th us in b oth cases (i) and (ii) we have a n ambiguity in the choice of first orde raxiomatization. Therefore special interest attaches to results that are as independentof these choices as possible. Below I shall try to give such formulations, e.g., inconnection with the role of e0 in arithmetic.Analysis. The structure considered is (N, $(N), 0, S, E). The familiar secondorder characterization adds to Peano's axioms: extensionality for E and the socalled comprehension axiomA more convenient formulation uses the structurewhere 0 denotes application, sometimes called function evaluation: NN x N --f N,end f, g . ar e function variables (of lowest type), an d second order binary rela-tions X on N x N, N x NN eip. NN N N . Besides the usual axioms relatingfunctions and their graphs, we have the axiom of choice (AC) in one of the forms

wqVx(x E x7 x X)4- x(0 E x+ E X)].

(CA) VX3XVx[x E X t. X(x)].(N, 9 ) ", 0, , 9 o>,

(ACoo)(ACOl) ~ 3 f P X 3 g m) -+ VXX(X, fJl(DC11)

VX3f PX ~Y X (X ,) +VXX(X, x)]VX%3kPf3gX(f, g)4Vx(koX = hx A X(kx,k, + I)]

5 + and x are particularly elementary: they are implicitly definable in (N, 0, S).


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A SURVEY OF PROOF THEORY 327where f, is defined by: f,(y) = f((x, y)), (x , y) being a term that defines a pairingfunction.It is well known that DCll j ACo, -j ACoo jCA. By means of quite elemen tary(first order) closure conditions we also have CA jACoo.But since DCll is true inthe unique structure satisfying the axioms for analysis (CA), also DCll is a secondorder consequence of CA (together with the elementary axioms) Fo r some c o mments on nterpretations for which DCll or even ACol is not plausible, cf., NoteIV(b).First order systems, where X is replaced by explicitly defined relations B in thesecond order axioms (of induction, comprehension, choice respectively). Oneconsiders subsystems for reasons explained in the introduction; but also if onesimply wants a faithful picture of the arguments actually used in existing analysis.One way of obtaining such subsystems is by restricting the syntactic form of 8 ;e.g., by the number of alternating quantifiers in its prefix (in prenex form) familiarfrom Kleenes hierarchy. Two points should be noted: (i) Inspection of informalanalytic arguments suggests that the induction schema and the other schematashould be treated asymm etrically;one applies freely induction t o arbitrary form ulae(containing variables of all sorts in the formalism considered), but it turns o ut tha tvarious branches of analysis only use (AC) or (DC) in quite restricted form. Amore theoretical reason for this asymmetry can be given in terms of w-models inNote IV(a): if a formula A is not derivable because induction is restricted, amodel of iA must contain nonstandard integers, i.e., it cannot be an &model.(ii) Though certain levels of (the sy ntactic analogue to) Kleenes hierarchy lead tosignificant subsystems, e.g., fo r 8 E 2: (cf., Note V) and almost certainly E E C,it should not be assumed that this necessarily remains interesting for all CActually it will turn out that, for present day proof theory, a completely diyerentclassificationhas so far been much more successful.This will be considered in $9 below. Roughly speaking, one there avoids afundamental defect, formulated and established in $8, of the subsystems above.There are canonical definitions R of quite familiar well orderings of the naturalnumbers, e.g., of ordinal co, such th at the least element principle(*> 3xE(x) -+ 3u[ (u) A Vv{R(v, u) -+ iS(v)}],cannot be derived in the subsystem for su5ciently complicated 8. Since thesecanonical definitions define the well ordei-ings considered in all w-models, onceagain as in the case of restricted induction, a model negating (*) is bound to benonstandard (even with respect to the natural numbers).A word on Significance of subsystems. Philosophically, the criterion is clear:given9 one wants t o find a subsystem which can be reduced to 9 in the sense ma deprecise in this introduction) and, at the same time, permits a convenient formulationof mathem atical practice without ad hoc tricks. Mathematically, the question is lessclear cut, and , often, a bit subjective, being relative to existing knowledge. (That isone reason why mathematical logic is so often frustrating without philosophicalinterests ) A subsystem is clearly interesting if it is satisfied by a familiar class of(sets and) functions which does not satisfy the full system: e.g., in the case ofZ:-ACol or Z:-DCll, the class of hyperarithmetic functions. (Evidently, to appre-


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328 G. KREISELciate this example one has to know the notion of hyperarithmetic function, and to.--cognize its value.) In particular, most results of hyperarithmetic (and also of socalled recursive) analysis are immediate corollaries of the fac t that these results canbe derived in suitable subsystems of classical analysis: people working in this sub-ject seem to enjoy rewriting classical arguments, having forgotten the axiomaticmethod altogether fo r specific examples, see, e.g., the review of [4]. (Naturally,independence results require a d hoc constructions [25] . )The crucial problem in theuse of subsystems is almost always the proper choice of definition for a conceptamon g thqse that are equivalent in full analysis or set theory. F or instance, differentdefinitions of the class of hyperarithmetic functions, i.e., different propertieswhich single ou t these functions in N N, are, in general, not provably equivalent inX+ACol, [Interestingly, it is often the earlier clumsier definitions for which morecan be proved in weak subsystems; for striking cases, cf., Note V(a).] People arefrightened of asking: which definition is correct? or which is f i i~da me ntal ? nd soinvestigate a dreary list of alternatives. Here is an obvious p roposal: One choosesthat definition of the concept studied for which the prin 5p al properties used in theinformal theory can be proved in a weak subsystem. (This suggestion is heuristicand research is needed to verify whether it holds in a particular case, i.e., for agiven concept and a proposed subsystem.) Of course this suggestion applies notonly to an axiomatic theory of the hyperarithmetic hierarchy, but, even more, todefinitions of more intuitive concepts such as the geometric concepts of open orclosed sets, now one requires of the definition that the intuitively evident propertiesof the concepts be provable (not merely the principal properties that happen tohave been found useful.) Another useful but less decisive criterion concerns thefollowing kind of stability; if one goes over to a different language and writes dow nthe axioms t ha t strike one as an alogous, the resulting system should be closelyrelated to the orig inal system: for more precise info rma tion, cf., Note VI(c).

$5. Classical first order predicate logic. This is, of course, the most familiarpa rt of proof theory an d th e area w here many of its fruitful ideas were first applied.But, though the philosophicdl interest (for Po of the work of e.g., Gentzen orHe rbr and is quite easy to state, the mathematical side is confusing: one feels tha tsomething is achieved, but many of the results seem to be got more easily by non-constructive methods, in particular, so-called semantic completeness and soundnessof the rules studied. (As said before, these are paradigm s fo r properties of formalsystems that have nothing to do with proof theory.)EXAMPLE. Consider systems of arithmetic with a purely universal axiom A.The n, if Vx3yB(x, y) is a consequence of A, there are terms tl, * , , built up fromthe function symbols in A such that B(x, tl) v . . . v B(x, t,) is a consequence o fA l A . . . A A, where each A, is a substitution instance of A. Furthermore thereis an operation f in Po iving a purely propositional derivation from a formalderivation d of A -+ Vx3yB(x, y) (and hence a bound on the complexity o f the ti interms of d ) .Philosophically, this is important bccause in the derivation d there may occurformulae which have no sense in Po (if no interpretation is given to quantifiedformula) or are simply false (for the natural interpretation of quantifiers in Po);


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A SURVEY OF PROOF THEORY 329but the result shows that if A is valid, for the natural interpretation, in Po,so is

Mathematically, this realization of existential quant9ers by explicitly definedcombinatorial functions has some algebraic applications, discussed in [22]; seehowever Note I(d). If, from a given d , one actually wants to get these realizationst one does have to go back to the combinatorial proof. But if one merely wants theexistence of such t, or of a recursive function f, on e has a very simple model theore-tic argument. (So if one is neither thinking of actual computations nor has a graspof Po ne will be ill a t ease with the old argum ents.)EXAMPLE. One of the proofs of the preceding result uses Gentzens so calledcut elimination. Consider, for the moment, any of the systems that you know ascut free. Gentzen gave a proof in Po ow to obtain a cut free derivation of aformula A from a derivation in the usual systems. But the existence of the cu t freederivation is an immediate consequence of (i) completeness of the cut fre e ruleseand(ii) soundness of the usual rules. Since, for many algebraic applications only theexistence of a cut free derivation is needed, we have the same situation as inExample 1. For refinements, see Note IIb(i).Of course, with a little sophistication one sees that cut elimination is quite adifferent theorem from (i), since the method of proving cut elimination was applieddirectly by Gentzen to intuitionistic systems which are patently not complete forordinary semantic validity.Towards the end of this survey we shall find it necessary to analyze what isessential about cut free systems. For the present (i.e., in the next 3 sections) themathematical applications will turn on the subformula proper ty, i.e., the fact thatin a cut free derivation of A only subformulae of A app ear ; the usefulness of thisproperty, in turn, depends on the existence of partial truth definitions for the sub-formulae of any fixed A, for a natural meaning of subformula. (As observed byTarski, we do not have unrestricted tr uth definitions.) App lications of this propertyare spelt out ad nauseam in [36].For differences between the usual rules and cut free rules in relation t oGodels second incompleteness theorem, cf., footnote s 8 and 16.From the point of view of length of derivations the situation is less clear cut.Thus, the addition of cut (modusponens)shortens calculations; o n the other handthe use of tr uth tables (in propositional calculus), e.g., formalized by the construc-tion of Posts normal forms, is generally longer.Note in passing that Gentzen [8] attributed a much more central significance tocut free rules; roughly tha t logical operations are defined by rules of deduction, andthat the cut constitutes a certain impredicativity (which he wished to eliminate;Lorenzens operative logic [45] and game theoretic analysis [44] are attem pts in thesame direction). There may be something to this: but the two best known logicaloperations, truth functional and intuitionistic ones, are certainly not defined byrules of deduction; on the contrary, one looks for rules that are valid for theseoperations. Also in analysis or set theory, the logical ope rations are used a s princi-

* With respect to validity, not consequence Since, e.g., from A A B one cannot derive A

VX3YB(X, Y>.

by a system of rules that possesses the subformula property.


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330 G . KREISELpal means of (impredicative) definitions in the comprehension axioms. I shallreturn to this point below.As so often, elaborate mathematics goes some way towards replacing a iirmgrasp of an intuitive conception such as 9 .The perhaps subtle distinctions madeabove become much clearer when applied to the case of injinitary languages, wherewe have natural (classical ) examples of cut elimination without completeness;also what I said abo ut length of derivations (which many logicians would regard aspiddling) is magnified enormously: for countable languages we dont even have(countable) Post normal forms even when we do have cut free rules of proof

Further, as Tait has pointed out [61], the study of infinitary propositional lan-guage (negation, infinite disjunction) is very suitable fo r a proof theoretic analysisof arithmetic provided only that one pa ys due attention (i) to the principles of defini-tion em ploye d in describing the (infinite) pro of trees and (ii) to the methods of prooff o r showing that the proo f trees so described are well foun ded. (F or the proof theoreticstudy of subsystems of analysis in place of arithmetic, one has a choice betweenusing infinitary languages with finite strings of quantifiers [3] or a more or lessdirect reduction to the propositional case [61].)The next sections will explain just what is involved in (i) and (ii). It should benoted that pxcisely analogous questions had to be considered before when oneused infinite proof trees but only finite (quantified) formulae a s in Schiitte [50], orinterpretations such as the no-counterexample-interpretation for arithmetic.

[54] 7

#6. Classical first order arithmetic Z (formulated by means of the schema ofinduction). The pro of theoretic results in (a) and (b) below establish formal relationsbetween Z and certain systems which formulate so called ro-induction. More pre-cisely, this induction principle involves a definition Q, i.e., a form ula with two freevariables, of a binary relation on the natural numbers which strikes one as a naturalordering of ordinal E,.The no tion of na tura l ,-ordering will be analyzed in (d) below an d shown to beunique up to isomorphism (in a suitable class of mappings). Among all definitionsof this ordering one picks out a canonical definition (unique up to provableisomorphism) as in [31, p. 154, 3.222].-NB. Th e forma l results are quite inde-pendent of this analysis since the definitions one na turally thinks of (or which havebeen though t of in the literature, e.g., Gentzen [lo], Hilbert-Bernays, Schiitte [50],Tait [59 ] ) all satisfy the criterion for a canonical natural ro-ordering. However,the significance of these results fo r 9, depends on this analysis : according to taste,the reader may read (d) before or after (a)-(c) (or skip it altogether if he finds thediscussion too discursive).(a) Main result. We consider Z, obtained, roughly speaking, from Z by restrict-ing the rules of logic to be cut free, extending the schema of induction to the rule:Sometimes the other extrem: Cfragrnenrs of predicate calculus) also illustrates the samepoint; but again, fragments would generally (I think, wrongly) be considered piddling. Inci-dentally, the advantages of Gentzens cut free rules over methods based on the intuitively muchclearer no-counterexarnple interp:etation (see the remark in Note VII) also show up in theinfinitary case or with fragments that do not possess prenex normal forms; f. Note VIII.


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A S U R V E Y OF P R OOF T HE OR Y 331for each numeral f i , infer (Vx 4 fi)Ax from (V X 4 i)[(Vy Q x)Ay Ax], an dadding the schem a of definition by recursion 4fi (for formal details of this schema,see, e.g., [59]). Then Z and 2 hatie the same set of theorems in the language of 2.Though this simple formulation does not seem to occur in the literature, someversion of this sort follows directly from any of the familiar proof theo retic analysesof Z given in (b ); we shall sketch an argum ent using Schu tte's [50].-NB. Th erules of Z' will be indicated at the end of the sketch. They are 'cu t free' in the senseof p . 9, i.e., they have the subformula property, but are not elegant: specifically,they permit certain cuts, 'men tioned below, and substitution of num erals for freevariables. It might be interesting to see whether these modifications of the usual cutfree rules a re essential. I am indebted to C.D. Parsons and L. H. Tharp for con-structive criticism of an earlier formulation.Note that some extension of the schema of ordinary induction is necessary,e.g ., by [31, p. 163 , 3.331.First step. Given a formal derivation of the formula A in Z, and using theinformal instructions in [50] we get a description of an infinite tree of (finite)formulae and a purely quantifier fr ee derivation in Z' establishing the followingresults (cf. [31, pp . 163-164, 3.331). (i) The tree is a locally correct proof figure,i.e., for each node m, he formula AN t R s related to the formulae a t the imme-diate neighbours ofmaccording to the rules of inference in [50], (ii) th e tree is wellfounded in the (strong) sense that we give an explicit order preserving m apping ofthe tree into a segment of the canonical c,-ordering. This step can be formalizedin primitive recursive arithmetic : he explicit mapping is necessary to avoid quan-tifiers in the definition of well foundedness.Second step. We use a coding of our infinite proof trees, and the proof predi-cate Prov, for them (as in [31, p. 1641). By the fundamental subformula property,we have a formula A(N) with variable N that enumerates the subformulae of A,i.e., A t (( ;>)where ( ) denotes the top no de of the tree, and, fo r each node R,AN + A@) are formally derivable. Let n N) be the (proof) tree below m, nd letR(N) be the formulawhere s,(N) defines the number of the formula a t the node R.Thirdstep. Let < be the par tial orderin:: between nodes already used in (ii)of the first step above. We get an elementary cu tfr ee derivation of the implication(VN' . N)R(N') -+ R(N), and hence, via the mapping ls., a derivation in Z' ofand so of Prov, [n )), sA(( ))] +A(( )). Since we already have a (quantifier free)proof in Z' of Prov, [n >),sA(( ))I, applying cut to this formu la, we get A(( )).This gives a derivation of A itself modulo the step A(( )) A ; the enumerationA(N) can certainly be so chosen tha t cuts need only be applied to form ulae of com-plexity less than A itself.

ProvI [ x ( N ) , SA(N)I A(N)

Prov1[dN), sA(N)] --t A(N)

Note tha t the reduction can be given by free variable co-recursion.eI have not checked whether the equivalence proof can be formalized in 2 tself, or even inprimitive recursive arithmetic. Philosophically, the question is not particularly interesting, aselaborated in No te IIb. However, technically, i t is of interest in connection with Godel's second


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332 G. REISELTh e reduction of Z to Z is well know n; it can evidently be established in primi-tive recursive arithmetic, e.g., Schutte [SO, pp. 202-2091. More important, heanalyzes conditions , namely the axioms A1-A12, pp. 202-203, on Q together withcertain (ordinal) functions on 4 , which ensure that the rule of induction bA canbe formally derived in Z for each n. Thou gh in Schuttes exposition this analysisplayed a purely formal role, we shall be able to use it in (d)(vi) below in a moresignificant way.Corollaries to the main result. (i) Analysis of the cut free rules shows that, if auniversal formula VxA (A quantifier free) is derivable in Z then it is derivable bya-induction for some a < c0. And if Vx3yA(x, y) is derivable in Z there is a term txrepresenting a definition by a-recursion for which A(x, tx) is derivable by a-induc-tion. (For a precise description of free variable a-induction and a-recursion,see, e.g. [59].)(ii) Let Prov be a canonical definition of the proof relation for Z and let S,X bethe canonical definition of the Godel number of A(%).The n (cf. [31, p. 165,3.3322]),adding to Z the reflection principle, for all A :

Vx[3y Prov (y, sAx) -f A Jis equivalent to adding, again for all A

Vx[(Vy 4 x)Ay -+ Ax] -+ VxAx.Note in passing that the reflection principle is equivalent over Z to the rule:infer VxA from Vx3y Prov (y, sAx ) .(b) How does e0 come into the proofs of the main result ? (i.e., of the reduc tion ofZ to Z). Gentzen assigned ordinals to formal derivations, ix., to finite syntacticstructures. I like to see ordered structures to which we simply assign their ordertype. I do not believe that one has any hope of getting a significant ordering offorma l derivations if one only looks a t their syntactic structure. W hat is needed isto look at the intuitive proofs described by formal derivations, and to use proper-ties of these proofs to discover a useful ordering of the formal derivations.Cut elimination (either infinitary propositional calculus or [SO], already used in(a) above). If we sta rt with a formal derivation corresponding t o a d erivation in Z ,and apply cut elimination in a natura l way, we finish up with a cu t free derivationof ordinal < c o . This bound is minimal in the sense tha t for each a < co there is aderivation in Z which this method transforms into a cut free derivation of ordinal

> a . (This is independent of whether we consider the ordinal of the total orderingof the branches of a proof tree ordered from left to right, or the partial orderingby means of t he ancestral of the nodes.)No counterexample interpretation. Here one associates with each arithmetic Aits nci: 3FVfA(F, f) , A quantifier free, f of type NN of type NN +N (F in-incompleteness theorem applied to cut free systems. For the usual systems S (see, e.g. [31,p. 155, 3.2331 and also footnote 16 below) for which, besides the two other derivabilitycorditions of Hilbert-Bernays, closure under cut can be formally proved in S itself, Gadelstheorem simply implies that the consistency of S is not derivable in S. For cut free systems wehave the further possibilities that (i) consistency is derivable, but not closure under cut and (ii)neither is derivable. Clearly, if the equivalence proof under discussion can be formalized in Z,the consistency of the cut free system Z cannot be proved in Z, and closure under cut can ; atpresent I do n ot know which of the three possibilities applies to 2.


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A SURVEY OF PROOF THEORY 333tended to range over constructive operations continuous for the p roduct topology).With each F is associated the ordering of its unsecured sequences. If, for A formallyderived in Z , one associates natu ral definitions of FA:VfA(F,, f), it turns out againtha t the corresponding order types fill up the segment < c0. (For explicit analysis,cf. [60].)eREMARK.Both methods of proof generalize and explain how c0 is connectedwith Z in the followingformal sense: if we add to Z the schema of induction for (aformula < a instead of


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A SURVEY OF PROOF THEORY 335(i) Z and go. ecall the corollaries to the m ain result of (a), which imply that Zcan be reduced to go rovided the principles of proof by induction with respect to

4 , espectively definition by recurs ion with respect to Q, are valid in Pofor eachproper segment of 4 .Recall the precise formal descrip tion of these principles in [59]; note that recur-sion implies induction, and hence, loosely, the existence of a function defined byrecursion implies uniqueness; 1 s . p. 161. Roughly, the definition princ iple is this:if g and h have been introduced in go, e also introducef by the rulef n = c if n = O or i h n 4 n or i n 4 p

= g(n, fhn) otherwise.We have two questions: Are all such definitions, for given p, acceptable in go?

Is some such definition (with primitive recursive g and h) not valid without therestriction n 4 p ? in short, is a-recursion valid for each a < co, and ro-recursionnot valid in go?t is, of course, not assumed that these questions will be settledby methods of go.Comments. (a) By footnote 2 one must expect that different (though com-patible ) assumptions about (or: properties of) gowill be used in these answers.What one would like, of course, is a set of evident properties of go ufficient todetermine completely the set of formulae expressing assertions which are provablein Po and are expressible in some given language, e.g., of primitive recursivearithmetic. (For a very simple paradigm, when provable in gos replaced byintuitively valid, and primitive recursive arithmetic by predicate calculus offirst order see axioms I1 and 111for Val on p. 190 of [35].)

(b) Note that what we have to justify in gos the assertion that the equations fora-recursion are well defined rules. The notion of well ordering or ord inal does notappear explicitly. But the equations are plausible because we think of them asdefinitions by transfinite recursion.1(i) go nd the notion of well ordering; a fallacy. To ustify a-recu rsion we haveto see that, for n 4 j5 and h10 = n, h,(m + 1) = h(hlm), the sequence hlO, hll,h12, - . turns back, it., for some m, h l ( m + 1) 4 hlm. Since h, E Po,4 neednot define I well-ordering (containing no infinite descending sequence), only alo f I interpret the early literature at the time of Fe nn at correctly, the principle of proof by

rrunsfinite induction applied t o quuntificr-freeexpressions (equations), was accepted quite earlyand called method of inf inite descent. Thus, e.p., in the proof of xB + y3 # zB (for positiveintegers x, y, z), we have prima facie a choice between (i) ordfnury induction applied to thrquunrfficdfoxmula F u : VxVy(Vz < u) (x3 + y3 # z3) nd (ii) trunsfinire induction applied to theequation E(x, y, z): x3 + y3 # z3 n the lexicographic ordering of (z, x, y), say. It is true that,since only pairs (x, y) with x < z, y < z need be considered, we get back to an w-orderingcontrary to first impressions. m e ssential difference turns out to be that F(u + 1) is inferredfrom Fu, while E(x, y, z + 1) is inferred from E[f(x, y, z), g(x, y, z), h(x, y, z)] withh(x, y, z + 1) S z, and, roughly speaking, th e functions f, g, h cannot be too simple; for adefinitive discussion, see Shepherdsons [ 5 5 ] . ] It seems clear that at the time of Fermat oneneither thought of applying induction to quantified formulae nor did people reafizc that t&olexicographic ord erin g of ((2, x, y>: x < z, y < z} is an w -ordering. Gentzens result shows tha tthe uscs of induction in Z for proving arithm etic identities can be replaced by transfinite induc-tion applied to equations only (first impressions were right in the sense that induction on theordinary ordering is not enough ).


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336 G. KREISELquasi-well-ordering, i.e., one containing no infinite descending Po-sequence, which,bv principle 11, only requires that 4 do not contain any descending recursivesequence.Fallacy : it should be easier to establish that Q is a quasi-well-ordering tha n tha tit is a well orderingWhat is overlooked here is that the notion of quasi-well-ordering is not evenformulated in Po; nd whatever is to be established will have to be done by the methodrof Bo.-NB. My formulation, p. 295 of [23] is unsatisfactory as it stands, becausethese points were not considered properly.EXAMPLE. n the theory of ordinals we think of eo as the limit, or , equivalently,the sum of w , w, w o a , * and use the results: (a) if each order type wl, w2, wg,is well founded, so is their union, (b) if w1 and W 2 are well founded, so is wlWa,for the usual definition of exponentiation of (not necessarily well founded) ordertypes. This way of reasoning, in particular (b), is patently false if quasi-well-orderings are m ean t; we use (I) and (11). Fo r Parikh [48] has constructed a primi-tive recursive ordering w2 which has no recursive descciiding sequence such thatfor a two element order w l, wlwz has a primitive re am iv e descending sequence.Th is w2 is a quasi-well-ordering, so is, of course, wl, but not wlwa.Note that, if we replace (11) by the assumption th at the P 0-acceptable functionsall belong to a recursively enumerable collection of recursive functions, by T heorem5 of [48], not even multiplication preserves quasi-well-ordering. Even additionneeds care F or instance, the proof of Theorem 4 (a) of [48] is nonconstructive inthe following sense: suppose we know that the sequence h descends in w1 + w,;we do not necessarily know whether it stays in w2 or goes ultimately into wl, andso, given such an h we do not necessarily know how to find i and an hi whichdescends in wl (for i = 1 or i = 2).Digression. The use of the example above is typical of a very general principlethat can be applied to a wide class of informal notions ofp roo f B which do not admitthe general notion of function. Suppose it is evident that all functions which areadmitted in B, belong to nd all functions in F1 re admitted in B (e.g., B =predicative mathematics in the sense of [5], class of hyperarithmetic functions,Sl class of arithmetic functions). Then a function F defined on orderings w,does not preserve quasi-well-orderings with respect to B, f there is a w with charac-teristic function inFluch that w contains no infinite descending sequence inF,but Fw contains a n infinite descending sequence even i n S l . (Such a n F orrespondsto exponentiation to the base 2 in the example above.)(iii) Boan d th e notion of PO-well ordering; a fresh start. Instead of trying toexploit the fact t ha t we need on ly quasi-well-orderings, as in (ii) above, let us d othe oppo site:Formulate a stronger notion of g0-ordinal, which implies quasi-well-orderingtrivially, and then see whether such operatioits as addition or multiplication preserveBo-ordinals. (Specifically, the property of being a Sa,-ordinal applies to orderingsdefined in Po,nd we consider operations on orderings.)Th e general idea, m entioned already, is tha t a Po-o rdinal is built up by combi-natorial operations which La. preserve quasi-well-orderings. To express this ideaone needs not only the ordering, butfunctions defined on it corresponding to these


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338 G. KREISELthe result is again a finite ordering.-NB. Of course, for any given conce ption B tis not necessary that arbitrary F be considered, but only those accepted in B itself;so the closure condition above is particularly strong. One ca nnot expect an equallystrong closure property when finite order is replaced by L?o-well-order; the needfor care in the generalization offini te is familiar from [28] and, particularly, [30],an d will be taken up again in $13.(v) Po:terating the process offinite iteration. Perhaps only the study of specificfinite configurations as in (iv) deserves the namefinitist. Be this as it may, thereasoning in Po ertainly is not confined to such configurations since one usesvariables over such configurations (cf., p. 1); more precisely, Poconcerns theprocess of building up finite configurations, in the first place, in w-order. Once thisprocess is grasped (or accepted) it can itself be iterated, as, e.g., in building up anw + w -t . . , .e., an w w , order. The problem of analyzing Pos to give a theoreti-cal analysis of what is implicit in accepting w iterations. Applied to the case ofordinals, we ask: what orderings can be built up from w by iterating the process ofw-iteration ? These orderings are then, by definition, the Po-ordinals.I believe the general type of analysis required here is quite well illustrated by thediscussion of finite orderings in (iv); but I do not know how to say it correctly.Roughly speaking, it is clear that the sort of detail needed is given in the prooftheoretic study of subsystems of first order arithmetic Z; but the interesting questionis to fornzulute the significance of such detail. The autonomous progression de-scribed in [31,3.42, pp. 171-1721, seems to be directly relevant here, in particular t omaking clear th at PO -ordinalsare quasi-well-orderings.

But leaving aside the exact philosophical, or perhaps, psychological analysis ofthe w-iteration process, we can use the general ideas above to give precise mathe-matical results as follows.(vi) Natural a-orderings for a c e0. We shall extend the functorial analysis offinite orderings (with first element, successor and predecessor) in (iv). The ordinalfunctions used now will be the constants 1 (first element) and w ; addition (and,in some formal results, exponentiation to the base 2), together with their inverses,the latter, as always, being needed to avoid alternating quantifiers. In the firstplace we shall assume the existence of such ordinal functions, i.e., of functionssatisfying the recursion equations given, e.g., in [50, pp. 202-2031; their uniqueness,which depends on the fact that the orderings actually considered have no auto-morphisms, will be established later.An isomorphism result (up to recursive, not necessarily Po-mappings). Supposee i s an ordering of the natural numbers of ordinal a a < e0) together with theordinal functions mentioned above, including exponentiation (we here supposethe ordering to be given together with an enumeration of all its elements: so theleast number operator can be ap?lied and the inverse functions are not needed).Th en there are order preserving mdppings, recursive in these functions and in Q,between +IFi and the corresponding segment aij of the canonical co ordering.The proof is quite straightforward; the given functions allow one to constructCantors normal form t(n) for n + p, recursively in the given functions on Q ;andthe functions on eprovide the element in 4 defined by the fo rm t.As a corollary we get, of course, an isomorphism between any two algebraic


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A SUKVEY OF PROOF THEORY 339structures QP, Q pand the corresponding ordinal functions, by mapping eachon QP. For the given functions this is possible only for a < co.ll

Since not all recursive functions are accepted in Po, somewhat different iso-morphism result is more pertinent. Fo r ci = 28, we consider the algebraic structuresabove and the inverse functions: then we can define mappings w hich are no t onlyrecursive in these structures, but defined by 8-recursion.Finally,a closure propert y of eo. We call a collection 8 of orderings closed for anoperation F on orderings if, for w E 0, lso Fw E 8. et 8 be the collection of order-ings


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340 G. KREISELof logical operations in (i) constructing assertions, and (ii) defining objects;

For the formulation of some results it is convenient to introduce a third sort ofvariable of the type of functions (f, g, h, * .) and the usual axioms relating func-tions a nd (the sets which are) their graphs.This system is equivalent to the version of formal ramified analysis of level one,where induction is applied only to first order formulae (not containing set norfunction quantifiers); in contrast, e.g., to [50] where induction is not so restricted.Our main interest here is to give the improved formulations of the role of Q,,promised at the end of 6c, which need function or set variables. It might be ofinterest t o extend the work of 96a, i.e., to give an elegant cut free formulatio n of thepresent system using onlyfinite formulae a nd finite derivations with a suitable ruleof ei5 induction, fo r p = 1, 2, a - .NB. The interest of a cut free formulation is limited if for an arbitrary term Tbuilt up fro m the functors F, the formula A(T) is considered to be a subformula ofVXA(X) or 3XA(X). Fo r, in tha t case, there is no form ula TAwhich can beprovedto be a t rut h definition for all subformulae of A (unlejs all the quantifiers of A ar enumerical).Well foundediiess and proof by transfinite induction (two senses of : well-founded-ness). Let R be a formula containing the two variables x and y (i.e., R defines abinary relation)

(i) WF(R): VX[3x(x E X) + 3x(x E X A Vy[R(y, x) -+ i y E XI)].In terms of function variables we have, more simply, for arithmetic R(ii) The schema, for any formula A, containing the variable x , but not y :

Vx(Vy[R(y, x) --f Ay] +Ax) +VxAx.Note that TI is the contrapositive of the least-element-principle on p. 327; thesecond order formulation , fro m which the schema is derived, will be discussed in $9.MAINRESULTS. 1. If WF(R) is a theorem, so is TI(R, A) for each forniula Acontaining only numerical quantifiers. (The proof is immediate from the fact thatfor such A: 3XVx(x E X t x).)2. Even for the ordinary definition Ro of the natural ordering of the integers,

there is an A, for which TI(R o, A,) cannot be derived.(One way of seeing this is the consistency proof for arithmetic given in [15,pp. 366-3671; this consists essentially in deriving WF(R) --f WF(R#), and thenapplying ordinary induction t o the IXI-formula WF(W,) where Wo is the orderingof o + W, W,,, = W,# and , generally, R# is the canonical definition of 2R.)The asymm etry between WF(R) and the schema TI(R, A) will pl ay an importantrole in the next section.3. Let R be an essentially Xi formula, i.e. (in prenex form) containing onlyexistential quantifiers, numerical quantifiers not being counted, and suppose thatthe free variables of R are x and y : f WF(R) can be proved, then R defines (in thestanda rd model) a well foun ded relatior, of type < c0 [26]. If, further, R is quantifierfree, then there is a term tx for which (i)

Cf. , top of p. 33 1.

Vf3x i [f(x + l ) , f(x)]. (This is used in [17].)TI(R , A):

R(x, y) + (X) Q t(y) Q A R


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A SURVEY OF PROOF THEORY 341can be derived, where, as above, 4 is the canonical o rdering of c o ; (ii) it is a theoremthat tx is formally computable.

Here we have a negative result which expresses that co is the limit of provablewell orderings without assumptions on the existence of definable ordinal functions onthe ordering considered. (Only the purely syntactic, and, obviously necessary,C:-condition is imposed.)Note that the bound e0 is no t changed if any true C:-sentence is added, inparticular one asserting the existence of a function satisfying a recursion eq ua tio n;for the significance of this, cf., p. 326.Reflection principles and consistency statement. Now that we have functionvariables available (o r set variables satisfying a functionality condition) the advan-tages of a reflection principle over mere consistency can be stated as follows.(NB. We need the reflection principle only for first order formulae with set para-meters.)Let S be a system including arithmetic and let Prov, be natural, in particular,suppose it satisfies the conditions fo r Liibs theorem [42]. Then if

Vn[3y Prov, (y, sAn)+An](sAa canonical definition of the Giidel number of A ii where ii is the nth numeral)can be derived by adding WF(R) to S or an arithmetic R and WF(R) can be derivedin S then(*I i3fVxy[R(x, y)t (fx, fy)]is derivable in S .e., there is no order preserving mapping of R nto R ; in otherwords, if R defines a well-ordering its ordinal is less than the ordinal of R.(The proof is in [31, 3.3421, pp. 166-1671. For, if i *)s added to S, WF(R) --tWF(R); so, taking (*) for A and s, for sA,we havei *)3y Prov, (y, s*) --t (*)I,hence 3y Prov, (y, s*) + *) are both theorems of S. By Ltibs theorem, (*) itself isderivable in S.)Thus in c ontra st t o the consistency statement [cf., $6c(iii)] which can be formallyderived by induction on some ordering of ordinal w , there is a nontrivial lower boundfor the ordinals (of recursive well orderings) which permit a proof of the reflectionprinciple by means of transfinite induction applied to first order predicates. Thisbound is exact, e.g., if S contains the arithmetic comprehension axiom (withparameters).Note in passing that Gbdels original proof of his first incompleteness theoremfor systems S of arithmetic shows underivability of the reflection principle appliedt o the formula expressing I am not derivable. Underivability of the consistencystatement requires significantly stronger conditions on S and on the representationof its proof predicate; cf. [31, p. 154, 3.2211, or llb(i) below.

$8. Elementary analysis 8.This is a two sorted formalism with variables forindividuals and functions, and induction formulated as a schema applied to allformulae of the formalism; the main existential axioms, which are weak, assertthe existence of the successor function, pairing and projection functions and c losureunder primitive recursive operations. For a precise description, see [17, App. 1(p. 350)], where this system is called Z,.


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342 G. KREISELIt is evident that 6 s a conservative extension of fm t-o rde r arithmetic since itsaxioms can be proved in Z to be satisfied if the function variables range over the

(primitive) recursive functions.MAINRESULT cf., $6(a)]. For the ordering 6 f the natural num bers of ordinalc0 considercd in $6,B is equivalent to a (two sorted) system obtained by restrictingthe rules of logic to be cut free and by extending the schema of induction to therule :For each numeral R, derive (Vx


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A S U R V E Y OF PROOF TH EO RY 343orderings < E,~). We shall return to the above in 10. Trivially, for every definableR,well founded ness in sense W F is expressed by a siirgle formula.

By com bining these results with very pretty model theoretic constructions, Fried-man has obtained interesting results on Ci-AC: cf. Note V. The situation remindsone of proofs of recursive undecidability in the style of [ 6 5 ] : here is a basic prooftheoretic result (corresponding to their basic result from recursion theory) and tilerest of the argum ent is model theoretic a nd easy to follow.$9. A formulation of full analysis (in terms of well foundedness). We interrupthere proo f theoretic analysis to give a straightforward formal equivalence, straight-forward in the sense that we have implications (which are valid for a rbitrary exten-sions of the systems considered) an d not only proof theoretic rules of the form: if

A can be derived by means of the given rules then so can B ,(a) For a variable Y , write Y(x, y) for (x, y) E Y.vXw[(3X(X E x) A Vf3X 1 Y[f(X + I), f(X)]) -+3X(X Ex A v)"y E x + lY(X, y)])]

is equivalent to ACoo when the latter is formulated as a single axiom in the firstorder (two-sorted) theory of arithme tic properties in $7.(The proof in [17, p. 3521, applies unchanged although it is formulated there forschemata instead of axioms.)COROLLARIES.he schema WF(R) --f TI(R, A), for all explicitly definablerelations R and properties A , is equivalent t o th e schema of ACoo,because the classof formulae considered in the schemata is closed under arithmetic quantification.(For refinements concerning relations between this schema and the correspondingrule, see No te 111.) Mo re interesting:Th e second order axiom ACooof p. 326 is equivalent toVXY(l\df3x1 Y[f(x + I) , f(x)] A 3xx(x)] + x[x(x) A vy{x(y) --f iY(x, y)}]),

because of course the class of all properties of n atura l numb ers satisfies the closureconditions in the theory of arithmetically (definable) properties,(b) Extend the theory of arithmetic properties by adding a new sort of variableX for properties of functions, extend the use of E, but use only axioms fo r closureunderfirst order operations applied to the ncw sort of variable.F o r a variable f, write f* for the sequence of fu nctions coded by f, i.e., f*(x) =AYf(0, Y>)*

vkv?[(3f(f E A) A Vf3X 1 q[f*(X + I), f*(X)]) --f3f(f E k A vg[g Ek+ q ( f , g)])]is equivalent to DC,,.(The proof is also in [17, p. 3521, and the remarks and corollaries above apply.)Applications of this reformulation will be made in @10and 11 ,Subsystems. If we consider the schemata in the notatio n of elementary analysis

corresponding to (a) and (b), we can choose syntactic restrictions on R and Aindependently; analogously to the familiar asymm etric treatmen t of induction a ndcomprehension axioms. The next section will show that, at the present time, thesubsystems obtained by such restrictions are more appropriate than the conven-tional ones mentioned p. 327.


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344 G. REISELDiscussion. Philosophically, the reformulation above is not significant at all.For the intuitive conception of the structure of analysis, (CA) is evident and the

equivalence above is derived.Technically, there is something to be said for it at least as long as the prooftheoretic analysis involves a reduction to intuitionistic methods of proof. The follow-ing formal results, which will be briefly discussed below, summarize the main factsabout the systems obtained from B by replacing the classical rules by intuitionisticrules a nd adding (i) (AC) and (ii) WF(R) -+ TI(R, A) (for arbitrary R and A in thelanguage of 8). et 6, denote the intu itionistic version of 8,which is called H in [17].(i) (AC) in any of its forms is reducible to arithmetic (e.g. [12] applies withoutmodification).(ii) Th e schema above (for intuitionistic logic) reduces, via continuity axioms, tothe schema restricted to quantifier-freeR (by [21] or [17]): it is satisfied i f f , g,range over free choice sequences.Note that if we also add set variables to di hen (CA) together with the axiomVX3fVx(fx = 0 t+ x E X) s certainly not valid, when thc variables X, Y, . rangeover species and f, g, .. over either constructive fJnctions or over free choicesequences : for (CA) requires the existence of undecidable species.and (CA) is added, theresulting system is satisfied by th e most general notion of species of na tura l num bers(as will be elaborated in my paper at Amsterdam cited in footnote 1). Now astraightforward extension of Godels old translation (see, e.g. [20]) of d u (CA) tothis system provides an intuitionistic consistency proof of classical analysis.Discussion. Quite naively, this easy proof in no way reduces the interest of amore detailed proof theoretic reduction in the next section; just as Godels originalintuitionistic consistency proof for classical arithmetic Z did not make Gentzensreduction superfluous.Fo r the logician a principal problem is to formulate the reasons for this naiveimpression; e.g., in the case of go his was done by Godel [12]: Po nly admitsconstructions on combinatorial objects while the laws of intuitionistic logic areimmediately evident only if ccnstructions on a bstract objects (such as constructionson unctions o r even on proofs) are accepted.The next section gives a reduction to certain definition principles for operations

on sequences of natural numbers, in particular, higher types than this will bereduced. For further discussions including some open problems, see end of $11below, and, particularly, my paper at Amsterdam 1.c.

(iii) If set variables replace function variables in

$10. Well foundedness of elementary relations. We consider now elementaryanalysis 6 extended by the schema WF(R) TI(R, A) for arbitrary formulae Aand elementary R: pecifically canonical definitions R of primitive recursiverelations [for the (obvious) precise formulation of canonical definitions, unique upto provable equivalence, cf. [31, p. 154, 3.22211. For details on the effect of para-meters, see Note V.Properties of the system. (i) Th e schema implies Z:-ACol (for number theo reticrelations R) and Z:-DC,, (for relations R between functions). T his is seen by inspec-tion of the equivalence in $9 above; cf. [16].


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A SURVEY OF PROOF THEORY 345(ii) The converse of (i) fails in the strong sense that the collection of hyper-arithmetic functions is an w-model of X:-AC,, by [29] (and also of Zi-DC,, by

the same method), but not of the schema. In fact there are parts of practicalanalysis that can be developed in the present system, bu t not from Z i- D C ll, e.g., thetheorem of Cantor Bendixson in the theory of sets of points [25].(iii) A simple model satisfying the schema is given in Note V : the propertiesneeded are proved by use of Zi-AC,, (or, equivalently, II:-AC,,, or again A:-CA).But a more delicate proof theoretic argument, via reduction to intuitionisticsystems, shows that the consistency of the schema can actually be proved indu (Z:-CA}. In cont rast:(iv) The schema is not included in any finite extension of elementary analysis.(One first shows WF( 4 ) directly, and then applies p. 342 $8(3).)Thus (iii) shows that the assumptions required for establishing the consistencyof the schem a are relatively weak, an d (iv) indicates why certain consequences ofthe schema seem to assum e strong existential axioms, nam ely: if we insist on deriv-ing the schema from comprehension axioms we do need essentially all instancesIncidentally, this is the kind of situation in which new axiomatizations of aninformal branch of mathematics can be foundationally really efficient (cf., e.g. [29,to p of p. 3281). An d, it may be adde d, this possibility is not nearly a s often exploitedas it could be. This failure of contemporary logic to use the axiomatic method isperhaps related to the situation described on p. 328, and ultimately to the philo-sophical problems presented by the choice of subsystems (p. 323).Proof theoretic analysis of the system. The three principal results are, roughly,these:(i) The system is interpreted, by use of Godels method in footnote 9, in aquantifier free formal system, sayY or functions of finite type containing besideselementary (primitive recursive) schemata, the following so called bar recursionschema. whose arguments a re functionals F oflowest type, and finite sequences c of natural numbers, by :F o r G, H given, define the c onsta nt

+o.H(F,) = G(F, C) if FUcI) < 1~$G,H (F, ) = H[F, c, Xx$,,H(F, c * x)], otherwise,where lc is the len gth of c, c = (c,, . ,cl,. . .) or i c c, [c]is the function definedby [cJ(x) = c, for x < Ic, [c](x) = 0 beyond, and G, H have types for which thesubstitutions ma ke sense.(This schema is a particular case of a general schema formulated explicitly bySpector [58].)(ii) T he next step is to define a m odel ofF y interpreting the functions of highertypes as neighborhood functions of continuous functions in the sense of [24]. Inthis way the axioms ofY re translated into statemen ts in the language of analysis.The main principle needed to prove these statements is the schema WF(R)-+TI(R, A) itself, but using intuitionistic logic only.(iii) This latter system, say Y, has been interpreted ([31, p. 140, 2.6211) in thepurely arithmetic system (only variables for natural numbers) consisting of intui-tionistic first order arithmetic and a monadic predicate constant for the notion of


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346 G. KREISELrecursive ordina l, i.e., Church-Kleenes set 0 of recursive ordinal notations,together with the corresponding principle of proof by induction, namely:

Let D be the quantifier free formula such that the inductive definition of 0takes the form(*I VX[VUD(X,, 0 -t OX]we take as axioms (*) itself and the schema, for all formulae A (the predicate letter0 being included)

VxFuD(x, u, AyAy) Ax ] +Vx(0x --f Ax).(Conversely this system can be interpreted in 9 y using a n explicit definition for0 ; note that in [31, 2.6211, the set K replaces 0.(iv) By analysis of F Howard has shown that Bachmanns ordinal pSnc1(1),which is recursive, is a bound for all provable well orderings, and his studentGerber has verified that it is exact. (For a detailed description, see [1I].) Roughlyspeaking, one sta rts with Bachmanns definitions [ l ] involving the cardinal Ci(= th e first uncountable ordinal, =en), the first -number > R, i.e., en+*, etc. Onethen considers the terms used for these definitions with names of the cardinalsinvolved, and considers the ordering of these terms t defined by the order of magni-tude of the mlues oft. It turns out th at this ordering is recursive. (For a systematicformulation, see footnote 11 and, of course, [6].)Since Bachmanns notationsinvolve normal functions, unique characterizations follow automatically.A consequence of any one of the results (ixiii) is that the consistency of theschema cannot be proved in Fefermans (IR) [ 5 ] , and hence not by predicatiuemethodr.12Discussion. In (i) the system is reduced to currently formulated intuitionisticprinciples. Th e use made of variables of type higher than that of F i s quite elemen-tary. As far as actual understanding is concerned this reduction is significantsimply because functiona ls of lowest type have been stud ied more. But there is alsoa more theoretical reason: Suppose given the elementary notion of function offinite type as described in [12], i.e., constructive arithmetic functions (given byrules), functions whose arguments and values are such functions, etc. We nowpick out from the m, first the constructive arithmetic functions (ES. ,hen those Ffor which, roughly, recursion on their unsecured sequences is valid (EF~),ndafter that we only admit the functions obtained by applying primitive recursiveoperations and pa,a to .Fl 9;: ll that has t o be verified then is th at the opera-tions of type .E2 o obtained a re again in 3Ts.(Of course, the verification should useonly elementary facts abo ut the notion in [12].) Th e appropriate technical means forpicking out F2s by use of constructive ordinals. (The reduction to arithmeticinvolves the further restriction to recursive ordinals.)The reduction to (ii) is prima facie significant in that higher type variables arehere avoided altogether; in particular there is no appeal to the (allegedly specially

11 I proposed the characterization of predicotive proof here employed in (231 and then againin (271. But my specific technical conjecture on the actual limit given in the review of [50,bottom of p. 2461, was false.


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A S U R V E Y O F P R O O F T H E O R Y 347problematic) notion of free choice sequence. 3 Concerning a coherent notion ofordinal which is independent of the more abstract constructions involved in thebasic intuitionistic no tions, cf. $12.The full significance of result (iii), and, in particular, of the ordina l qen ,(1) hasnot been established. Specifically, granted the notion of ordinal, formally strongerprinciples than those of (iii) are justified, in particular definitionby recursion on 0.It should be mentioned tha t the systems here discussed cover the bulk of actualmathematics that can be formulated in full analysis at all (of course, not the theoryof larger cardinals). So, at the present stage of knowledge, the reduction of analysisto familiar intuitionistic principles, even excluding the theory of species, is prob-lematic only for possible, not for actual uses of classical analysis: cf. Gentzensremark on the situation in number theory in 11.4 on pp. 532-533 of [9].811. Partial results on full analysis. Consider first the extension of the pre-ceding paragraph to full analysis: actually, it was the extension that was firsttreated (by Spector in [ 5 8 ] ) from w hich th e special case above w as isolated(a) Well founded relations between objects of finite type. As pointed out in [24],Godels interpretation [121 extends formally to full analysis; furthermore, theequivalence A H 3sVtAl(s, t) (in the no tation of foo tnote 9) is proved by a pparently quite elementary instances of the axiom of choice (this is the principalresult of [24], cf. Note VII).Spector discovered that, for theorems A of full analysis (based on AC,,), thefunctionals s needed can be generated by extending the principle of definition (in10) of recursion on the unsecured sequences of F to allfinite types: specifically,while P m a p s sequences of natural num bers into natural numbers, we now considerFmapping sequences of type T objects into natural numbers. Put differently, weconsider elementary well-founded relations between objects of type T (and notonly between numbers o r functions). Spectors proof was a bit difficult, because heconsidered the interpretation of ACol directly ; he matter becom es routine if oneconsiders the interpretation of WF(R) -+ TI(R, A ) instead and by $9, one evengets DCI1.l4The higher types come in because one now considers R defined by useof function quantifiers.No notion of constructive function of finite type is known for which the extended

schema of bar recursion can be established. Qf course, the schem a can be shown tobe consistent by use of the theory of species, bu t then it is unnecessary (for a con-sistency proof of classical analysis) since, by p. 344, Godels old method is hereavailable. Itsformal similarity to th e restricted schema of 10 s certainly no guaran-tee for its constructive validity: just as formal similarity is not a reliable index ofproof theoretic strength.l 3 More precisely, it is quite correct that, after the elim ination of higher types, all the objectsconsidered can be named or listed; but, contrary to some traditional assumptions, such listingis not essential for constructive reasoning; see especially the notion of constructive function o ffinite type in [12].l4 It is quite usual that interpretations are much easier to verify for one of several formallyequivalent axiomatizations.


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348 G. KREISEI,REMARK.When I originally considered the extension of [I21 to analysis (in [24])I believed that the particular notion of functional of finite type there described

could be proved by intuitionistic methods t o satisfy 3sVtA,(s, t) fo r theorems A ,having shown that this could be done nonconstructively. Put differently, I thoughtthat the existing intuitionistic theo ry of free choice sequences, especially if m e usedthe formally pow erful continu ity axioms, was of essentially the same proof theoreticstrength as full classical analysis In other words, I thoug ht the relation w as analo-gous to tha t between formal intuitionistic arithm etic and first-order classical arith-metic mentioned at the end of $9, or, as pointed out in the discussion of $9, betweenformal classical analysis and the intuitionistic theory of species. In terms of $9(which, of course, was not known then) I had not realized the result (ii) on p. 344.(b) Notions of subformula in higher order fogic. (Recall $7,) It has been ob-served (e.g. [31, p. 167, last paragraph], without proof) that a formal system S ofanalysis containing the schema WF(R) --f TI(R, A), does not allow a cut freeformulation using recursive derivations in the sense of p. 329, even if R is restrictedto be primitive recursive.(The proof is straightforward, as soon a s the sense of p. 329 is analyzed . Speci-fically, we suppose the cut free, po,sibly infinitary syntactic rules to be defined in S,and for each formula A we suppose that the validity or soundness of the rules, whenapplied to subformulae of A , can be formally proved in S; here the partial truthdefinitions of p. 329 are needed. But then o ur schema perm its a proof of the reflec-tion principle for the cu t free derivations and any finite extension. Tak e the exten-sion obtained by adding the sentence that the given system S is formally equivalentto the cut free system, an d apply Godels first incompleteness theorem as stated a tthe end of $7.)Takeuti [63] considers the pure theory of types, which can be described (fortypes I ) as follows: instead of using a two-sorted formalism as in $7, one has alsoa third sort k f variable for properties of properties, and functors for the corre-sponding second-order closure operations, in particular for econd-order projec-tions:

VXVX[X e F,X t Y((X, Y) E X)],(extensions t o all finite types a re clear).

Takeutis notion of subform ula is as in $7; for an arbitrary term T built up fromthe functors considered, A(T) is regarded as a subformula of VXA(X) or 3XA(X).Th e rules of pro of ar e then essentially those of ordinary predicate logic. Evidently,when we apply the system (e.g., to the type structure on the natu ral numbers) wed o not have partial tru th definitions for the subformulae of a formula A unless A is(equivalent to) a formula without higher order quantifiers.(i) Compfeteness of cut free rules, in the sense that any theorem of the fullsystem can also be derived without cut (not, of course, in the sense of completenessfor validity in all principal models), Tait [62], by essential use of earlier work ofSchutte [51], gave a very short proof of completeness using the principles of third-order arithmetic in which the existenr,e of a standard model of analysis can beformally proved. (Recall the related situation in $ 5 . ) (There is even an unpublishedmanuscript of Takahashi: A proof of cut-elimination theorem in a simple type


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A SURVEY OF PROOF THEORY 349theory, and of Prawitz: Hauptsatz for higher order logic, where the theory of allfinite types is con ~id ere d.) ~

As Takeuti observed [63], the consistency of his cut free system of second-order logic together with a certain first-order axiom (asserting the existence of thesuccessor function) can be proved in the system itself. So one of the derivabilityconditions in Godels second incompleteness theorems is violated, in particularclosure under cut of the system cannot bep roved in the system (even if cut were added).So within the general framework of Taits proof, the use of third-order existentialaxioms is necessary.16(ii) Cut elimination. Leaving aside the fac t of closure under cut of Takeutisrules, the problem of cut elimination has had heuristic value, at least for him. Hewas led to longer, but more explicit proofs of cut elimination for a series of sub-systems of analysis, the last of which having a simple description in fam iliar terms,namely Z+CA (or, equivalently, XIi-CA) added to elementary analysis (and thuscontaining the unrestricted schema of induction) [64].The cut elimination isdescribed by means of definition by recursion on certain primitive recursiveorderings which he calls, somewhat colorlessly, ordinal diagrams. (One certainlyhas the impression that they are more natural than the pointless orderings of $6c,but I do n ot know w hat property m akes them specially interesting; if I did I couldpropose a better name for them )1. Taits proof [62] does not establish cu t elimination fo r subsystemsof analysis (even for Takeutis notion of subformula). I have not checked whether,in [64], akeuti gives cut elimination for derivations of arbitrary formulae and Ido not know if full cut elimination holds for the subsystem considered.2. It has been suggested that cut elimination may be connected with explicitrealizations of existential quantifiers in the sense that there are explicitly definedXo, v A&) is derivable if 3XA(X) is derivable.This is false. (Essentially, the same type of counterexample as to the analogue in

Warnings.

,Xk uch that A(Xo)

Taits result was asserted in 1431, but the idea of 1431 is certainly not c orrect. Th e suggestionis that th e formulae, in particular the te rms of the theory of simple types should be interpretedwith quantifiers ranging over th e constructible sets, an d the elimination of cuts sho uld proceedaccording to the order (in the constructible hierarchy) of the sets so defined. It is quite correcttha t in this way o ne will get a proof figure with atom ic formulae T E T as axioms. But there is noreason to suppose that these formulae are theorems of analysis (or even true in the universe ofall sets of finite type, since the reduction only ensures that T E T holds for the sets defmed by Ta n d T in the collection of constructible sets). Taken literally, th e stateme nt of [43] was in anycase suspect: if cut elimination can be proved under the assumption V = L, t can be provedwithout, since the statement of cut elimination is purely arithmetic. The following possibilityhas not yet been excluded. We may find a natural reduction procedure on formal derivationsand discover that it reduces the ordinal associated to the derivation roughly as follows: takethe m aximum of the orders in the ramified analytic hierarchy of the sets defined (in this hierarchy)by the abstraction terms occu rring in the derivation. B ecause of the absoluten ess (invariance) ofA:-definitions, this kin d of thing is particularly promising for the subsystem Ai-C A of classicalanalysis.

l6 This situation is sometimes formulated in the literature by saying: Takeutis conjecture(closure under cut of his system) implies finitistically the consistency of analysis. In these terms.i.e., in terms of consistency proofs, Taits argument would only have proved the consistency ofclassical analysis in third order arithmetic


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350 G. KREISELfirst-order logic: let A(X) express that X is not constructible; we have of course3XVY[A(X) v iA(Y)]; but thcre can be no sequence Xo, ,Xk f the typerequired because there is a model [41], even of set theory, in which every definableset is constructible, but not all.)(c) Conjectures and Problems. Of course, it would be interesting to use thc resultsin (a) and (b) above in a more constru ctive proof theory of classical analysis. At thepresent time the principal difficulty seems to be philosophical; cf. the remarks atthe end of 99 and in Note Ib. Meanwhile, here are some formal mathematicalproblems, suggested by the two remarks at the end of (b) above.1. Le t Prov,, an d Prov,, den ote the proof predicates for classical analysis withand withou t the rule of cut. Let us assign a measure of complexity to formulae ofanalysis; for examples of such measures, see [36]; the complexity of a derivationwill be, by definition, the maximum complexity of the formulae occurring in it.We know that(*I VpVa[Prov,* (P, a) f 3q Prove, (9, allis true, hence there is a recursive function such tha t(**I V P V ~ P ~ O V C AP, ) + ProvcF (+P, 41(since a is detcrmined by p, $may be taken t o depend only onp , and is determineduniquely if we take the minimum); further, we know that (*) cannot be formallyderived in analysis.Question 1. Fo r each numeral 5, let p < i5 mean that p is of complexity


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A SURVEY OF PROOF THEORY 351On the other hand, a formal proof of (ii) would strengthen the corollary toQuestion 2; replacing w-consistency by simple consistericy .2. Let us consider, instead of classical analysis, the theory of species, i.e., thecomprehension axiom together with intuitionistic logic, as on p. 344. (I owe thissuggestion to a conversation with D. Prawitz.)Prawitz suggested that a suitable extension of h is work w ould lead to the result:if 3XA(X) is formally derivable then there is an explicitly defined Xo uch thatA(X,) is derivable; and hence if A v B is a closed theorem, so is either A or B; if3xAx is a closed theorem, there is a numeral ii such that Aii is a theorem. Finally(cf. [31, p. 160, 3.3221) one expects: if Vx(Ax v i A x ) and i i 3 x A x are, not neces-sarily closed, theorem s, so is 3xAx; in particular, exactly the same equa tions can beproved to be recursion equations (for total functions) in classical analysis and inthe theory of species.It might be added that a formalizatim in analysis of the proofs suggested byPrawitz would have the consequence:If Vx3yA(x, y) is a closed theorem of the theory of species there is a numeralsuch that VxA[x, {t}(x)] is also a th eorem ; cf., Churchs thesis.For, any given derivation of Vx3yA(x, y) is of bounded co

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G. Kreisel - Mathematical Logic - A Survey of Proof Theory