The Foundations of Intuitionistic Mathematics (Studies in Logic and the Foundations of Mathematics, 39)

THE FOUNDATIONS OF INTUITIONISTIC MATHEMATICS ESPECIALLY I N RELATION TO RECURSIVE FUNCTIONS

STEPHEN COLE KLEENE Cyrus C. MacDuffee Professor of Mathematics The University of Wisconsin, Madison, Wis.

R I C H A R D E U G E N E VESLEY Assistant Professor of Mathematics Case Institute of Technology, Cleveland, Ohio

1965

N O R T H - H O L L A N D P U B L I S H I N G COMPANY AMSTERDAM

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PREFACE

In his intuitionistic analysis, L. E. J. Brouwer created a theory which diverges from classical mathematics and which in its details has not been widely known or understood. Hence it has seemed to us that this theory especially has presented a challenge to the metamathematical and model-theoretic methods. In Chapter I we formalize it, differently than Heyting did in 1930. This formalization includes a continuation of the formal development of (intuitionistic and classical) elementary number theory, which should interest readers of the senior author’s “Introduction to Metamathematics”, where the development of formalized mathematics in Part I1 ended somewhat abruptly. The interpretation, or model-theoretic treatment, is undertaken in Chapter 11, using the theory of recursive functions from Part I11 of “Introduction to Metamathematics”. This chapter contains two new realizability interpretations (one briefly announced in another form in 1957), with variations. Vesley’s Chapter I11 formalizes Brouwer’s theory of the continuum, and Chapter IV applies both metamathematical and model-theoretic considerations to some moot points in that theory. Nothing is presupposed of the reader except familiarity with some material which can be found in Chapters I-XI1 of “Introduction to Metamathematics” or elsewhere. The research now definitively reported in this monograph was begun essentially in 1950, while the senior author held a John Simon Guggenheim Memorial Fellowship granted for the purpose. The final stages in the research and editing fell in a period (from July 1, 1963) during which his research was supported by the National Science Foundation of the United States under.Grant G P 1621. The-senior-author was in residence at the University of Amsterdam from Januari to June 1950, and is indebted to L. E. J. Brouwer, Arend Heyting, Johan J. de Iongh, and the late David van Dantzig

VI

PREFACE

and Evert W. Beth, for orienting him in intuitionistic mathematics, and for many fruitful conversations then and later. He must take the sole responsibility for the direction in which his research led him and for the interpretations he reached. Richard E. Vesley joined the project as a Ph.D. student. Each author’s contributions are identified by chapters; but each has influenced the other’s chapters (as noted on pp. 88, 150, 179, and in other details). A proof by Joan Rand Moschovakis enters on p. 68. We thank Douglas A. Clarke, John Moschovakis, Joan Rand Moschovakis and Georg Kreisel for many helpful suggestions on the manuscript. Joan Rand Moschovakis also read the printer’s proof. We lament the death, during the printing, of Evert W. Beth, who gave us so much encouragement in this undertaking. June 1964

S. C . KLEENE

CHAPTERI

A FORMAL SYSTEM O F INTUITIONISTIC ANALYSIS by S. C. KLEENE

tj 1. Introduction (to the monograph). The constructive tendency in mathematics has been represented, prior to or apart from intuitionism, in the criticism of “classical” analysis by Kronecker around 1880-1890, and in the work of PoincarC 1902, 1905-6, Bore1 1898, 1922 and Lusin 1927, 1930 (cf. Heyting 1934 or 1955).1 Modern intuitionism, founded by Brouwer, constitutes a vigorous manifestation of the constructive tendency. In his thesis 1907 Brouwer treats of the intuitive genesis of the natural number series and argues the distinctness of mathematics from the language in which mathematics is expressed. In his famous paper 1908 on the untrustworthiness of logical principles, he rejects the traditional belief that the classical logic derived from Aristotle has an a priori validity (cf. I M 5 13). His writings, such as 1912, 1921, contain many examples proposed as 1 Dates in gothic numerals (e.g. 1898) refer to the bibliographymofthis monograph, and in medieval tailciphers (e.g. 1902) to the bibliography of Kleene “Introduction to Metamathematics” 1952b. The latter book we cite simply as “IM’, and furthermore we cite things in IM simply as they were cited within IM when like designations do not exist in the present monograph. Some errata in early printings of IM are given in the bibliography below. Also: (a) A t the top of p. 51 of IM, the seeming circularity that not B is used in explaining not A is to be avoided thus. Sameness and distinctness of two natural numbers (or of two finite sequences of symbols) are basic concepts (cf. p. 51 lines 20-24). For any B of the form m=n where m and n are natural numbers, not B shall mean that m and n are distinct. The explanation of not A in lines 5-8 then serves for any A other than of that form, by taking the B in it to be of that form. Equivalently, since the distinctness of 1 from 0 is given by intuition (so not 1 =O holds), not A means that one possesses a method which, from any proof of A , would procure a proof of 1=0 (cf. lines 8-9). (b) On p. 506, the notation in 10 and 11 overlooks the possibility that t contains x; but the reasoning in that case is just the same. (Similarly on p. 155 of the addendum to Kleene 1962a.)

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confuting classical arguments and results. In 1918-9 he began the construction of an intuitionistic mathematics divergent from the classical; the results are collected in 1924-7, 1927. Subsequently he was joined by Heyting I925,1927,1927a, I929,1941,1953a, de Loor 1925, Belinfante 1929, Freudenthal 1936, van Dantzig 1942, Dijkman 1948, 1952, van Rootselaar 1952, 1960, van Dalen 1963 and others in continuing the development of intuitionistic mathematics ; cf. the bibliographies in Heyting 1955 and 1956. (For “negationless intuitionistic mathematics” Griss 1944, 1946-51, 1955, cf. Heyting 1956 8.2.) Besides the approach to intuitionism through Brouwer’s original papers (including those already cited, and e.g. 1928a, 1929, 1948, 1952, 1954), there are the approaches through Heyting’s expositions (1934, reedited 1955, and 1956) and Beth’s (1955, 1959). Meanwhile there arose in the 1930’s a different manifestation of the constructive tendency, in the form of a general theory of constructive processes, or algorithms, for calculating functions or deciding predicates (i.e. properties and relations). This theory is commonly called the “theory of (general) recursive functions”, after one of the mathematical definitions of a class of number-theoretic functions which has been proposed as coinciding with the intuitively-understood class of the number-theoretic functions which can be calculated effectively by a uniform method. The definition of the general recursive functions was given by Godel in 1934 building on a suggestion of Herbrand (cf. I M 9 55). The identification of the general recursive functions with the effectively calculable functions, which we call Church’s thesis, was first proposed by Church in 1936 (cf. I M p. 300 and 5 62, Kleene 1958 9 2). Whether or not one accepts the thesis, the considerations adduced in support of it bear on the significance of the theory. Simultaneously, the class of the A-definable functions, which had been introduced and studied by Church and Kleene beginning in 1932, was proved by them to be the same as the class of the general recursive functions (Church 1936, Kleene 1936a; cf. Church 1941). Turing independently proposed the like thesis for the computable functions, introduced by him in 1936-7 and independently in less detail by Post 1936; and in 1937 Turing proved them to be the same as the other two classes of functions just mentioned. Subsequently other equivalent formulations have been introduced, as by Post 1943 and by Markov 1951c. On this basis an extensive theory has been developing in a variety of directions, e.g. decision problems, hierarchies, degrees

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of unsolvability, recursive equivalence types, recursive analysis, constructive ordinals, recursive functionals. Among the books are Peter 1951, Tarski-Mostowski-R. M. Robinson 1953, Markov 1954, Davis 1958, Trahtenbrot 1960, Dekker-Myhill 1960, Uspenskii 1960, Hermes 1961, Klaua 1961, Sacks 1963, Rogers 1964. We have counted (by October 15, 1963) 150 contributors to the subject.2 The proceedings of several recent gatherings indicate the scope of the work that is going on (Rosser 1957, Heyting 1959, Mostowski 1961*, Nagel-SuppesTarski 1962, Dekker 1962). The theory of general recursive functions had antecedents in work of Skolem 1923, Hilbert 1926, Ackermann 1928, Godel 1931,Peter 1934 dealing with special classes of recursive functions, which has been continued by 35 investigators.3 2 The 20 cited already, and J. W. Addison, S. I. Adyan, Giinter Asser, Gilbert Baumslag, N. V. Beljakin, Bernays, William W. Boone, R. E. Bradford, Donald Bratton, J. L. Britton, J. R. Biichi, C. S. Ceitin, V. S. Cernjavskii, Andre Chauvin, C. R. Clapham, Douglas A. Clarke, Alan Cobham, J. N. Crossley, V. K. Detlovs, V. H. Dyson, Andrzej Ehrenfeucht, Calvin C. Elgot, Herbert B. Enderton, A. P. Ersov, Walter J. Feeney, Solomon Feferman, Patrick C. Fischer, Frederic B. Fitch, Roland Fra'issB, A. A. Fridman, Richard M. Friedberg, A. Frohlich, D. C. Furguson, R. 0. Gandy, Seymour Ginsburg, Andrzej Grzegorczyk, William P. Hanf, Harrop, Hasenjaeger, Graham Higman, Louis Hodes, P. K. Hooper, Hu Shih-Hua, A. Janiczak, Ju. I. Janov, JaSkowski, A. S. Kahr, KalmAr, L. A. Kaluinin, Carol Karp, Clement F. Kent, Akiko Kino, Kolmogorov,V. S. Koroljuk, Donald L. Kreider, Kreisel, Kuznecov, Daniel Lacombe, Azriel Levy, Samuel Linial, Shih-Chao Liu, Loh Chung-Wan, Paul Lorenzen, David C. Luckham, M. Machover, Maehara, A. I. Mal'cev, Werner Markwald, D. A. Martin, B. H. Mayoh, S. Mazur, John McCarthy, Thomas G. McLaughlin, Medvedev, Marvin L. Minsky, Moh Shaw-Kewi, John N. Moschovakis, S. Mrbwka, A. A. MuEnik, A. A. Mullin, N. M. Nagornyi, Nelson, Anil Nerode, B. H. Neumann, P. S. Novikov, Walter Oberschelp, 8. S. Orlovskii, Rohit J. Parikh, Porte, Marian Boykan Pour-El, Hilary Putnam, Michael 0. Rabin, Wolfgang Rautenberg, H. G. Rice, W. H. Richter, William E. Ritter, Julia Robinson, Alan Rose, Gene F. Rose, Rosser, N. A. Routledge, Ryll-Nardzewski, $anin, F. J. Sansone, Schutte, Scott, Claude E. Shannon, Norman Shapiro, J. C. Shepherdson, J. R. Shoenfield, Skolem, Smullyan, Ernst Specker, Spector, Peter H. Starke, Yoshindo Suzuki, M. A. Taiclin, W. W. Tait, Gaisi Takeuti, Stanley Tennenbaum, Tosiyuke TuguB, J. S. Ullian, Robert L. Vaught, Vladeta VuEoviC, Wang, Jesse B. Wright, C. D. Wyman, C. E. M. Yates, P. R. Young, I. D. Zaslavskii. 3 Ackermann, David R. Anderson, Paul Axt, James H. Bennett, Bereczki, Church, John P. Cleave, Paul Csillag, R. J. Fabian, Feferman, Fischer, Joyce Friedman, Godel, R. L. Goodstein, Grzegorczyk, James R. Guard, Juris Hart-

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But, although intuitionism began twenty-five years before the theory of general recursive functions and so contributed to the climate of research in foundations in which the theory of general recursive functions arose (cf. Church 1936 Footnote 10, 1936a p. 102), the latter theory in its details developed initially quite independently of the intuitionistic mathematics. On the other side, in the next twentyfive years after the theory of general recursive functions had appeared, intuitionism under Brouwer continued on its way without taking explicit notice of the theory of general recursive functions. However it is natural to look for connections between intuitionism and the theory of general recursive functions, which (at least until recently) could be characterized as the two principal manifestations of the constructive tendency in modern mathematics. (This characterization no longer fits as well, with the recent rapid proliferation of notions of constructiveness ; cf. the volume “Constructivity in mathematics” ed. by Heyting 1959.) So Kleene conjectured (in 1939-40) that only number-theoretic functions which are general recursive can be proved to exist in the existing formal system of intuitionistic number theory. This conjecture was confirmed, and other metamathematical results concerning the intuitionistic formal number theory were obtained, in his 1945 with Nelson’s 1947 ,(expositionin I M 3 82; also cf. Sanin 1958, 1958a, Kleene 1960). Kleene (in 1941) and Beth 1947 proposed to interpret the “laws” in Brouwer’s definition of a set or spread (1918-9, 1919, 1924-7, 1954) to be general recursive functions. Proposals for a thoroughgoing use of recursive functions in interpreting intuitionistic analysis were elaborated by Kleene in 1g5oa, which is a prospectus for the present monograph; and Kleene in 1957 gave a progress report. (Ideas from intuitionistic analysis were applied in recursive function theory by Kleene in 1955a pp. 417, 420 or 1955b p. 203.) In that prospectus 1g5oa it was suggested that use of recursive functions might help to make Brouwer’s analysis more accessible. The appearance meanwhile of new expositions of intuitionism, especially Heyting 1956, has diminished the need for such help. In these expositions one can follow the development of intuitionistic mathematics in a quite straightforward manner, and we shall not mans, H u , Kalmhr, Kaluinin, Kent, Kleene, Kreisel, A. H. Lachlan, R. S. Lehman, Moh, Nelson, Peter, Robert W. Ritchie, Julia Robinson, Raphael M. Robinson, Rogers, H. S. Shapiro, R. E. Steams, Tait.

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duplicate them here except to a limited extent. We do aim in the present monograph, among other things, to clarify the differences between intuitionistic and classical mathematics. These differences consist in part in the intuitionist’s merely refraining from affirming classical results, and in part in the intuitionist’s affirming results contradictory to classical ones.

Q 2. Status of the formal system. We shall begin by setting up a formal system, in which, so far as we have pursued it, the existing intuitionistic mathematics exclusive of the theory of species of higher order (Brouwer 1918-9, 1924-7) can be developed. This investigation is continued in Vesley’s “The intuitionistic continuum”, constituting Chapter I11 below. Formalization may at first seem inappropriate. For one of the Brouwerian tenets is that mathematics should consist in intuitive constructions, and intuitive reasoning on the basis of the meaning of the propositions as about those constructions, rather than in formal deductions from formally stated axioms. However this need not forbid the use of formal processes of deduction to save the labor of going back to the meaning at each step, when the axioms and rules of inference have once been justified in terms of the meaning so that we know we always could go back to it. As is familiar from the historical development of elementary algebra e.g., such formal manipulations add much to the rapidity with which mathematical deductions can be performed. The Brouwerian objection is surely only against formal reasoning that could not be performed in terms of the meaning. (Cf. I M p. 62 last paragraph, or Heyting 1953 pp. 59-60.) Brouwer took the position on philosophical grounds that the possibilities of construction cannot be confined within the bounds of any given formal system, long before this position received confirmation in the famous proof by Godel 1931 that formalisms adequate for a certain portion of number theory are incomplete. So it is to be understood from the outset that our formal system for intuitionistic mathematics is not complete, even for the portion of intuitionistic mathematics without species of higher order which is expressible in its symbolism. It, or any intuitionistically correct extension of it, could be extended by the Godel process (which is valid intuitionistically), i.e. by adding a formally undecidable but true formula (cf. I M §§ 42, 60). We shall indeed frequently think of our formal

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system as extensible, any extension to satisfy certain general conditions. There are advantages in describing formally a portion of intuitionistic mathematics, even though that portion be only a fragment. Indeed the intuitionist Heyting did exactly this in 1930, I930a, in response to a challenge by Mannoury to do so. The formation rules and postulates help the outsider by separating out for him a relatively small number of concepts he must grasp, and of propositions and methods of inference to which he must assent, in order that in principle nothing else should remain problematical for him in the body of the intuitionistic mathematics deducible within the system. Furthermore, metamathematical investigations are made possible, as well as investigations, which we may call model-theoretic or semantical (cf. I M pp. 63, 175, 176, 501), employing interpretations not necessarily elementary or finitary. Thus Heyting’s formalization 1930, 193oa $5 5, 6 of the intuitionistic logic led t o a variety of metamathematical and semantical investigations and applications, of interest in diverse connections not all intuitionistic, by Glivenko 1929, Kolmogorov 1932, Godel 1932, 1932-3, 1933, Gentzen 1934-5, zegalkin 1936*, JaSkowski 1936, Stone 1937-8, Tarski 1938, Wajsberg 1938, McKinsey 1939, Garrett Birkhoff 1940, Kleene 1945 with Nelson 1947, Heyting 1946, McKinsey-Tarski 1946, 1948, de Iongh 1948, Ohnishi 1953, Mostowski 1948, Rieger 1949, Henkin 195oa, Curry 1950, Kleene 1948, 1952, 1952a, I M (1952b), Ridder 1950-1, Schiitte 1962, Kuroda 1951, Eukasiewicz 1952, Maehara 1954, Umezawa 1955, 1959, 1959a, Nishimura 1960, Schroter 1956, 1957, Porte 1958, Schmidt 1958, Skolem 1958, Leblanc-Belnap 1962, McCall 1962, Vesley 1963, Rasiowa 1951 , 1954, 1954a, Rasiowa-Sikorski 1953, 1954, 1955, 1959, Sikorski 1959, Pil’Cak 1950, 1952, G. F. Rose 1953, GB1-Rosser-Scott 1958, Medvedev 1962, Kabakov 1963, Harrop 1956, 1960, Beth 1956, 1959a, Dyson-Kreisel 1961, Scott 1957, KreiselPutnam 1957, Kreisel 1958a, 1958b, 1959a, 1959c, 1961, 1962, 1962a, Kleene 1962a. (Glivenko 1928 partially anticipated Heyting 1930. Kolmogorov 1924-5 first gave, for a related system, a result like Gijdel 1932-3, I M p. 495.) Gentzen in 1934-S gave a formalization of the intuitionistic logic in which it differs from the c1ai;sical logic in just one axiom schema, that for “negation elimination” ; cf. I M p. 101 , and Remark 1 p. 120 reading “intuitionistic” for “same”. (Glivenko I929 had already observed

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that adjoining the law of the excluded middle to the intuitionistic propositional calculus produces the classical.) This facilitates comparison of the two systems. The formal system of intuitionistic arithmetic and number theory which is obtained by adjoining Peano’s axioms and recursion equations for suitable number-theoretic functions t o the intuitionistic logic (in Heyting’s or Gentzen’s formalization) has been studied metamathematically or semantically by Godel 1932-3, Kleene 1945 with Nelson 1947 (as indicated in 4 1 above), Nelson 1949, Kleene 1948, IM, Sanin 1953, 1954, 1955, 1958, 1958a, Kleene 1960, Kreisel 1958*, 1959, 1959a, 1959b, 1959d, 1962, 1962d, 1962e, 1962f, Harrop 1956, 1960, Godel 1958, Kleene 1962a, T. T. Robinson 1963. Incidentally, the formal system of intuitionistic arithmetic and number theory does not occur as a subsystem readily separated out from Heyting’s full system of intuitionistic mathematics rg3oa (with 1930). At the time this investigation was begun, and up through our progress report 1957, there were so far as we are aware no other metamathematical or semantical studies in print of the Heyting 1930, 193oa system of intuitionistic mathematics including set theory, or of any other formal system for a like portion of intuitionistic mathematics. Kreisel 1959a, 1959c, 1962, 1962a refer to our 1957. In 1958b he describes another such formal system, and treats some of the same matters as the present monograph. Kleene, after unsatisfactory attempts to use recursive functions in the interpretation of the Heyting 1930a system, came to feel it not well suited for the metamathematical studies proposed. I t employs somewhat bizarre primitives, namely symbols ‘‘d’ and “z” where “opzq” is to be read “from the species q the element p is chosen” and “pzq” to be read “the fi chosen from q”. In any case, such symbolism is an impediment to the comparison of the intuitionistic with a classical system. So in 1950a Kleene proposed the innovation of using simply one-place number-theoretic function variables for Brouwer’s “choice sequences”. Thus the part of intuitionistic mathematics up through intuitionistic set theory excluding higher-order species, which we call intuitionistic analysis, comes to be expressible in the same language as a portion of classical analysis. The differences between intuitionistic and classical analysis are then to be found formally in the different treatment of the function variables under the deductive postulates rather than in formation rules. Kreisel 1958b also uses one-place function

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variables, under a restriction (citing on p. 372 our 1957). Spector 1962 takes into account the present formalism (cited on p. 9), details of which were made available to him in November 1959 and January 1961 ; and Kreisel 1962d, 1962e, 1962f cite Spector 1962. Our postulates will begin with ones which will be common to (or hold in) both the intuitionistic and classical systems, namely those of Postulate Groups A-C in fj 4, of Postulate Group D in 9 5 (to which additions may be made later), and Axiom Schema X 2 6 . 3 in 3 6. This gives what we call the basic system. Strengthening the intuitionistic negation elimination schema 81 I M p. 101 (included in Postulate Group A) to the classical one 8” IM p. 82 for the classical system, and for the intuitionistic system adding a single non-classical postulate X27.1 in 5 7, will then produce the divergence between the two systems. Of course other additions could be made to the classical system, including further formation rules (cf. Hilbert-Bernays 1939 Supplement IV) ; but this would take us further afield than we wish to go from our primary subject, the foundations of intuitionistic mathematics.

Q 3. Formation rules. 3.1. We must now assume familiarity with the concept of a formal system. A plethora of references could be given; but, when we can, we shall confine our references for this and related purposes to IM. We now cite I M $9 1 5 f f . 3.2. The formal symbols of the system shall be the logical symbols . . . then . . .”), & (“and”), V (“or”), 1 (“not”), V (“for all”) and 3 (“there exists”), number variables a, b, c, . . ., x, y, z, . . ., function variables u, p, y, . . ., the predicate symbol = (“equals”), certain function symbols fo, . . . , f, where fo is 0 (“zero”) and f l is ’ (“successor” or “plus one”), Church’s I , commas and parentheses. The number variables are variables over the natural numbers 0, 1, 2, . . .. The function variables are variables for one-place numbertheoretic functions, i.e. one-place functions from the natural numbers to the natural numbers. We suppose potentially available a countable infinity of variables of each sort. In IM a special font of type (lower case script “a”, “b”, “c”, . . .) was used for the number variables themselves as distinguished from letters (lower case Roman “a”, “b”, “c”, . . ., “x”, “y”, “z”) standing

=I (“implies”, or “if

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as names for number variables. We simplify the notation here by using the latter only, as there is little further need (after the basic discussion of formal systems in IM) for exhibiting specimens of the number variables themselves.4 We can likewise understand that the lower case Greek letters in the font “a”, “p”, “y”, . . . (Roman) are names for function variables.5 When we use letters as names for variables, we have to be clear when different letters name variables required to be distinct and when they may be names for the same variable. The same problem was present in IM, though there in some cases we could avoid it by using specimens of the variables themselves. The problem is complicated by the use in I M and here of metamathematical expressions to name formulas etc. that may contain variables not shown or named within those metamathematical expressions. We shall be explicit in these matters at the outset and in our more formal statements. Elsewhere the simplest blanket rule is that variables not required by the notation to be the same (i.e. not named alike, or at corresponding positions in alike named formulas etc., within a connected context) should be distinct whenever it makes a difference for the discussion in progress whether they are the same or distinct. The passages in which we are explicit about the stipulations should provide ample illustration for the application of this blanket rule. The function symbols f t for i > 1 will be specified later. It will be somewhat a matter of convenience which ones we introduce. The formation rules regarding them, and some other features of the system, we state first in general terms to leave the way open to other applications besides to the specific system considered below. Under the interpretation, each function symbol f t (i = 0, . . ., p ) shall express a “primitive recursive” function ft(u1, .. ., uk,, cq, . . ., q,)of a specified number ki of natural numbers and a specified number Zg of oneplace number-theoretic functions (kt, 12 2 0; in particular, ko = 10 = 11 = 0, k1 = 1). We could admit here “general recursive” functions of such variables; but we shall not have occasion to introduce any that are not “primitive recursive”. For the notion of ‘primitive 4 We shall retain the script letters, capital and lower case, in writing particular formulas of the pure propositional and predicate calculi. 5 Italic Greek letters OL,b, y , . . . will be used here as names for functions; in IM Roman Greek letters were so used, and elsewhere, e.g. Kleene 1955, 1955a, 1955b, italic Greek letters.

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(‘general) [‘partial] recursive’ function, we refer to I M Part 111, especially 0 43 (0 55) [3 631. But we are following Kleene 1g5oa 3 2 and 1955 0 1 in allowing one-place function, as well as number, variables; that fi(a,, . . ., a,{, a,, . . ., azt) with li > 0 is #rimitive (general) [partial] recursive then means that fi(a,, . . ., a,,, al,. . ., azc) as a function of a,, . . ., a,< is primitive (general) [partial] recursive uniformly in al,. . . , al,I M 3 47 (3 55) [363]. When we come to specify the function symbols, we shall introduce such special symbols, not always prefixed, as will be convenient; so the “f6” can be considered as names for the symbols to be introduced later, or for i = 0, 1 already introduced .

3.3. We define ‘term’ and ‘functor’ inductively, as follows. The terms are to be the formal expressions in the role of nouns expressing natural numbers, and the functors those expressing one-place numbertheoretic functions. 1. The number variables a, b, c, . . ., x, y, z, . . . are terms. 2. The function variables a, @, y, . . . are functors. 3a. 0 is a term. 3. For each i (i = 0, . . ., p ) , if t,, . . ., t,{ are terms, and ul, . . ., ulr are functors, fi(tl, . . ., tkr,ul, . . ., uzf) is a term. (Clause 3a is the case of Clause 3 for i = 0.) 4a. ’ is a functor. 4. For each i (i = 0, . . ., p ) such that ki = 1 and 12 = 0, f t is a functor. (Clause 4a is the case of Clause 4 for i = 1.) 5. If u is a functor, and t is a term, (u)(t)is a term. (When u is a functor f i by Clause 4, Clause 5 duplicates Clause 3, and (u)(t) will be written in whatever manner may be adopted for fi(t), e.g. for i = 1 as (t)’.) 6. If x is a number variable, and s is a term, ;Ix(s) is a functor. 7. A formal expression is a term or functor only as required by Clauses 1-6. Next we define ‘formula’ inductively as follows. The formulas are to be the formal expressions in the role of sentences, i.e. expressing propositions (but they may contain “free” variables). 8. If s and t are terms, (s)=(t) is a formula. 9. If A and B are formulas, so are (A) I) (B), (A) & (B) and (A) V (B); if A is a formula, so is -(A). 10. If x is a number variable, and A is a formula, Vx(A) and 3x(A) are formztlas. 11. If u is a function variable, and A is a formula, Vu(A) and 3 a ( A ) are formulas. 12. A formal expression is a formula only as required by Clauses 8-1 1. In writing terms, functors and formulas, parentheses may be omitted under the familiar conventions when no confusion can result (134 9 17), or they may be changed to braces or brackets. (They can

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always be omitted in using Clauses 6 and 8.) We adopt the abbreviations “#”, ‘‘2”, ‘‘3”, . . . I M p. 75, “-” p. 113. Instead of including among our formal or primitive symbols just the one predicate symbol =, we could have allowed others, each one Pj expressing a primitive recursive predicate Pi(ul, . . .,a,, a,, . . .,an,). We shall get the same effect by providing, when we want a symbol for such a predicate, that Pi(a,, . . ., a,,, a,, . . ., unJ be an abbreviation for sj=O where sj is a term (containing at most the variables a,, . . ., am,,u l , . . ., un5) which expresses the representing function pi(ul, . . ., um5,al,. . ., an,) of Pj (IM p. 227). For the Pj (as for the f g ) we shall introduce such special symbols, not necessarily prefixed, as will be convenient.

3.4. Free and bound occurrences of variables in terms, functors and formulas are distinguished in the familiar manner (IM 9 18). The operators which bind variables now are the A-pefixes Ax and the quantifiers Vx, 3x, Vu, 3u where x is any number variable and a is any function variable. As in IM, our definitions of ‘term’, ‘functor’ and ‘formula’ do not exclude the application of one of these operators to a scope containing bound occurrences of the same variable; then in the resulting expression it is only the occurrences free in the scope which are bound by that operator. The result of substitzcting a term for (the free occurrences of) a number variable, or a functor for (the free occurrences of) a function variable, or of several such substitutions performed simultaneously for several variables, and freedom at the substitution positions or of the substitution, are defined as before (IM $ 18). We use the same notation as before. Thus if we are to substitute for x in a term, functor or formula (or sometimes simply for emphasis), we may introduce a composite notation “E(x)” for it, after which for any t “E(t)” will denote the result of substituting t for (the free occurrences of) x in E(x). There will be no conflict between “r(x)” as a composite notation for a term and “r(t)” as expressing the result of a substitution into r(x) on the one hand, and “r(x)” and “r(t)” as obtained by omitting the first pairs of parentheses in writing terms (r)(x) and (r)(t) introduced under Clause 5 in 3.3 on the other. For in the first case the “r” separately from the composite notations “r(x)” and “r(t)” will have no status, while in the second the “r” by itself will already denote a functor (it will be one of “u”, “p”, “y”, . . ., or an fi, or a letter either

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explicitly introduced, or indicated by the context, as denoting a functor). Similarly, conflict will not arise between composite notations with several argument places and terms formed under Clause 3 (the f i identify the latter). “Pf(tl,. . ., t,,, ul, . . ., un,)” (cf. end 3.3) will denote the result of a free substitution into sj=O after any necessary changes of bound variables in s j (cf. IM end 9 33). A term, functor or formula is closed if it contains no variable free (and following Tarski, a closed formula is a sentence), otherwise open. A formula is prime if it contains no logical symbol. LEMMA 3.1. If a1, . . ., ak are number variables, ct1, . . ., a1 are function variables, tl, . . ., t k are terms, u1, . . ., ul are functors, and E(a1, . . . , ak, 011, . . . , q)i s a term (functor) [formula],then E(t1, . . .,tk, u1, . . . , U Z ) i s a term (functor) [formula]. PROOFis by induction corresponding to the inductive definitions of ‘term’, ‘functor’ and ‘formula’.

3.5. A term u(t) introduced by Clause 5 of the definition of ‘term’ and ‘functor’ expresses the value of the function expressed by u for the number expressed by t as argument. A functor Ixs introduced by Clause 6 expresses the one-place number-theoretic function whose value for a given argument is expressed by s when x expresses that argument, or briefly it expresses s as a function of x (Church 1932, I M p. 34). LEMMA 3.2. If the functions y(a1, . . . , ak, 011, . . . , 011, @) and X(a1, . . ., a k , al, . . ., 011, x ) are primitive (general) [partial] recursive, so i s the function y(a1, . . ., ak, 011, . . ., 011, Ax X(a1, . . .,a&,011, . . .,011, x)). (Kleene 1955 1.3 [I956 p. 2791.) PROOF.By I M Lemma I p. 236 with the obvious transitivity of the relation ‘primitive recursive in’ (Lemma VI p. 344 with Theorem I1 p. 275) [Lemma VI with Theorem XVII (a) p. 3291. LEMMA 3.3. Let s be a term (u be a functor) [ P be a prime formzlla] containing free no variables other than the number variables al, . . ., ak and the function variables ax, . . ., al. T h e n under the intended interpretation s (u(x) where x i s another number variable) [PI expresses, as the ambiguous value I M p . 33, a primitive recursive function of al, . . ., ak, a l , . . ., 012 (function of a l , . . ., a&,q, . . ., 011, x ) [predicate of a1, * * * , ak, O 1 l t * * f , 4. PROOF is by induction corresponding to the inductive definitions

§4

POSTULATE GROUPS A - C

13

of ‘term’, ‘functor’ and ‘formula’, using Lemma 3.2 for Clause 3. (A functor u not of the form Ixs is equivalent under the interpretation to the functor of this form Ixu(x).) REMARK 3.4. Instead of a prime formula P in Lemma 3.3, we can allow one merely containing no quantifiers, or none except bounded number quantifiers. (Cf. I M #D, #E p. 228.)

Q 4. Postulates for predicate calculus, for number theory, and concerning functions (Postulate Groups A-C). 4.1. In giving the transformation rules or postulates (IM 5 19) for the basic formal system or the intuitionistic formal system, we begin with postulates for the intuitionistic predicate calculus; but instead of one sort of variables we employ our two sorts. GROUPA. Postulates for the two-sorted intuitionistic predicate calculus. GROUPA 1. Postulates for the intuitionistic propositional calculus. For these postulates, A, B, C are any formulas; and the postulates are la, lb, 2, 3, 4a, 4b, 5a, 5b, 6,7, 8I as in IM top p. 82 except: 81.

i A 3 (A 3 B)

(as in I M p. 101).

GROUPA2.. (Additional) Postulates for the two-sorted intuitionistic predicate calculus. The first four postulates 9N, 10N, 1 lN, 12N appear exactly as 9, 10, 11, 12 in I M middle p. 82 with the x in the stipulations bottom p. 81 now a number variable. For the second four as follows, a is any function variable, A(a) any formula, C any formula not containing a free, and u any functor free for a in A(a). 9F.

C 2 A(a)

C 3 VaA(a). 11F. A(u) 2 3aA(a).

10F. VaA(a) 3 A(u). A(a)3C 12F. 3aA(a) 2 C.

The classical formal system arises from the basic system by adopting 8”.

i i A 3A

(as in I M p. 82)

as an additional postulate (or equivalently, since 81 then becomes a derived rule IM p. 101, as an alternative to 81).

14

FORMAL INTUITIONISTIC ANALYSIS

CH. I

4.2. Next we take over the additional postulates for the usual system of intuitionistic number theory. We specify that the next two f Z , f 3 of our function symbols be * (with k2 = k3 = 2, 22 = 23 = 0).

+,

GROUPB. (Additional) Postulates for intuitionistic number theory. The postulates are 13-21 as in I M bottom p. 82, with the x now a number variable, and the a, b, c particular number variables. 4.3. Anticipating that in Group D in !j 5 we shall postulate the recursion equations for the exponential function (a6 or a exp b), we now let f 4 express this function (with kq = 2, 14 = 0), and use it in Axiom Schema x2.1 of Group C. (Beyond Group B our postulates do not correspond to ones in IM, and we change to another system of numbering, which will distinguish references within this monograph from ones to IM.)

GROUPC . Postulates concerning functions. For Axiom Schema XO. 1, x is any number variable, r(x) is any term, and t is any term free for x in r(x). For Axiom XI. 1, a and b are particular distinct number variables, and a is a particular function variable. For Axiom Schema x2.1, x and y are any distinct number variables, M is any function variable, and A(x, a) is any formula in which x is free for M.

xO.1 . {Axr(x)}(t)=r(t). XI.]. a = b z) a(a)=a(b). x2.1. Vx3aA(x, a) 3 3aVxA(x, hya(2x.3Y)). 4.4. The intuitionistic and classical systems of predicate calculus are treated in I M (Chapters IV-VII, XIV, XV), mostly simultaneously, those results which are established only for the classical system being marked with (cf. p. 101). This material can be taken over for the purpose of proving formulas now containing no function variables or no number variables. When both sorts of variables appear, we can still use the methods developed there, only exercising care not to violate the definition of formula; the restrictions appear in Postulates ION, 1 IN where t must be a term, and in Postulates ]OF, 1 IF where ‘lo”

§4

15

POSTULATE GROUPS A<

u must be a functor. For example, *79 p. 162 gives now I- VxVyA(x, y) 2 VxA(x, x), t- VaVpA(a, p) 3 VaA(a, a), but not I- VxVpA(x, p) 2 VxA(x, x) nor I- VaVyA(a, y) 3 VaA(a, a). The consequences of the postulates of Group B, taken with the predicate calculus with number variables, are likewise developed in I M (Chapter VIII and elsewhere). We assume enough familiarity not only with the results of IM, but also with the methods underlying them, to permit us here to adapt them to the development of the present formal system without detailed explanation in cases when the adaptation is straightforward. We must stay attentive to the precautions in the handling of variables (IM beginning 3 32 p. 146). REMARK 4.1. Since each prime formula of the present system is an equation s=t between terms, it is immediate by substitution in I M *158 p. 192 that I- P V 1 P for each prime formula. Using also *150, *151 p. 191 and Remark 1 (b) p. 134, likewise t E V YE for any formula E built from prime formulas (or formulas for each one P of which t P V 1P) using only the propositional connectives (2, &, V, 1)and bounded number quantifiers. By Remark 1 (a) p. 134 with Theorem 9 p. 128, all provable formulas of the classical propositional calculus (e.g. the equivalences *56-*61 p. 119) are provable in the intuitionistic system when t- P V 1P for each of their distinct components P prime for the propositional calculus p. 112. 4.5. Our formation rules provide = as a primitive symbol only between terms. Now, when u and v are functors, “u=v” shall be an abbreviation for Vx(u(x)=v(x)) where x is any variable not occurring free in u or v. In the following, x is any number variable, r(x) and s(x) are any terms, b is any number variable free for x in r(x) and not occurring free in r(x) (unless b is x), and a is a function variable. *0.2.

*0.4.

r(x)=s(x) I-X Axr(x)=Axs(x). t IX~(X)=E.

“0.3.

t Axr(x)=Abr(b).

twice and * 101, * 102 p. 183 (cf. end 3 26 p. 118),r(x)=s(x) t {Axr(x)}(x)= r(x) = s(x) = {Axs(x)}(x) P Vx[{Axr(x)}(x) = {Axs(x)}(x)],i.e. Axr(x)=Axs(x) (by V-introd. I M § 23). *0.3. Similarly I- {Axr(x)}(x)= r(x) = {Abr(b)}(x)(cf. I M Example 9 p. 80), whence by V-introd. t Axr(x)=Abr(b).

PROOFS.*0.2. Using Ax. Sch.

XO. 1

16

FORMAL INTUITIONISTIC ANALYSIS

CH. I

The reflexive, symmetric and transitive properties of equality between functors are immediate by the predicate calculus from the same properties of equality between terms ("100-*lo2 p. 183). The replacement property of equality (IM Theorem 24 and Corollaries pp. 184-185) requires additional care because of the presence of the A-operator. 4.2. (Replacement theorem.) Let ER be a term or functor LEMMA [formula] containing a specified occurrence of a term {functor} R not as the variable of a &prefix [A-prefix or quantifier], let Es be the result of replacing this occurrence by a term {functor} S , and let XI, . . ., xn [XI, . . ., x,, u1, . . ., urn] be the free variables of R or S which belong to a A-prefix [A-prefix or quantifier] having the specified occurrence of R within its scope. T h e n

provided that each of the equality axioms (IM p. 403) for each function symbol f g occurring in ER with the specified occurrence of R in its scope i s provable. (The proviso will be met for the way we select the axioms of Group D, by Lemma 5.1.) PROOF. The part for ER a formula will follow as before, after we prove the part for ER a term or functor. We do this (taking both ER and the specified occurrence of R to be variable; cf. I M p. 90) by course-of-values induction on the depth of (the specified occurrence of) R in ER, defined to be 0 if ER is (CASE 1) R itself, and to be r f l if ER is as follows: (CASE2) UR(t) where R is at depth r in UR,(CASE3) fi(t,, . . ., tkt,ul, . . ., ulr) where R is in one of t,, . . ., tkr,ul, . . ., ult at depth Y, (CASE4) U(tg)where R is at depth Y in tR, (CASE 5){Axs(x)}(tR) where r is the maximum of the depths of R in t R and of each of the free occurrences of x in s(x), (CASE 6) AXSR where R is at depth r in sR. Case 1 is trivial. In Case 2, by the hypothesis of the induction, UR=US, which unabbreviated is VX(UR(X)=US(X)) with x not free in UR or US; thence by V-elim., UR(t)=US(t), i.e. ER=Es. The equality axioms for fo, . . ., f, (cf. the proviso) with the hyp. ind. (and substitution I M "66 p. 147, and 3-elim.) take care of Case 3;Axiom Xl.1 similarly of Case 4; and "0.2 of Case 6. In Case 5, say s(x) contains n free occurrences of x, and let a, b be distinct number variables not occurring in s(x). By n successive applications of the hyp. ind., a=b t s(a)=s(b); thence using Axiom Schema XO.l twice, a = b I{Axs(x)}(a)={Axs(x)}(b); and thence by 2-introd., substitution and

§4

17

POSTULATE GROUPS A-C

D-elim., tR=tS k {~xs(X)}(tR)={~xS(X))(tS). Also, by hyp. ind. R=S F X I ...X" tR=tS. The definition of congruence I M p. 153 extends obviously to the present system in which variables may be bound by A-prefixes as well as by quantifiers. Since by Lemma 4.2 we (will after we obtain Lemma 5.1) have the replacement property of equality as well as of equivalence (also the symmetric, reflexive and transitive properties), and *0.3 as well as "73, "74, congruent terms or functors are equal and congruent formulas are equivalent. (This adapts IM Lemma 15b.) If x is any number variable, A(x) is any formula, and a is any function variable free for x in A(x) and not occurring free in A(x): "0.5.

t VXA(X)

-

VaA(a(0)).

*0.6.

-

t ~ x A ( x ) 3aA(a(O)).

PROOFS. *0.5. Assume (preparatory to 3-introd.) (a) VaA(a(0)). Thence by V-elim., A((Ayx)(O))(y a variable distinct from x) ; by XO. 1 and replacement (Lemma 4.2), A(x) ; and by V-introd., VxA(x). (Since VaA(a(0)) does not contain x free, x has not been varied.) By the 2-introd. (discharging the assumption (a)),VaA(a(0))3 VxA(x). Using simply the predicate calculus, VxA(x) 3 VaA(a(0)). By &introd. (*16), VxA(x) VaA(a(0)). N

4.6. For all our essential purposes, we could have postulated in place of x2.1 the following consequence of it "2.2 (cf. 7.15 below). If x and y are any distinct number variables, A(x, y) is any formula in which x is free for y, and a is any function variable free for y in A(x, y) and not occurring free in A(x, y):

"2.2.

t Vx3yA(x, y) 2 3aVxA(x, ~ ( x ) ) .

PROOF.Assume (preparatory to 3-introd.) Vx3yA(x, y). Thence by "0.6 and replacement Vx3aA(x, a(O)),whence by X2.1 3aVxA(x, (Aya(2X.3Y))(0)). Assume (preparatory to 3-elim.) VxA(x, (Aya(2xe3Y))(0)). By XO. 1 and replacement VxA(x, {Ax(Aya(2X.3Y))(O)}(x)), whence by 3-introd. 3aVxA(x, a ( x ) ) (which does not contain CI free). Completing the 3-elim. and 2-introd., Vx3yA(x, y) 3 3crVxA(x, a ( x ) ) . Simply the predicate calculus gives the converses of X2.1 and "2.2; so they can be strengthened to: *2.la. "2.2a.

k Vx3aA(x, a) I- Vx3yA(x, y)

--

3aVxA(x, Aya(2x-3y)). 3aVxA(x, a(x)).

18

FORMAL INTUITIONISTIC ANALYSIS

CH. I

4.7. We say a term, functor or formula E is convertible into one F, and write “E conv F”, if E can be transformed into F by a series of zero or more replacements (I) of a part Axr(x) by Abr(b) under the conditions of “0.3, or (11) of a part {Axr(x)}(t)by r(t) under the conditions of XO. 1, or (111)inversely. The relation ‘E conv F’ is reflexive, symmetric and transitive. Since we (will after proving Lemma 5.1) have the replacement theorem, etc., if E conv F, then t E=F if E, F are terms or functors (t E F if E, F are formulas). A term, functor or formula is (lambda) normal if it contains no part of the form {Axs}(t)where x is a variable, s is a term, and t is a term (not necessarily free for x in s). If E conv F, and F is normal, F is a normal form of E. LEMMA4.3. Each term, functor or formula E has a normal form. PROOF.It clearly suffices to prove this for any term or functor E, by (course-of-values) induction on its rank defined thus: 0, if E consists of a single symbol; the maximum of the ranks of t,, . . ., tk,, u,, . . ., u ~ , ,if E is fi(t,, . . ., t,,, u,, . . ., q , ) ; the rank of t, if E is u(t), unless u is Ixs, in which case it is I+max(s, t) where s, t are the ranks of s, t ; the rank of s, if E is ilxs. E is normal exactly if its rank is 0. If E is not normal, its rank r > 0 is the maximum of the ranks of its parts {Axs}(t), and is possessed by n > 0 such parts, which are outermost, i.e. not contained in the s or t of another such part. Within the induction on r we use one on n. Select a part {Axlsl}(tl) of rank r. By hyp. ind. on I , s1 has a normal form q l ; pick a term rl(x1) congruent to q1 in which tl is free for XI, and replace the part {lxlsl}(tl) of E successively by {Axlql}(tl), {Axlrl(xl)}(tl) and rl(t1) to obtain El. Since rl(x1) is normal, the parts {ilxs}(t) in rl(t1) will be exactly those in the occurrences of t l that replace the free occurrences of XI in rl(x1) (no part u(t) of rI(x1) with u a one-symbol functor becomes a part {hxs}(t) by the substitution of tl for XI, since x1 is a number variable), so their ranks will be < Y. Outside the part {Axlsl}(tl) replaced by rl(tl), nothing will have been changed (no part u(t) of the whole with u a one-symbol functor becomes a part {Axs}(t) by the replacement). So the number n of the parts {Axs}(t)of the maximum rank r will have been decreased by 1, unless n was 1, in which case the maximum rank r of such parts will have been decreased. So by hyp. ind. on n or on I , El has a normal form; and hence E does. REMARK 4.4. In the general theory of &conversion (Church 1932, N

§5

POSTULATE GROUP D

19

Kleene 1934;exposition in Church 1941,Curry-Feys 1958), where no type distinction is enforced between terms and functors, an expression E may have no normal form; e.g. {~xx(x)}(~xx(x)) does not. REMARK 4.5. By the Church-Rosser theorem 1936 Theorem 1 Corollary 2 p. 479 for their third kind of conversion p. 482, any two normal forms of a given expression are congruent. REMARK 4.6. An application of (11), with zero or more preliminary applications of (I), we call a reduction. The proof of Lemma 4.3 shows that a certain sequence of reductions starting from E will lead to a normal expression. By Kleene 1962 Theorem 1 (b) (adapting ChurchRosser 1936 Theorem 2 p. 479), there is a number m such that any sequence of reductions starting from E will lead to a normal expression in at most rn reductions.

Q 5. Postulates for certain primitive recursive functions, and consequences (Postulate Group D). 5.1. In this section we shall specify more of the list fo, . . ., f, of symbols expressing primitive recursive functions, and introduce as axioms the recursion equations for, or equations defining explicitly, the functions expressed. (The symbols fo-f4 were already specified in 3.2, 4.2, 4.3, and the recursion equations for fz, fa are already included in Postulate Group B.) The further axioms thus introduced will constitute Postulate Group D, but we leave it open whether the list fo, . . ., f, and Postulate Group D are concluded in this section or are to be augmented as may be convenient later. Properties and applications of suitable primitive recursive functions beyond 0, ’, +, * are an essential part of our subject matter in developing intuitionistic mathematics in the formal system. It is simplest now to add the symbols as primitive symbols, and the recursion equations as axioms, which are entirely acceptable from the intuitionistic standpoint (as well as classically). We could avoid these additions and still in effect develop the same theory in the system (adapting I M 5 74, especially pp. 415-416), but that can better be left till later. We begin with some generalities, independent of the particular list of symbols and axioms to be added. The recursion equations need not be in the standard form of I M (Va) or (Vb) p. 219, but may have steps of explicit definition lumped with them (as in IM bottom p. 221). Now that we have function variables, the steps of explicit definition

20

FORMAL INTUITIONISTIC ANALYSIS

CH. I

may include substitutions. of functions formed using the I-operator (cf. Lemma 3.2). Specifically, the axiom(s) for a function symbol f i for i 2 4 will be (and for i = 2, 3 already are) of one of the following two forms, which we illustrate for the case of three variables y, a, a : (a) (b)

a, .)=p(y, a, a ) , fz(0, a, a ) =q(a, a ) , f&’, a, a)=r(y, fi(y, a, a ) , a, a ) ,

f&,

{

where p(y, a, a ) , q(a, a ) and r(y, z, a, a ) are terms containing only the distinct variables shown and only function symbols from among fo, . . ., fg-1, and y, a, a are free for z in r(y, z, a, a ) . 5.2. LEMMA 5.1. T h e equality axioms I M p . 403 are provable for each of the function symbols fo, . . ., f, introduced successively with axioms as described (Postulate Group B , and 5.1). PROOF, by (informal) induction on i. For i = 0 there are no equality axioms, and for i = 1 there is one, which is postulated as Axiom 17 of Group B . For i > 1 , assume the equality axioms for fo, . . ., fz-1 provable. Those for q(a, a ) , i.e. (with b, fifree for a, a, resp.)

a=b

I)

q(a, a)=q(b, a ) ,

a=P

q(a, .)=q(a,

=l

P),

and for r(y, z, a, a ) , or for p(y, a, a ) , are provable by applications of Lemma 4.2 and 3-introd., since by the hyp. ind. on i the equality axioms for the function symbols in them are provable. The equality axioms for f g introduced by (a) follow immediately. Those for f g introduced by (b) follow by the method used for a+b (our f 2 ) in I M *104, “105 pp. 183-184, namely: The conclusions of the formulas a = b 3 fi(y, a, a)=ft(y,b, a ) ,

a=P 2

ft(y, a, a)=fz(y, a, P)

expressing the replaceability in the position of either parameter are deducible from the antecedents by (formal) induction on the recursion variable y. To prove the replaceability in the position of the recursion variable, write “A(y, x)” for the formula y=x

I)

fg(y,a, a)=ft(x, a, a)

to be proved. We prove VxA(y, x) by induction on y. BASIS:to prove VxA(0, x). We prove A(0, x) by ind. cases on x (IM bottom p. 186), and VxA(0, x) follows by V-introd. IND.STEP.: assuming VxA(y, x),

§5

POSTULATE GROUP D

21

to deduce VxA(y’, x). We get A(y, x) by V-elim. from the hyp. ind. Thence we deduce A(y’, x) by ind. cases on x, and VxA(y‘, x) follows by V-introd. 5.3. Next we generalize I M *(175)-*(177) p. 201 (cf. pp. 195, 199, 200), and extend the theory to function variables. As in I M p. 298 but with the function variables al, . ., ul as the function letters gl, . . ., gl and with uniformity, we say a formula A(a1, . . ., ak, a1, . . . , at, w) containing free only the (distinct) variables shown numeralwise represents a function s(a1, . . ., ak, al, . . ., al), if, for each al, . . ., ak, 011, . . at,

.

.,

(v)

if s(a1, . . ., a k , 011, . . ., 01z)=w, then El;:::;: 1A(a1, . . ., a k , a1,

. . ., uz, W ) ,

(vi) E;;::::: I- 3!wA(al, . . ., a k . a1, . . ., at, w), with a1, . . ., a2 held constant in each deduction. LEMMA 5.2. Consider any term s(a1, . . ., ah, al, . . ., U Z ) (briefly s) containing free only the (distinct) variables shown, and let s(a1, . . ., ak, orl, . . ., al) be the function expressed by it (Lemma 3.3). The formula s(a1, . . ., ak, 011, . . ., al)=w (where w is another number variable) numeralwise represents s(a1, . . ., ak, al, . . ., 011). PROOF. I. (vi) is immediate by I M *I71 p. 199. 11. If (v) holds for each of the function symbols f i i n s (i.e., if, for each a,, . . ., ak,, a,, . . ., al,,whenever fi(al, . . ., ak,,al,. . ., az,)=w then E;::::;: I- f,(a,, . . ., akt,ul, . . ., uZf)=w with al, . . ., al, held constant), then (v) holds for s itself. By Lemma 4.3, we can without loss of generality take s to be normal. Now we use induction on the number of (occurrences of) function symbols and function variables in s. If s is a single symbol, (v) is trivial. Next take the case s is fi(tl, . . ., tkf,ul, . . ., uz,). Consider fixed values a l , . . ., @k, al, . . .,at of al, . . ., ak, ul, . . ., ul, and say that for these values t,, . . ., tkt,ul, . . ., uZr,s express t,, . . ., tkf,bl, . . ., bit, w.By hyp. ind.,

( j = 1, . . ., K g ) EZ::::;: I- t;=ti with al, . . ., al held constant, where * indicates the substitution of al, . . ., a k for al, . . ., ak. By the hypothesis that (v) holds for the f i in s, Ei::::;: I- fi(tl, . . ., t,,, P1, . . ., Pz,)=w with P,, . . ., Pz, held constant, whence by &-elims. and 3-introd., (1)

(2)

I- F(P,,

-

* *>

PZ,)

3 fi(t1, *

* *

* tk{r

P1,

*

- .)Pz*)=W,

22

FORMAL INTUITIONISTIC ANALYSIS

CH. I

.,

where F(p,, . . pl,) is a conjunction of the finitely many equations of EI::::;: used in a given such deduction. But F(p,, . . .,pz,) is a 6 conjunction of equations of the form pm(a) =b where /9m(a)= b (1 m < Zi). If um is lavm(a), E:::::;: t V ; ( U ) = ~ t (U;)(a)=b with cc1, . . ., al held constant, by hyp. ind. and XO. 1. If um is a one-symbol functor, similarly E;:::::: t (uz)(u)=b (uz being urn simply), by hyp. ind. So (3)

E;::::;: t F(uT, . .., u:)

with a1, (4)

. . .,cq held

t F(uT,

. .., u:)

constant. By substitution in (2), 2

* . ..,u *~ , ) = w ,

fi(t,, . .., t,,, u l ,

whence by replacements using (l),

with cc1, . . . , cq held constant. Now (v) comes from (3) and (5) by 2-elim. The remaining case that s is a(t) for u one of a1, . . . , uz is similar but simpler. 111. (v) holds for each of fo, . . ., f, as the s. By induction on i, using I1 and substitution into the axinms (a) or (b) 5.1 introducing ft, and in the case of (b) induction on y. 5.4. Similarly (using IM p, 298 with function variables and uniformity), the notion of when a formula numeralwise expresses a predicate (IM p. 195) extends to the present situation. We elected (end 3.3) to use symbols of abbreviation for primitive recursive predicates other than =. Using Lemma 5.2 and Ax. 15 I M p. 82, each predicate Pi(al, . . ., urn,, al,. . ., an,) for which we introduce a symbol Pj of abbreviation will be numeralwise expressed by Pj(a,, . . ., amJ,al,. . . , anJ). 5.5. There are in the literature a number of formal treatments of number-theoretic functions beyond 0, ’, -,particularly in Skolem 1923, Hilbert-Bernays 1934 and Nelson 1947. We give another, in order that the present monograph combined with I M be (nearly) self-contained, to confirm that the proofs can be given in an intuitionistic system (as in Nelson 1947),and to keep close agreement between the formal notations and certain informal ones.

+,

§5

POSTULATE GROUP D

23

The informal notations are those which we have been using in the theory of recursive functions. In particular, we have been following Godel I931 in correlating single natural numbers to finite sequences of natural numbers via the unique factorization of positive integers (the “fundamental theorem of arithmetic”). It is possible, and advantageous at least for purposes of machine computation (which isn’t an objective here), to use a correlation based instead on n-adic representation of numbers, which does not require as rapidly increasing functions as the exponential function, as in Smullyan 1959. By using this alternative, we could probably spare the proof of “Euclid’s first theorem” and the fundamental theorem of arithmetic; but we would have to develop some other things instead. However the work of proving these theorems is not so very great, and puts us in a position where we can use formally the notations used in the informal literature to which the present investigations are most closely related, which has its advantages too. (Our choice to do this is anticipated in Axiom Schema X2.1; thus “2. l a formalizes Kleene 1955 (5).) Accordingly we now introduce formal notations for functions (and symbols of abbreviation for predicates), the same as in I M except with formal variables, taking over I M $5 44, 45 #3-#21, #A-#F and also as #22 Seq (Kleene 1955a p. 416), #23 Z(x) (Kleene q 5 o a p. 680), #24 Z(x) (IM $ 46). We know a priori that we can have function symbols and predicate symbols-of-abbreviation expressing, and numeralwise representing or expressing, all these functions and predicates. What we are to add now is the verification that, with the recursion equations selected (usually) as in I M but now formalized, the formulas expressing various useful properties of the functions and predicates are provable. We already have # 1, #2 (addition and multiplication) with Axioms 18-21, and properties in I M § 39. At least until # 15 is treated, we use “a
N

3c(c+a=b).

“145~. t- a l b 3 a c l b c .

For #3 (exponentiation), we already have the symbol (as f4), we now take the recursion equations as Axioms X3.1, X3.2 (with a, b particular distinct number variables), and we give further properties as the provable formulas of “3.3-”3.13. Then for #4 (the factorial function), we introduce the symbol a! as f 5 (with k5 = 1, Is = 0),

24

FORMAL INTUITIONISTIC ANALYSIS

CH. I

the recursion equations as Axioms x4.1, x4.2, and a property in "4.3; and similarly through # I 3 [a/b] as f14. In #I4 no new symbol is involved, and in #15, #16 predicate symbols of abbreviation. x3.1. "3.3. "3.6. "3.9. *3.11. "3.13.

a0=1. X3.2. ab'=aba. I- abac=ab+c. "3.4. t (ab)c=abc. "3.5. I- (ab)C=aCbC. I- al=a. *3.7. t b>O-Ob=O. "3.8. k Ib=l. t a>O 2 ab>O. *3.10. I- a > 1 2 ab>b. t a, b> 1 3 ab>a. "3.12. t- a> 1 3 (bO 3 (a
-

-

PROOFS. "3.3-"3.5. Induction on c (the necessary replacement properties are available, by Lemmas 4.2, 5.1). "3.7. I. Prove O C ' = O (cf. the proof of I M *143b p. 189).11. ByX3.1. "3.9. Ind. on b (using * 143a). "3.10. Ind. on b (using *143b, *138b). "3.11. We prove a > l 3 ac">a, using "3.10 (with *145a), *143a. "3.12. Assume a > l . I. Assume d'+b=c (for 3-elim. from bO & a < b 2 ac'
0!=1.

x4.2.

a'!=a!a'.

"4.3. I- a!>O.

x5.1.

pd(0) =O.

X5.2.

pd(a')=a.

"5.3. k pd(a)
a"O=a.

X6.2.

a I b ' =pd(a

b).

x6.1. "6.3. "6.5. "6.7. "6.9.

"6.4. t O l a = 0 . t- (a+b) Ab=a. *6.3a. t a-a=O. "6.6. I-b>cx t a-(b+c)=(a-b)-c. ta>b

t b>c-

-

(a+b) I c = a + ( b l c ) . (aLb)+b=a.

a A (b 1-c)=(a+ c) b . "6.11. t-aO 2 ( a L b > c -a>b+c). "6.13b. I- c>O 3 (aLb=c -a=b+c). "6.15. I - a - b l a . *6.17. t a 2 b 2 a-c>b-c.

"6.8. "6.10.

t (a+c)A(b+c)=aLb. ka2b N a ~ ( a ~ b ) = b .

-I_

"6.12. t a > b alb>O. *6.13a. t aAb>c -a>b+c. N

"6.14.

t c(alb)=calcb.

"6.16. "6.18.

t a l b 3c A a 2 c l - b .

l a , b>O-aLb
§5 "6.19. "6.21.

25

POSTULATE GROUP D

t a>b, c

a.-c>b-c. I- a-c<(a-b)+(blc). N

"6.20.

I- ac'b.

PROOFS.*6.3. We prove Va((a+b)Lb=a) by ind. on b. IND. Assume Va((a+b) -b=a), whence (a'+b) l-b=a'. Now (a+b')-b' = pd((a+b')-b) = pd((a'+b)-b) [Ax. 19, "1181 = pd(a') = a, whence Va((a+b') Ab'=a). *6.5. Ind. on c. *6.6. b r c gives 3d(d+c=b); assume d+c=b. Now (a+b) -c = [*6.3] = a+(b-c). (a+d+c)Lc = a+d [*6.3] = a+((d+c)'c) "6.7. I. By "6.6, *6.3. 11. By *137a (above). *6.9. I. a-(b-c) = (a+c)l-((bl-c)+c) [*6.8] = ( a + c ) I b [*6.7]. 11. b b 1 al-b>O. The converses then follow by * 139, * 140. *6.14. By *139, a > b V a l b . CASE 1 : a>b. Use *6.3. CASE 2: a l b . Use *145c (above), *6.11. *6.17. Use cases (bc). STEP.

X7.1. min(a, b)=bl-(bl-a). X8.1. "8.2. *7.2. t a l b min(a, b)=a=min(b, a). "7.3. t min(a, b)=min(b, a). "8.3. "7.4. t min(a, b) c-min(a,b)>c. "8.5. *7.6. t a , b > c ~ m i n ( a , b ) > c . *8.6. *7.7. t min(c+a, c+b) *8.7. =c+min(a, b). "8.8. I- min(a, b) <max(a, b). *8.9. I- min(a, b)+max(a, b)=a+b.

-

max(a, b)=(a-b)+b. I-a>b max(a, b)=a=max(b, a). t max(a, b)=max@, a). t max(a, b) >a, b. ta,b
PROOFS. "7.2. I. min(a,b) = b L ( b 2 a ) = a [*6.10] = al-0 [X6.1] = a l ( a l - b ) [*6.11] = min(b, a). *7.3-*7.7. By cases ( a l b , a 2 b ) from *7.2.

-

X9.1. sg(0)= 1. X9.2. sg(a') =O. *9.3. t %(a)=O a>O. *9.4. I- %(a)= 1 a=O. *9.5. t g(a)=O VSg(a)=l.

N

x10.1. sg(O)=O. x10.2. sg(a') = 1. *10.3. I- sg(a)=O a=O. "10.4. I-sg(a)=l -a>O. "10.5. tsg(a)=OVsg(a)=l.

-

26

FORMAL INTUITIONISTIC ANALYSIS

CH. I

PROOFS.By induction cases (IM p. 186). ~ 1 1 . 1 . la-bl=(a-b)+(b-a). "1 1.2. t- Ia-bl=O-a=b. *11.3. 1 la-bl>O W a f b . *1 1.4. t- la-bl=jb-al. "1 1.5. t- la-cl a , b x I(cAa)-(clb)l= la-bl. [(a2 c ) - (b -c) I = la-bl. *11.12. t - a l b , b1-ac a>b+c V b>a+c. * 1 1.14a, b. Similarly with >, =. *11.15. t- (a-bl
-

PROOFS.11.2, 11.3. Using *6.3a, a = b 3 la-bl=O. Using cases (ab), *6.12 and *142a, a#b I la-bl>O. The converses follow by *158, *140. "1 1.5. Use "6.21. * I 1.10. Use *11.7, *6.7. *11.11. Use "6.5, "6.10. "11.14. I. By cases ( a k b , a
rm(0, b)=O. rm(a', b)=(rm(a, b))'.sglb-(rm(a, b))'\. [O/b]=O. [a'/b]=[a/b]+glb-(rm(a, b))'l. *13.3. t- [a/O]=O & rm(a, O)=a. t- b>O rm(a, b)O 2 [ab+c/b]=a+[c/b] & rm(ab+c, b)=rm(c, b). t- b>O 2 [ a b ~ c / b ] = a ~ ( [ c / b ] + srm(c, g b)). t- a < b 2 [a/c]<[b/c]. *13.11. t- a>b>O I [c/a]<[c/b]. N

PROOFS.*12.3. I. Ind. on a. In the ind. step, use cases by "10.5: sglb-(rm(a, b))'/=O, sglb-(rm(a, b))'l=l (then by *10.4 and *1 1.3, b#(rm(a, b))').

§5

27

POSTULATE GROUP D

* 13.4. Similarly. *13.5. By *146b with “13.4, “12.3. *13.6, “13.7. By *13.5. *13.8. By *12.3 (assuming b>O), rm(c, b)O. CASE 1 : rm(c, b)=O. Then [ab-c/b] = [ab~-b[c/b]/b] [*13.4] = [b(a- [c/b])/b] [*6.14] = a 2 [c/b] [*13.8, X13.11 = a ~ ( [ c / b ] + s grm(c, b)). CASE 2: rm(c, b)>O. SUBCASE 1: [c/b]l C”6.12, *138b], whence ab--b[c/b]>b [*145c, *6.14]. So [ a b ~ c / b = ] [abL(b[c/b]+rm(c, b))/b] [*13.4] = [(ab~b[c/b]) -rm(c,b)/b] [*6.5]=[(((ab~b[c/b]) ’b)+b)Lrm(c,b)/b] [*6.7] = [(ab-!-(b[c/b]+b))+(b~rm(c,b))/b] [*6.5; *6.6 with *12.3] = [b(a-~-([c/b]+l))+(b-l-rm(c,b))/b] = a - ~ ( [ c / b ] + l ) [*13.8; *13.7 with *6.16] = a ~ ( [ c / b ] + s g r m ( cb)). , SUBCASE 2: [c/b]>a. Then b[c/b] >ab. So [ab -c/b] = [ab 1(b[c/b]+rm(c, b))/b] = [O/b] [*6.11] = 0 [X13.1] = a-~([c/b]+sgrm(c, b)) [*6.11]. *13.10. By ind. on p, [a/c]<[a+p/c]. *I3.11. Assume a>b>O and [c/a]>[c/b], and use *13.4 to deduce rm(c, b) >b, contradicting “12.3. By a standard formula, we mean a (prime) formula of the form t = O where t- t=O V t = 1 (equivalently, t- t < 1). The following result *14.1 (by “1 1.2, *10.3) with *10.5 shows that any prime formula t l = t z is equivalent to a standard formula. This will be useful in building other standard formulas to express primitive recursive predicates under the constructions of #D- #F.

* 14.1. t- a = b

-

sgla-bl =O.

Hitherto “a
PROOF.sg(a’-b)=O [*138b].

N

a
N

sg(a’Lb)=O.

a’Lb=O [*10.3]

-

a ’ s b [*6.11]

N

a
28

FORMAL INTUITIONISTIC ANALYSIS

CH. I

In I M p. 191, “alb” abbreviates 3c(ac=b). Now *16.8 gives an alternative. (The proofs of * 16.1-* 16.5 are good in the system of IM.) “16.1. * 16.4. “16.6. *16.8.

-

I-aIO. *16.2. I- Olb b=O. *16.3. I- 1la. * 16.5. I- a ( b3 (alc alb+c). I- alc & bjd 2 ablcd. I- b>O 3 slab. *16.7. t O
*16.2. By *16.1, *156. *16.5. To show alb, alb+c t alc, assume ad=b and ae=b+c, whence c = ae-b [*6.3] = ae-ad = a(e>d) [*6.14]; or (avoiding A) we may proceed as follows. CASE 1: a=O. Use “16.2. CASE 2: afO. Assume for 3-elim. ap=b and aq=b+c. Then ap = b < b+c = aq, so by *145b, p l q . Assume p+r=q. Now ap+ar = aq = b+c = ap+c, and by *132, ar=c. *16.6. Prove alac’. *16.7. Ind. on a (using *138a). “16.8. I. Assume alb, and for 3-elim. aq=b. CASE 1 : a=O. Then b=O, so rm(b, a ) = O by x12.1; etc. CASE 2: a>O. Then b=aq+O & O
#A, #C need no comment. To take over f B , we let the next function symbol f15 express finite sum (k15 = 115 = l), with the axioms : XBla.

f15(O, a)=O.

xB2a. f15(~’,a)=fl5(z, a)+a(z).

This meets the requirements of Lemma 5.1. So, introducing “CY
flG(0, a)= 1.

XB4a. f16(z’, a)=fIG(z, a).a(z).

For any variable y and formulas A(y), R(y), we may use ‘‘VyA,,)R(y)” as abbreviation for Vy(A(y) 3 R(y)), and “3yA~g,R(y)” for 3y(A(y) & R(y)). We also want versions of *B5-*B13, *B16-*B21 with the inequality y
§5

POSTULATE GROUP D

29

for Z y<sLvtr(v+y) where ti(y) is congruent to t(y) and v is free for y in tr(y)). These versions follow from those given using !- VY,,,
*B2. tC 5aa(y) =~~<~a(y) +a(z). 1 &O - ny4(y) 9. k %d+) >O - &+a(y) 9. t jyyd(y) =O - fl~
P ROOFS . *B5. Ind. on z. BASIS. Use -,y0. B y “136, CyO. B u t i f &O, contradicting the assumption. II. Assume &O. By Remark 4.1, 3yY0 V -r3yYO. But if +lyY<,a(y)>O, then (using “86, *58b) Vy,<,a(y)=O, and by *B5 CY
30

FORMAL INTUITIONISTIC ANALYSIS

CH. I

*B12. In the ind. step a(z)= 1 or *B9 would be contradicted. *B13. Note *131. *B14. Assume w l z , and for 3-elim. z=w+c. Then *6.3 gives c=z ‘w. So we have to deduce Xy<w+ca(y)=Xy<wa(y)+X,<,a(w+y), which is straightforward by ind. on c. *B21, *B23. Using a < b 3 a c l b c and (for *B23) *145a. We treat #D, #E as schemata concerning certain forms of definitions. The results will enable us, for any formulas PI, . . ., P, to each of which we have chosen an equivalent standard formula and any formula E composed out of PI, . . ., P, by propositional connectives and bounded quantifiers, to build a standard formula equivalent to E. By Remark 4.1 now, it suffices under #D to treat V and 1 (as in IM). Via the alternative versions of *B5 etc., we have alternative versions of *E1-*E4 with v
-

-

*D1. I- Q V R qr=O. Sg(q)=O. *D3. I- iQ N

*D2. t q r l l . *D4. I- Sg(q)I 1.

Let y be a variable, R(y) a formula, and r(y) a term, such that I- R(y) -r(y)=O with I- r(y)
PROOFS. *D1. By “129 etc. *D4. By *9.5. *El. This is essentially *B8 (merely ilyr(y) has been substituted for u.) *E2. By *B10. *E3. By *B5 with “10.3.

§5

31

POSTULATE GROUP D

Before proving *E5-*E7, we obtain by *B13 with *B11 (4

I- VYY
Also by Remark 4.1 (or * 150), *86 and *58b,

"

(b) I- 3YY
*E5. Assume 3yY<,R(y). By *149a (with Remark 4.1, *159, or #D, #15), 3v[v
-

*F1. I- Q.j Z)p=pi.

*F2. I- -Q1&

. . . & 1Qm 2 p=O.

#G (IM $46) will be taken over by Lemma 5.3 (c) in 5.6.

*H 1-*H4 are equivalent to axioms xH 1a-xH4a (with function symbols f17, f18), as *Bl-*B4 to xBla-XB4a. *HI. I- min,50a(y)=a(0). *H2. k min, 5zra(y)=min(min,,,a(y), a(z')). *H3. I- may,,a(y)=a(O). a(z')). *H4. k may,,,a(y)=max(may,,,ct(y), *H5* Vy,,,a(yl 2min,,,a(y). *H6. 3yy,,a(y)=min,,,~(Yf. *H7. I- VY,&Y) smax,,za(Y). *H8* t 3Yy,z4Y)=maXy&4Y). By #15, #16, #E and #D (3c(l
-

a > l & d c ( l < c < a & cla).

Thus the prime formula Pr(a) is equivalent to the composite formula taken as Pr(a) in I M p. 191. (The proofs of *17.3, "17.4, "17.5 are good in the system of IM.) By 5.4, I- lPr(O), I- l P r ( l ) , t Pr(2), I- Pr(3), t lPr(4), I- Pr(5),

. . ..

32

FORMAL INTUITIONISTIC ANALYSIS

CH. I

As we now have the function a ! formally, we can improve "161 to:

"17.2. I- 3b(a
PROOF.By "4.3 and "16.7, I- a!>O & Vb(O
* 17.3. t Pr(a) & Pr(b) & a f b

2 lalb.

PROOF. By cases (ab), using "17.1, *156. "17.4. t a > 1 2 3p[Pr(p) & pla]. PROOF.Assume a > 1. Using * 153, a > 1 & ala, whence 3b[b> 1 & bla]. Hence by "149a (with Remark 4.1, or in the system of I M Remark 1 (b) p. 134 with "158-"160, *150), 3p[p>l & p l a & Vc, 1 &cla)]. Assume for 3-elim. p> 1 & p J a & V c c < p ~ ( c1>& cla). Now assume 3c( 1 < c < p & clp), and for 3-elim. 1 1 & cla; but also using c

1 & cia), l(c> 1 & cia). By reductio ad absurdum (cf. I M Remark bottom p. 188), i 3 c ( l
* 17.5.

I- Pr(p) & plab 2 pla V plb. (Euclid's first theorem.)

PROOF.We use the method of Hardy-Wright 1954 p. 21 2.1 1 (but only to get Euclid's first theorem, not the fundamental theorem of arithmetic). Thus informally $la V filb holds trivially for ab = 0. So we show the absurdity that there exist n, p , a, b such that n=ab & n#O & Pr(p) & pjn &$la& plb. But if there exist any four such numbers n, p , a, b, we can in particular pick the least n for which the other three 9, a, b exist ; then pick the least fi for which with this n the other two a, b exist; and finally pick the least a for which with this n and p the b exists; then the b is determined by n=ab & n f O . This start of the proof is formalized, with the precautions necessary for working in an intuitionistic system, in 1-111 below. The deductions about the n, p, a, b by which we then reach a contradiction, in IV-VII, are hardly different formally than informally, except for our having to verify meticulously that various familiar propositions we need to apply are expressed by formulas previously shown to be provable in the present system. I. For ab=O, use "129, "16.1. It remains for us to prove ab#O& Pr(p) & plab pla V plb. By the classical propositional calculus

=I

§5

33

POSTULATE GROUP D

(applicable by Remark 4.1, or in the system of IM by Remark 1 (a) p. 134 with "158-*160, *150), this is equivalent to l(ab#O & Pr(p) & plab & l p l a & lplb). So it will suffice, using t- 3nn=ab (by "100) and 3-elim., to deduce a contradiction from n=ab & nfO & Pr(p) & pin & Tpla & l p J b ,

(1)

call this R(n, p, a, b), or by 3-introds., from 3n3p3a3bR(n, p, a, b).

(2)

11. We show that R(n, p, a, b) t- p l & b > l , and also by *143b, a l . Using n=ab, nfO and pln in "156, p l a b . But if p=ab (so alp), then by "156 (with p > l , a > l ) , l < a < p ; but a = p contradicts l p l a by "153, so 1
-

(3)

N

-

3p3a3bR(n, p, a, b) & Vmm<,-.r3p3a3bR(m, p, a, b).

Repeating the procedure twice, we further assume (4)

3a3bR(n, p, a, b) & Vqq<,4a3bR(n, q, a, b),

(5)

3bR(n, p, a, b) 8~Ve,
Preparatory to 3-elim. from the first member of (5), assume (6) (= (1))

R(nJ

p,

b).

Finally, preparatory to 3-elim. from pin (in (6)),assume (7)

pd=n.

By the anticipated 3-elims., a contradiction deduced from (3)-(7) will result in one deduced from (2) and thence from (1) (cf. IM Remark bottom p. 188).

34

FORMAL INTUITIONISTIC ANALYSIS

CH. I

IV. We deduce p l d . By (7) d#O (since nfO is in (6)), and d#1 (since p 1. To show pd. By d> 1 with "17.4, 3q[Pr(q) & qld]. Assume for 3-elim., Pr(q) & qld. By *156, q l d , so q 1 from Pr(q)); so plalb, but lplal (or we would have pla). Also by *143b with q > l and dl#O, dl < qdl = d ; so by *145a, alb = pdl < pd = n. Thus alb1, afO) would make p < a and p>a absurd, and thus would imply p=a; but p#a by l p l a with *153. So p p < n & a a < n or p p < n & a a < n , each of which leads by *145a, *145c to p a p a a m , whence (by contradicting p a r " ) pa
* 17.6.

I- Pr(p) & plan 3 pla.

PROOF.Ind. on n. BASIS. Assume Pr(p) & pIa0. By X3.1 and *156, p = l . But lPr(1). IND.STEP. Use *17.5 with X3.2. We use #4, #15, #17, #D, #E, #A to select the term ,~b,,,!+~[a?. *18.6. 1i < j ,-pi
§5

35

POSTULATE GROUP D

PROOFS. "18.3. By *17.2 (and *138a), 3b,,,,,+,[pi a' > a,so3jpj>a. By*149a, 3j[pj>a & Vii<jipi>a]. Assume pj>a & Vii<jipi>a, whence Vii<jpi a, which with lPr(O), ,Pr(l) contradicts Pr(a). CASE 2: j>O. Write j=i'. Then i<j, so p i l a . But if pi
* 19.2. *19.3. "19.5. * 19.7. *19.9. *19.11. *19.12. *19.13. *19.14.

-

(a)i=,uxxO (a)iO & h>(a)i l p ih la. k a>O & p"a & l p y ( a 3 (a)i=h. *19.6. k (O)i=O. k (a)i>O a>O & pila. * 19.8. I- i 2 a I) (a)i=O. I- (p,h)i=h. *19.9a. I- (l)i=O. *19.10. ki#j D(pp)j=O. k ab>O 3 (ab)i=(a)i+(b)i. I- V Y Y < Z ~ ( Y ) 3 > ~(n,,a(y))i=~y k 3 (rIh5i
-

-

*19.15. k OO - a = ni
PROOFS. *19.2. I. Assume a>O. The conclusion will follow from X19.1 by *E5 if we get 3Xxl. By *3.10 pr>a, so by *156 lptla, whence 3ylprla. By *149a, 3y(lpTla & Vx,<,llp?la). Assume 1p;la & Vx,<,llpfla, whence Vxx<,pfla. CASE 1 : y=O. Then 1p;la contradicts p!la (obtained by "16.3, X 3 . 1 ) . CASE 2: y>O. Put y=x'. Then pfla by Vx,<,pfla. Also xa, then by "3.12 pf>pt>a, so by "156 lpfla. Thus x < a & pfla & 1pF'la. Use 3-introd. "19.3, *19.4. By *16.1, *19.2, *154 and h<m '3 chlcm (from *3.3). "19.6. From X19.1 by *E7. *19.8. CASE 1 : a=O. By "19.6. CASE 2: a>O. By *19.7 (for i 2 a with "18.5, *156 gives -pila). "19.9. By *18.5, p i > l ; SO $>O by "3.9, and ppO. By *19.2 (with *129), p?)'la, lp?)'+'la,

36

FORMAL INTUITIONISTIC ANALYSIS

CH. I

p{'))l\b,-tpl""+llb. Hence (by * 16.4, *3.3), p!a)l+(b)llab.The conclusion will follow by *19.5, if we deduce -,p?)l+(b)f+llab. So assume p!n)lf(b)l+llab, and for 3-elims., p?)*c=a, p:b)ld=b J p(a)L+(b)i+le=ab. 1 Then lpilc (or we would contradict lp?)'+'la, by "16.4, "153, X3.2); similarly, lpild. Now p?)icp!b)'d = ab = p?)l+(b)i+'e, whence by *133 (with X3.2, "18.5, "3.9) cd=pie, SO piled. By *17.5 (with "18.4), this contradicts -.Ipi/c& l p i / d . *19.12. Ind. on z, with *19.9a, "19.1 1, *B6, etc. (There is an alternative version with v 1, and by "17.4 3p[Pr(p) & plb]; assume Pr(p) & plb. By "18.7, 3j(p=pj); assume p=pj, so pjlb. By "19.7, (b)j>O. By "18.5 and "156, j < j' < pj < b ; SO by Vii&(a)i=(b)i, (a)j=(b)j. Thence by (b)j>O and "19.7, pjla. Assume pjc=a and pjd=b. By *145a, Oa) using "19.14, "19.13 and "19.8, Vi(a)i= (ITiO. *20.2. t- a > 1 "20.4. t- a>O a22lh'"). N N

-

"20.3. t- lh(p:+') = 1. *20.5. t- a>O lh(a)
PROOFS."20.2. I. Were lh(a)=O, by *B5, "10.3, X3.1 and *B13, we would have ITiO. To infer the conclusion from *19.16, deduce niO) and *18.5, "3.12, *3.13 etc.

§5

37

POSTULATE GROUP D

*20.5. I. Immediate for a= 1 ; and for a> 1, lh(a) >a would contradict "20.4 via *3.12, *3.10. x21.1. a*b=a.ni
Let Seq(a) be a standard formula. such that *22.1 holds. "22.1. "22.2. "22.3. *22.4. "22.5. *22.8. "22.9.

--

I- Seq(a) a>O & ViiO. I- Seq(a) a>O & Viirlhca)(a)i=O. I- Seq(a) a=II.IO & ViiO & Vi,,,(a)i=O lh(a)=k & Seq(a). I- Seq(2*+1). "22.6. I- a*l =a. "22.7. I- Seq@) l*b=b. k Seq(a) & Seq(b) 2 lh(a*b)=lh(a)+lh(b) & Seq(a*b). I- Seq(a) & Seq(b) & Seq(c) 2 (a*b)*c=a*(b*c). N

N

N

PROOFS."22.2. When a>O, then by X20.1 and *B14 with "20.5 lh(a) = ziO & ViiO. Then by *B11 (with x10.2), ziO & Vii,lh(a)(a)i=O. Then by *B5 (with X10.1) the second term is 0, hencethe first term is lh(a), hence by *B12 (with "10.5, *10.4) ViiO. *22.3. When a>O, then by *19.16 and *B15 with "20.5, a = Hitap!&)'= (&h(a)Pi (ah)'H]h(a)si 0. So by *133 the second factor is 1. Hence by *B13 with "3.10 and "19.8 Vilh(a)si(a)i=O,and so by *22.2 Seq(a). *22.4. I. Assume the hyp. By *19.8, k j a ; so by ~20.1,*B14, *B11, *B5 etc., lh(a)=k; so by "22.2, Seq(a). "22.7. By *22.3 (with lh(l)=O). *22.8. Assume Seq(a) & Seq(b). Then a*b#O (*B6 etc.). Using "6.3 (with *B 9, 'ilh(a)+lh(b) 2 (a*b)i=O. Hence by "22.4, lh(a*b)=lh(a)+lh(b) & Seq(a*b).

38

FORMAL INTUITIONISTIC ANALYSIS

CH. I

"22.9. Assume the hyp. By '22.8 and the evaluations in its proof, *6.17, *6.3 and *6.19,

i
i>lh(a) +lh(b) +lh(c)

3

((a*b)*c)i=O=(a*(b*c))i.

Hence by *19.15, (a+b)*c=a*(b+c). X23.1. "23.2. *23.3. "23.4. *23.5. *23.6. "23.7. "23.8.

X24.1. E(x) =rIi<xpr(i). I- i<x (E(x))i=.(i)+ 1. *24.2. k i<x 3 (E(x))i=a(i). I- i 2 x (E(X))i=O. *24.3. I- i 2 x 3 (C(x))i=O. I-Y<x -E(y)=ni
N

-

PROOFS. "23.2, "23.3, "23.4. I. By *19.14, *19.13, *B19 (with *23.2), respectively. *23.5. By *22.4. *23.6. I. Assume Seq(a). Then a = &
+

5.6. We may want a function defined by recursion for temporary use only (so that we prefer not to add a new formal symbol and axioms), or even for use only in constructing a deduction under assumption formulas preparatory to using a subsidiary deduction rule (cf. I M Chapter V). We can in effect adjoin such a function temporarily by assuming the formula A(a) expressing the recursion equations, preparatory to 3-elim. from the proved formula 3aA(a) of Lemma 5.3 (b) (a and any other variables free in A(&)to be held

§5

39

POSTULATE GROUP D

constant during this temporary use of a). An illustration occurs in the proof of Lemma 5.3 (c). This serves the purpose of I M Example 9 p. 415;but this comes at a later stage of development, and the greater resources of the present system make it easier to obtain. Using instead Lemma 5.3 (c), the recursion can be of the courseof-values type (IM $ 46 #G). We may even find it cdnvenient notationally to introduce similarly a variable to stand for an explicitly defined function, preparatory to 3-elim. from Lemma 5.3 (a). An illustration occurs in the proof of Lemma 5.3 (c). (The same effect could be gained by introducing a symbol of abbreviation for the functor Lyp(y), for which however we prefer not to use a Greek letter.) LEMMA 5.3. Let y, z be distinct nmmber variables, and a a fmnction variable. Let p(y), q, r(y, z), r(z) be terms not containing a free, with a and y free for z in r(y, z) and in r(z). Then: ( 4 I- 3aVYa(Y)=P(Y). (b) 1 3a[a(O)=q &Vy.(y')=r(y, 4Y))l. (c) I- 3aVya(y)=r(ii(y)) and I- 3aVya(y)=r(y, E(y)).

(If q contains y free, some occurrences of y in the proof and applications of (b) will have to be changed t o other variables. By *23.5, r(y, ii(y))=r(g(y)) when r(z) is r(lh(z), z).) PROOFS. (a) By *lo0 and xO.1 {lyp(y)}(y)=p(y), whence by V- and 3-introd. 3aVya(y)=p(y). (b) Let B(c, i, w) be the formula (c)i=w. By I M *171 p. 199, I- 3!wB(c, i, w).

By *19.9 I- (p&=w,

whence

(4

I- 3cB(c, 0, w).

Using *19.11 etc., *19.14,*19.10, "19.13, *19.9, I- Vii<,t (c1)i =((Ui
i

& ((ni<, cp?)~).p;)

y, =w,

whence

(PI

13cz(Vii
Let Q(w), R(y, z, w) be q=w, r(y, z)=w. By *171, I- 3!wQ(w),

I- 3!wR(y, z, w).

40

FORMAL INTUITIONISTIC ANALYSIS

CH. I

Form P(y, w) as P(y, x2, . . ., xn,w) was formed on I M p. 243, but with the present B, Q, R. By Remark 1 p. 244,

--

(1) I- P(0, w) Q(w), (2) I- P(Y’, w) 3Z[P(Y, z) 8L R(y, 2, w)l, (3) I-3!WP(Y,w),

whence t Vy3wP(y, w), and by “2.2 I- 3aVyP(y, a(y)). Assume for 3-elim., VyP(y, a(y)). Now as in I M top p. 416, a(O)=q & Vya(y‘)=r(y, a(y)), whence (b) follows by 3-introd. (and the 3-elim.). (c) WITH Z(y). Assume for 3-elim. from (a case of) (b), (i)

P(o)= 1 & Vyp(y’)=p(y)-p~Y,P(Y”.

Assume for 3-elim. from (a), (ii) VYa (Y) =(B (Y ’1 ) Y. Using *19.11 (with x24.1, *B6 etc.), “24.3, “19.9, (jii) (~i(y).p;yAy))) Y-r(Y> Z(Y))Now we deduce by induction P(Y)=Z(Y). = 1 [(i)] = z(0) [X24.1, *B3]. IND. STEP. p(y’) = p(y).p;(Y,P(Y )) “I = Z(Y).P,r(s,&s)) [hyp. ind.] = E(y).p$p(y’))y [(iii)] = L(Y).P;(~) [(ii)] = Z(y’) [X24.1, *B4]. - So a(y) = (p(y).p;(y~P(y)))y [(iv)] = r(y, B(y)) [(iii)]. By V- and 3[(ii), (i)] = (E(y)-p;(y~“y)))y introd. (and the two 3-elims.), 3aVya(y)=r(y, Z(y)). (c) WITH E(y). Apply (c) with E(y), for r(l-Ii
BASIS.p(0)

5.7. We bring together results permitting alterations of quantifiers, based on #19 in 5.5. Cf. “0.5, “0.6, *2.la, *2.2a in 4.5-4.6; I M p. 285; Kleene 1955 p. 315, 1959 2.1. For each m 2 0, let “, after 1955 9.2); let “” abbreviate ilx; and let “(a)l” abbreviate Ax(a(x))i (and “(a)i,j” abbreviate ((a)i)j, etc.). The letters here are subject to the obvious stipulations, under the blanket rule of 3.2.

-pz,

*25.1. I- (),=a, “25.2. I- ( < N O , . . ., a,>),=ai *25.3. 1 Vao. . .Va,A(ao, . . . , a,)

-

(i = 0, (i = 0,

VaA((a)o, . . ., (a)J.

. . .,m ) . . . .)m).

§5

POSTULATE GROUP D

--

41

k 3ao. . .3hA(ao, . . ., am) 3aA((a)o, . . . , (a),,,). k Vao. . .VamA(ao, . . ., am) VaA((a)o, . . . , (a),,,). I- 3ao.. .3hA(ao, . . ., am) 3aA((a)o, . . ., (a),,,). I- Vao. . .Va.,JbA(ao, . . ., am, b) 3aVao. . .VamA(ao, . . ., am, a()). *25.8. t Vao. . .VaJaA(ao, . . . am, a ) 3aVao. . . V h A ( a o , . . ., am, lya(
*25.4. *25.5. *25.6. "25.7.

- -

PROOFS.*25.1. By *19.11 (with '18.5, "3.9, *129), *19.9, *19.10 (and Lemma 5.2, by which pI=pi). *25.2. By XO.1, *25.1, *0.4. *25.3. I. Simply by the predicate calculus. 11. Assume VaA((a)o, . . ., (a),,,), and apply V-elim. with as the t (IM p. 99), *25.1 and V-introd. (IM *64). "25.7. I. Use *25.3 and *2.2; then, prior to 3-elim., assume VaA((a)o, . . ., (a),,,, a(.)) and continue as for *25.3. *25.8. I. Similarly, using X2.1, and transforming A(ao, . . ., am, lya(<,y))) into A(ao, . . ., am, ly{lza(<<(z)o,. . ., (Z)m>, (z)m*>)}()). *25.9. I. Assume Vii<,3xA(i, x). By cases @
The first assumption formula expresses that the cases are exhaustive. The other (y) (= 1 for m = 2) assumption formulas express that the cases are mutually exclusive. The conclusion formula shall be an abbreviation for

42

FORMAL AINTUITIONISTIC NALYSIS

CH. I

and

PROOFS. (a) In the SPECIAL CASE that Ql(y), Q2(y) are prime formulas, or equivalent to prime formulas by applications of #D and #E, we need only apply Lemma 5.3 (a) for p(y) the term p obtained by using #F with I- Qi(y) ,- qi=O and with pc(y) as the pg (i = 1, 2). However, the GENERAL CASE can be treated directly] thus. The first assumption formula gives two cases. CASE 1: Ql(y). Then -rQz(y).

so (QiMV Q ~ ( Y ) )& (Qdy) 3 Pi(Y)=Pi(Y)) & (Qz(Y)3 P~(Y)=Pz(Y)). By %introd., W Q i ( y ) V Qdy)) &

(QiM 3 a=pi(y))

&

(Q2M 3a = ~ z ( ~ ) ) l -

CASE 2: Q2(y). Similarly. - By V-introd. and *2.2,

3aVy[(Ql(y)V Q ~ Y ) & ) (QlW 3 a(y)=pl(y)) & ( Q ~ Y3) ~ Y ) = P ~ Y ) ) I * (b) Substituting ( y ) ~ilt(y)l , for y, a, and using xO.1: Qi((y)o,( ~ 1 1V) Qz((y)o,( ~ 1 1and ) -(Qi((y)o, (~)i) & Qz((y)o1( ~ ) i > SO, )using (a), we can assume for 3-elim.

(i)

VYdY)=

ri((y)oJ(Y)I) if Qi((y)o, { rz((~)o> if Q~((Y)o,

(~)i),

(~11)

(~)i).

Applying Lemma 5.3 (b) with p() as the r(yl z), assume for 3-elim. (ii)

m = q

VYdY’)=P().

Taking for y in (i) (by V-elim.) and using *25.1, the result

§6

T H E BAR THEOREM

43

with (ii) gives Qc(y, a(y)) 3 a(y’)=rt(y, a(y)). By &-, V- and 3-introds., we obtain the required formula. It does not contain p or a free, so the 3-elims. can be completed. (c) WITH ~ ( y ) .By *158 since Seq(z) is prime, Seq(z) V -&eq(z). Using cases thence, and in the first case *23.6 with Ql(E(y))V Qz(~(y)) : (i) (Seq(z) & Q1(z)) V (Seq(z) & Qz(z)) V iSeq(z). Using *23.6 with l(Ql(E(y))& Qz(a(y))): Seq(z) l(Ql(z)8~Qz(z)). Using this and *50, the three cases in (i) are mutually exclusive. Assume for 3-elim. from the result of an application of (a) with m = 3,

=I

I

ri(4 if seq(z) &z Qi(z), Vzp(z)= 4 2 ) if Seq(z) Qz(z), 0 if lSeq(z). Now use Lemma 5.3 (c) with p(z) as the r(z) (and later *23.5). ( Z h L l), (C) WITH L(y)- Apply (C) with E(y), for Q$(lh(z),fli
PROOF.Assume preparatory to 3-elim. from 3!wA(w), A(w) & Vx(A(x) 3 w=x). By *158, w=x V wfx. CASE 1 : w=x. Then A(x), whence A(x) V 1A(x). CASE 2: wfx. Then lA(x), whence again A(x) V iA(x).

Q 6. Postulate on spreads (the bar theorem). 6.1. In the intuitionistic set theory or analysis of Brouwer, a fundamental role is played by what he called a “set (Menge)” in his early papers on the subject (1918-9 I p. 3, 1919 pp. 204-205 or 950-951, 1924-7 I pp. 244-245) and more recently a “spread” (1954 p. 8). There are several versions of the notion of ‘spread‘, differing in details. We begin with a version differing from that of Brouwer’s early papers, reproduced in Kleene 1g5oa 3 1 (p. 680 end line 8, add ‘‘> O”), by the omission of what Brouwer called “sterilized (gehemmt)” sequences. In 6.9, we shall consider other versions. A given sfiread is generated by (i) choosing natural numbers in sequence, (either freely or) under an effective restriction which says, given the (numbers chosen in the respective) previous choices if any and any number, whether that number may be chosen next, and (ii) after each choice correlating effectively an object (depending on the

44

FORMAL INTUITIONISTIC ANALYSIS

CH. 1

previous choices if any and that choice) from a fixed countable set. Furthermore, under (i) it is effectively determined after each choice whether (depending on the previous choices if any and that choice) the sequence of choices is to terminate therewith or shall continue; in the latter case, the restriction governing the choices must allow at least one natural number to be chosen next. When a sequence of choices terminates, the element of the set or spread correlated to the sequence is the finite sequence of the objects correlated to the choices up to its termination. When a sequence of choices continues unterminated ad infinitum, the element correlated to the sequence is the infinite sequence of the objects correlated to the choices ; intuitionistically this element is not considered as completed, but only as in process of growth as the choices proceed. The word “effectively” in the foregoing is intended to convey what Brouwer expressed (in his early papers) by speaking of a “law (Gesetz)”; and indeed in 1924 9 1, 1927 5 2 he used “algorithm (Algorithmus)” in a related connection. What choices are permitted, and whether termination takes place, is determined by a law, which we call the choice law. What object is correlated is determined by another law, which we call the correlation law. (Cf. Kleene q 5 0 a p. 680, and Heyting 1956 p. 34,where the terminology is a little different.) These two laws each operate upon the finite sequence of the choices (natural numbers) up to and including the one which is under consideration (i.e. the natural number about to be chosen, when the question is whether the choice of it after the choices already made if any is permissible; the one just chosen, when the question is whether the sequence of choices thereupon terminates, or what object is thereupon correlated). A set or spread is not thought of intuitionistically as the “totality” of its elements, not even in the case all (permitted) choice sequences terminate so that the elements themselves become intuitionistically completed objects. To do so would (in general) involve the completed infinite (IM p. 48); e.g. the spread in which all choice sequences terminate after one choice which is completely free, with the number chosen correlated, is simply the set of all (unit sequences of) natural numbers. A spread from the intuitionistic standpoint is the pair of laws governing the generation process under which its elements grow. Through his notion of ‘spread‘, Brouwer found a way, while maintaining the standpoint of the potential infinite, to deal with collections

§6

T H E BAR THEOREM

45

some of which are even uncountably infinite (of classical cardinal number 2N9. The objects correlated to the choices in Brouwer’s applications may be, e.g., natural numbers, rational numbers, intervals with rational endpoints. Since for a given spread they must be chosen from a given countable class of objects, abstractly we can always take them to be natural numbers. When we do so, the notations available in the formal system suffice for the theory of spreads. Indeed, these notions include the fundamental constituents for dealing with spreads. These constituents can be combined under the formation rules of the system in a flexible manner, so that the particular way of combining them that gives a spread loses some of its preeminence in this formalism. Cf. however 7.8 below. 6.2. In this section, we shall concentrate on the choice sequences, which may underlie a spread, and which can be regarded as themselves constituting a spread by taking for the correlation law the trivial one which correlates the last natural number chosen. If then there is no restriction on the choices, the spread consists simply of all the infinite sequences of natural numbers in process of growth. This Brouwer called the universal spread. We study it now. When exactly t (‘20) natural numbers ao, a l , . . ., at-1 have been chosen successively, we have in other words chosen the first t values a(O),a ( l ) , . . ., a(t- 1) of a number-theoretic function a(x), the remaining values of which are still undetermined. Now we may associate with any finite sequence ao, . . ., at-l of natural numbers the natural number a = .-p::-;+’ = = [ao, . . ., at-11 = &(t), whereupon t = lh(&(t)), and a$ = a(;) = (&(t))iA 1 for i < t (cf. # 18-#23 in 5.5, and 5.7). This maps the finite sequences of choices 1-1 onto the natural numbers a such that Seq(a), which we call sequence numbers. The theory of choice sequences can now be dealt with in terms of the sequence numbers. The fundamental relation between sequence numbers is that of a sequence number a to the sequence numbers a*2s+l (s = 0, 1, 2, . . .) which represent the sequences ao, . . ., at-1, s coming from ao, . . ., at-1 by choosing one more number s ; the numbers a*2s+l are thus exactly the numbers &(t+l) for the various functions LY such that a = E(t). @+la..

46

FORMAL INTUITIONISTIC ANALYSIS

CH. I

6.3. Because the sequences of choices (“Wahlfolgen” in Brouwer 1918-9 and 1924-7, “infinitely proceeding sequences” in Brouwer 1952 and Heyting 1956) are considered intuitionistically as in process of

growing by new choices, especial prominence is given in intuitionism to those properties of choice sequences which if possessed can be recognized effectively as possessed at some (finite) stage in the growth of the choice sequence. Such a property of a choice sequence a is of the form (Ex)R(&(x))where R(a) is a number-theoretic predicate, effective at least when applied to sequence numbers a. With respect to such a predicate R(a), we say that, as a choice sequence a(O),a(l), 4 2 ) , . . . is generated, the finite sequence of the choices a(O), . . ., a(t- l ) , or the sequence number E(t) representing these first t choices, is secured, if it is known already from these t choices by the test of the predicate R that a possesses the property (Ex)R(&(x)),i.e. if (Ex).&(&(x)); 9ast secured, if this was known already without the last choice, i.e. if (Ex).
6.4. We have used function variables in expressing these notions. But there is a basic difference between the classical and the intuitionistic concepts ; for the intuitionists, the functions are not completed. The universal function quantifier (18) or (a),with its scope, in the expression for securability cannot be considered intuitionistically as a conjunction extended over all completed one-place number-theoretic functions, as it is classically. The intuitionistic meaning of (a)(Ex)R(&(x))is that, whenever one chooses successively natural

§6

THE BAR THEOREM

47

numbers a(O),a(l), a(2), . . . in any way, one must eventually encounter an x such that R(&(x)). How then can the intuitionists utilize the notion of securability ? To begin with, they can particularize, compatibly with their interpretation of (a),from (a)(Ex)R(S(x))to (Ex)R(&(x)) for such particular choice sequences a1 as they can specify; these, in connection with which Brouwer (1952 p. 143, 1954 p. 7) uses the term “sharp arrows”, are ones whose growth can be completely governed in advance by a law (after any t 2 0 choices, the law allows exactly one next choice). We have the formal counterpart of this in Axiom Schema 10F,where the functors u express primitive recursive functions in the case they contain no function variables (by Lemma 3.3). But it would seem that this makes rather weak use of (a)(Ex)R(&(x)). In fact, under the interpretation that an a1 giving a sharp arrow is a general recursive function, (a1)(Ex)R(&(x)) is in general weaker than (a)(Ex)R(&(x)) ; and the important “fan theorem” (in 6.10 below) fails when its hypothesis is weakened in the corresponding manner (Kleene 1g5oa 5 3, or Lemma 9.8 below). The intuitionists may refrain from adopting this interpretation, but they are in no position to refute it, since their actual constructions or laws conform to it (Chapter I1 below). Pursuing the matter further from the classical standpoint, while the fan theorem becomes true upon enlarging the class of a’s to the arithmetical functions (those such that a(x)=w is an arithmetical predicate IM p. 239; cf. Lemma 9.12 below), in order to exhaust the full force of (a)(Ex)R(Z(x))not even all the hyperarithmetical functions suffice (Kleene 1955b pp. 210, 208 with 1959 p. 48; or 1959b). REMARK 6.1. In the intuitionistic system, using * 158, we can prove VaVx(a(x)=O V a(x)#O), which seems to imply that any function a taking only 0 and 1 as values is recursive. (More generally, we can prove VaVxVw(a(x)=w V ct(x)#w), which seems to say that, for each a, the predicate a(x)=w is decidable, so presumable recursive, so by IM Theorem I11 p. 279 the function a, = Ax ,uwa(x)=w, is recursive.) On the other hand, as noted, we cannot interpret the universal quantifier (a)to mean “for all recursive functions a” without making the fan theorem of intuitionism false. This apparent contradiction is explained thus. As we choose the numbers a(O),a(l), a(2), . . . making up any choice sequence a, it will be known after each choice what number has been chosen; it is in this sense that VaVx(a(x)=O V

48

FORMAL INTUITIONISTIC ANALYSIS

CH. I

a(x)#O) is true. But as a choice sequence ar(O), a(1),a(2),. . . grows, in advance of each choice any number in the case of the universal spread (any number 1 in the case of the spread of choice sequences < 1 ) is eligible to be chosen; so the a is not governed by (%)a(%) restricted to be a recursive function. REMARK 6.2. In the present classical system with the same formation rules as the intuitionistic, the functors u available for Axiom Schema 10F are the same. (This is not so in classical systems like the ones in Hilbert-Bernays 1939 Supplement I V having a choice operator F or descriptive operator 6.) Fuller use of assumptions VclA(a) is obtained in the classical system via indirect proofs. 6.5. Brouwer found a solution to the problem of how to utilize an hypothesis of securability more fully than by Axiom Schema 10F. This consists in looking at the situation from the opposite direction, proceeding backwards from those sequence numbers G(x) for which I?(+)) to the other sequence numbers having such numbers in all their (sufficiently continued) extensions. To fix our ideas, let us confine our attention for the moment to sequence numbers not past secured (so that, in any sequence a of choices, we don’t overrun the first x at which we find R(Z(x))true). 7 to make it Then, slightly paraphrasing Brouwer 1927 FOOTNOTE read in our notation and terminology : Thought through intuitionistically, this securability is nothing else than the property which is defined thus. It holds for every sequence number a such that R(a). It holds for any sequence number a, if for every s (s = 0,1,2, . . .) it holds for u*28+1. This remark draws after it immediately the wellorderedness property . . .. In other words, Brouwer’s Footnote 7 says that securability is that property (of sequence numbers not past secured) which originates at the immediately secured sequence numbers, and propagates back to the unsecured but securable numbers across the junctions between a sequence number a and its immediate extensions a*2S+1 (s = 0, 1, 2, . . .). Let us review the situation using a geometrical picture (Figure 1). We can represent the universal spread 6.2 by a “tree”, with the sequence numbers a = fi?+’-. . .-p?E;+’ = [ao, . . . , at-11 at the vertices. The initial (leftmost) vertex is occupied by the sequence number 1 = [ ] = G(0). From any vertex, occupied by the sequence number a,

49

T H E BAR THEOREM

...

to, o,o, 11

11 Figure 1.

infinitely many arrows lead to the next vertices, occupied by the sequence numbers a*28+1 (s = 0, 1, 2, . . .). A part of this tree is shown in Figure 1 ;but the arrows for s > 1 are left to our imagination, as well as the vertices for t = lh(a) > 4 suggested by the dots. (The figure actually shows the “binary spread” or “binary fan” 6.10 as far as its vertices with lh(a) I 4 . ) An infinite choice sequence a or a(O),a(l ) , a(2),. . . is represented by an infinite path in the tree, starting at the leftmost vertex (occupied by) [ 3 and following arrows; a finite sequence of choices by an initial segment of such a path, or by the vertex &(t) at the (right) end of that segment. Thus, before a(0)is chosen, we are at the vertex [I; then if we choose a(0) = 1, we move to the vertex [l]; choosing next a(1) = 0, we continue to [ l , 01; choosing 4 2 ) = 1, to [ l , 0, 11; choosing 4 3 ) = 1, to [ l , 0, 1, 11; etc.

50

FORMAL INTUITIONISTIC ANALYSIS

CH. I

Consider a predicate &!(a), effective at least when applied to sequence numbers a. For each a, let us follow the corresponding path in the tree (starting from [I)until we first encounter a vertex ~ ( x for ) which R ( S ( x ) ) ,if we ever do, whereupon we underline that vertex. In the language of 6.3, we underline (the vertices occupied by) the immediately secured sequence numbers. Now (a)(Ex)R(E(x)), as we considered it in 6.3 and (intuitionistically) in 6.4, means geometrically that, along each infinite path starting from the leftmost vertex [ 3 and following arrows, we will encounter an underlined vertex. This is illustrated in Figure 1, so far as it can be shown with only the arrows for s = 0, 1. More generally, (/l)(Ex)R(a*p(x))or in words a i s securable (but not past secured) means geometrically that, along each infinite path starting from the vertex occupied by a and following the arrows, we will encounter an underlined vertex. Brouwer’s reversal of the direction consists in replacing this meaning of a i s securable (but not past secured) by that of belonging to the class of sequence numbers which is defined to include the ones underlined, and to include a whenever it includes all a*28+1 for s = 0, 1, 2, . . ., but to include no other sequence numbers. (This definition is an example of an inductive definition, in the terminology of IM 5 53.) In Figure 1, the securable but not past secured sequence numbers are those which are in bold face (heavy type), if we suppose appropriate behavior along paths containing arrows with s 2 2. But under the first meaning of securable (but not past secured), which we now call the explicit sense, a vertex’s being in bold face means that proceeding rightward from it in the direction of arrows along all possible divergent paths an underlined vertex will be encountered. Under the second meaning (the inductive sense), a vertex’s being in bold face signifies its membership in the class of vertices generated by putting into the class the underlined vertices, and proceeding in the leftward or convergent direction (reverse to arrows) to include a in the class whenever all a*28+1 (s = 0, 1, 2, . . .) are included in the class. Since the explicit sense, which our symbols in 6.3 directly express, had already been used before Brouwer’s 1927 Footnote 7 was introduced, that Footnote 7 must come to this: The two meanings of securable (but not past secured) are equivalent ; and this equivalence is given by intuition (by thinking the matter through intuitionistically). We agree with him.

§6

THE BAR THEOREM

51

In the figure, whether one puts vertices in bold face by the criterion of finding an underlined vertex at or to the right of them along all paths, or moves leftward across the figure putting vertices in bold face by the two principles generating a class of vertices, the result is the same. One of the implications in this equivalence is actually unproblematical, i.e. easily proved (cf. end 6.7). The other implication, that by securable (bat not $ast secured) in the explicit sense of secarable (but not $ast secured) in the inductive sense (or of the “well-orderedness property”, which the latter entails immediately) is essentially what Brouwer subsequently called the “bar theorem” (1954 p. 14, cf. Remark 6.3 below). In 1924 9 1 (cf. 1924a $3 1, 2), in the text of 1927 9 2, and in 1954, he used a more complicated analysis to prove the bar theorem. Footnote 7 of 1927 concluded, “The proof carried through in the text for the latter property [well-orderedness] seems to me nevertheless of interest on account of the propositions included in its line of thought. ” We shall simply introduce what is needed here by an axiom schema X 2 6 . 3 which gives the effect of the bar theorem for the case of the universal spread. This schema takes the form of a principle of induction which attributes to the securable (but not past secured) sequence numbers any property expressible in the symbolism of the system which originates and propagates in the same way as the securability property itself (under the inductive sense). Our procedure amounts to adopting Brouwer’s 1927 Footnote 7 in place of the more elaborate treatment in the text of 1927. We thus quickly get over a moot point in Brouwer’s deduction of his analysis by postulating an axiom schema. This may strike some as an evasion. But this axiom schema is independent of the other intuitionistic postulates, as we shall see in Corollary 9.9 (and 9.2, by which its negation is unprovable). So there can be a question of deriving the axiom schema (the bar theorem), only if we first substitute another postulate to derive it from. We are unconvinced that any known substitute is more fundamental and intuitive. However, in view of the attention which the proof in Brouwer’s text of 1927 has continued to receive, we shall also examine that, in 6.12.

6.6. We consider now just how to state the bar theorem in the formal symbolism.

52

FORMAL INTUITIONISTIC ANALYSIS

CH. I

The definition of a property in Brouwer’s Footnote 7 reads, under the restriction there to sequence numbers not past secured, as an inductive definition of the securable (sequence) numbers. If we substitute “a which is secured” for “a such that R(a)”, then without the restriction it reads as an inductive definition of all the securable numbers (Kleene 1955a p. 416). If we omit the restriction, but require R t o be a predicate such that, for any a, R(ol(x))for at most one x, it reads as an inductive definition of the numbers securable but not past secured. If we simply omit the restriction, it reads as an inductive definition of the barred numbers. I t makes little difference to us here which reading we use, and the last is the simplest. We also obtain some simplification by stating the induction principle corresponding to the inductive definition only for inferring properties of 1 (for which ‘securable’, ‘securable but not past secured’ and ‘barred’ are equivalent). We do not lose thereby, as we shall verify in 6.1 1. For securability in the explicit sense of 6.3 we now write “securable,”, in the inductive sense of Footnote 7 “securable,”. The bar theorem is then the implication (*)

securable,

--f

securable,,

when the right side is rendered by the principle of induction corresponding to the inductive definition (cf. I M 5 53). We want to formalize this, applied to 1, with respect to R. The left side of (*) is then simply (a)(Ex)R(ol(x)). Let 3 ( R ,A ) be ( a )[Seq(a) & R(a)--f A (a)]& (a)[Seq(a) & (s)A(a*28+1) --f A(a)] + A(1); and for any formulas A(a) and R(a), let S(R, A) be the correspondingly constructed formula. The principle of induction rendering the right side of (*) is ( A ) S ( R ,A ) . Thus we render (*) in informal symbolism as (a)(Ex)R(ol(x)) + ( A ) s ( R ,A ) . Expressing this in the formal symbolism as nearly as we can in the absence of predicate variables (cf. I M p. 432), we are led to Va3xR(ii(x)) 2 3(R, A), which (trivially rearranged) is “26.1.

6.7. Before postulating a slight restriction of this for the basic system or the intuitionistic system, we verify that it is provable in the classical system. The proof is a formalization of the classical proof of (*) in Kleene 1955a (E) p. 417. If a, s, x are any number variables (a and s distinct), cx is any

§6

THE BARTHEOREM

53

function variable, A(a) is any formula not containing s free in which s is free for a, and R(a) is any formula not containing cc or x free in which a and x are free for a: "26.1 '. t- Va3xR(E(x))& Va[Seq(a) & R(a) 2 A(a)] & Va[Seq(a) & VsA(a*2s+l) 2 A(a)] 3 A(1).

PROOF.By the classical propositional calculus, it will suffice to assume WSeq(a) R(a) 3 A(a)l, Va[Seq(a) & VsA(a*2s+l)2 A(a)], -A(l),

(a) (b)

(4

and deduce lVdxR(E(x)), which by the classical predicate calculus (*85, *86) is equivalent to 3ccVxlR(h(x)). Likewise (b) is equivalent to Va[Seq(a) & lA(a) 3 3slA(a*2S+1)], whence by *97 Va3s[Seq(a) & l A ( a ) 2 -1A(a*2*+1)],whence by *2.2 3oVa[Seq(a) & lA(a) 2 lA(a*2"(a)+1)].Assume for 3-elim. from this Va[Seq(a) & lA(a) 3 ~ A ( a * 2 ~ ( ~ ) + ' ) ] .

(d)

By Lemma 5.3 ( c ) , 3ccVxa(x)=o(E(x));so assume (e)

Vxa(x)=o(E(x)).

Now we deduce by induction iA(a(x)).

(f

BASIS. By X23.1 and *B3, E(O)= 1. So by (c), lA(E(0)). IND. STEP. E(x') [*23.8] = E ( X ) + ~ ~ ( ~[(e)]. ( ~ )So ) +by ~ (d) with the hyp. ind. and *23.5, lA(E(x')). - By (f) and (a) with *23.5, lR(E(x)). By V- and %introd., 3ccVxYR(E(x)). The converse implication 7

(**)

securable, -+ securable,

This holds (Kleene 1955a (D) p. 416) is (A)$(R, A ) --f (a)(Ex)R(&(x)). intuitionistically, a fortiori from $(R, A 1) --f (a)(Ex)R(G(x))where A 1 = Au (a)(Ex)R(a*&(x)). So *26.2 can be considered as giving (**) in the basic system. If a, s, x are any distinct number variables, cc is any function variable, and R(a) is any formula not containing x, s, cc free in which x, s, cc are free for a:

54 *26.2.

FORMAL INTUITIONISTIC ANALYSIS

CH. I

I- {Va[Seq(a) & R(a) 3 Va3xR(a*G(x))] & Va[Seq(a) & VsVa3xR((a*2s+l)*E(x))3 Va3xR(a*Z(x))] 2 Va3xR( 1*E(x))}3 Va3xR(E(x)).

PROOF.Using a=a*E(O) (by *22.6 with X23.1, *B3), (a) Va[Seq(a) & R(a) 3 Va3xR(a*~(x))]. Toward (b) below, assume (i) Seq(a) and (ii) VsVa3xR((a*2s+l)*E(x)). Using (ii), 3~R((a*2"(~)+')*{Lxa(l +x)}(x)). Assume preparatory to 3elim. (iii) R((a*2"(0)+')*{Lxa( 1 +x)}(x)). But (a*2"(0'+1)*{Axa(1 +x))(x) - a*(2"(0)+1+{Lxa(l +x)}(x)) [*22.9 with (i), "22.5, *23.5] = a*E(l+x) [23.7 with X23.1, *B4, *B3, "1271. So by +introd., (completing) the 3-elim., and V-introd., Va3xR(a*E(x)). By &-elim. and 3- and V-introd., (b) Va[Seq(a) & VsVa3xR((a*2s+l)*E(x))3 Va3xR(a*E(x))]. Assuming the antecedent of the main implication of *26.2, and using (a) and (b), we obtaifi Va3xR(I*E(x)), whence the consequent Va3xR(E(x)) follows by *22.7 with *23.5.

6.8. The restriction that R be an effective predicate, introduced beginning 6.3 (but immaterial from the classical standpoint), must be made explicit in postulating the bar theorem (*) for the basic system or the intuitionistic system. As expressed by *26.1 simply, (*) is inconsistent with the further intuitionistic postulate X27.1 to be introduced in $ 7 , by *27.23. We give four forms x26.3aJ26.3d of the new axiom schema. Whichever one is introduced now as the postulate, all axioms by each of the others become provable. When it is immaterial which one we cite, we call it simply X26.3. The stipulations for X26.3a and X26.3~are the same as for "26.1. For X26.3b, Q and p are any distinct function variables, etc. X26.3a. Va[Seq(a) 3 R(a) V -1R(a)1& Va3xR(E(x))& Va[Seq(a) & R(a) 3 A(a)] & Va[Seq(a) & VsA(a*2s+l) 3 A(a)] 3A(1). x26.3b. Va3xp(E(x))=O& Va[Seq(a) & p(a)=O 3A(a)] & Va[Seq(a) & VsA(a*2s+l) 3A(a)] 3 A(1).

§6

THE BAR THEOREM

55

X26.3~. Vd!xR(E(x))& Va[Seq(a) & R(a) 3 A(a)] & Va[Seq(a) & VsA(a*2+1)3A(a)] 3 A(1). x26.3d. Va3x[R(E(x))& Vyy<x-rR(E(y))] & VaVx[R(E(x))L% Vyy<xlR(E(y))3 A(z(x))l Va[Seq(a)& VsA(a*28+1) 2 A(a)] 3 A(1). OF x26.3b FROM X26.3a. Taking R(a) in X26.3a as DERIVATION p(a)=O, we have R(a) V l R ( a ) by *158, a fortiori Va[Seq(a) 2 R(a) V +a)]. X26.3a FROM x26.3b. Assume the four hypotheses (a)-(d) of X26.3a. By *158, because Seq(a) is prime, Seq(a) V +eq(a). Using cases thence, and in the first case subcases from (a), (Seq(a)& R(a)) V .-r(Seq(a)& R(a)). Using this with *50 to apply Lemma 5.5 (a), assume preparatory to 3-elim. from the result

Vap(a)=

-

0 if Seq(a) & R(a), 1 if l(Seq(a) & R(a)).

Now Seq(a) 3 (R(a) p(a)=O), using which and *23.5 the three hypotheses of x26.3b follow from (b)-(d). X26.3~FROM x26.3a. Assume Va3 !xR(E(x)). Assume seq(a), so via *23.6 we can put a=E(x) (i.e. we assume this preparatory t o 3elims.). Using Lemma 5.6, R(E(x))V -.R(E(x)), whence R(a) V l R ( a ) . By (completing) the 3-elims., 3-and V-introd., Va[Seq(a) 2 R(a) V lR(a)]. Also, Va3xR(E(x)). X26.3a FROM X26.3~.Assume the four hyps. (a)-(d) of X26.3a. Let R'(a) be R(a) & V y ~ t l h ( a ~ l R ( n i < Y pso~ )using l ) , *23.5 and *23.4 R'(E(x)) R(E(x))& Vyy<xlR(E(y)).By *23.5 Seq(E(x)),so (a) gives R(E(x))V lR(E(x)). Thence by * 149a and * 174b Va[3xR(E(x)) 2 3!xR'(E(x))], and by *69 Va3xR(E(x)) 5 ) Va3!xR(B(x)). So we have Va3!xR(E(x)). Since R(a) 2 R(a), we also have Va[Seq(a) & R(a) 3 A(a)]. Now we can apply X26.3~with R as the R. x26.3d FROM X26.3~.Use *174b.

-

6.9. The foregoing induction principle X26.3 takes care of the bar theorem for the universal spread. We should like it also for other spreads of choice sequences. So instead of dealing with the class of all the sequence numbers a, characterized by Seq(u), we shall now deal with any suitable subclass

56

FORMAL INTUITIONISTIC ANALYSIS

CH. I

of them, which we shall characterize by a(a)=O for some function

0.

For simplicity, we may omit from consideration terminated sequences of choices (cf. 6.1), so this CT will serve as the choice law (the other function of the choice law in 6.1, to say when a sequence of choices terminates, is suppressed). We may do this here without loss, since we are interested only in what happens up to an x such that R ( & ( x ) ) . Indeed in general, with a simplified choice law u that doesn't provide for termination, we can still obtain the effect of termination, either (a) by using a predicate K and considering a(O), a ( l ) , a(2), . . . to terminate at a(x-1) for the least x if any such that R ( & ( x ) )or , (b) for spreads with a non-trivial correlation law, by using positive integers as (or t o represent) the objects which we are interested in correlating, and correlating 0 otherwise (essentially Brouwer 1924-7 I Footnote 1). In our theory of choice sequences we have been using to advantage the empty sequence, represented by the sequence number &(O) = 1. (Brouwer employed neither the empty sequence, nor sequence numbers.) For spreads all of whose elements are to be sequences with the same first member, we find it convenient to correlate that first member to the empty choice sequence. Then the correlation law p operates simply on all sequence numbers a with .(a) = 0. When we don't want the elements all to begin with the same first member (correlated to I), we may simply ignore what p(1) is. But whether we du or do not wish to consider p( 1) as first member of the elements, it seems to us natural to take advantage of our empty sequence by letting the spread be non-empty exactly when the empty sequence is permitted, i.e. when o(1) = 0. Thus the choice law suffices itself for deciding whether a spread is empty or not. When we thus both omit terminated sequences and use the empty sequence to test for a spread's not being empty, we are led to the following formula Spr(o) expressing in the formal symbolism the restrictions on CT that it characterize the choice sequences for a spread. Spr(o): Va[o(a)=O ZI Seq(a)] & Va[o(a)=O 3 3so(a+2s+l)=O] & Va[Seq(a) & a(a)>O I)Vso(a*2s+1)>0]. In "26.4 we state the bar theorem for spreads generally, using this version of the notion of 'spread'. The second hypothesis o(l)=O expresses that the spread is not empty. If we were simply to omit terminated sequences (which would give the version of Heyting 1956 pp. 34-35, = essentially Brouwer 1924-7

§6

57

THE BAR THEOREM

I Footnote 2), we would use instead of Spr(o) the formula Spd(a) obtained from it by prefixing o(l)=O & and replacing the second Va by Va,,,. "26.4 would become *26.4' with 3s0(2S+l)=O replacing o(l)=O to express the non-emptiness of the spread. Under the version of 'spread' in Brouwer 1954, all spreads are non-empty. That a choice sequence a is permitted by the choice law cr of a spread is expressed formally by Vxo(C(x))=0, which we abbreviate as " U E ~ ' . The form *26.4a of *26.4 corresponds to x26.3a and is proved from it ; using instead X26.3b426.3d, corresponding forms *26.4b-*26.4d are obtained (not written out when clear). Also, from any one of *26.4a-*26.4d the others can be derived (using only Postulate Groups A-D), as with "26.3. Similarly with *26.6, "26.7 and "26.8 below. *26.4a. k Spr(a) & a( 1)=0 & Va[a(a)=O 2 R(a) V l R ( a ) ]& Vaa,03xR(E(x))& Va[a(a)=O & R(a) z) A(a)] & Va[a(a)=O & Vs{a(a*28+1)=0 2 A(a*28+1)}z) A(a)] 2 A(1). *26.4d. !- Spr(o) & o(l)=O & Vaa,,3x[R(E(x)) & Vyy<xlR(~(y))]& VUVX[C(E(X))=O & R(E(x))& Vyy<x-~R(E(y)) 3 A(E(x))] 8~ Va[a(a)=O & Vs{a(a*2s+l)=O 2 A(a*2s+l))2 A(a)] A(1).

=I

PROOF OF "26.4a. In I, we shall set up a mapping of the universal spread onto the spread characterized by CT.Thus, to each element a of the universal spread, the function a,, (= At (y(&(t')))tAl) will belong to the spread cr, as shown by (E). If a already belongs to 0, ay = a, as shown by (q). (We give (C) and (q) for use in proving *26.7a, *27.4 etc.) In 11, this mapping carries the bar theorem for the universal spread into the bar theorem for CT. I. Assume the first two hypotheses of *26.4a, call them (1) and (2). By cases from a(a)=OVo(a)#O (by *158), using ( l ) , 3s[o(a)=O z) a(a*2s+l)=O], whence by V-introd. and *2.2 3xVa[o(a)=O 3 ~~(a*2~(*)+') =O]. Assume (a) Va[a(a)=O 2 o(a*2"(a)+1)=0].

In the following formula (p), the case hypotheses are exhaustive (by cases from applications of * 158, since the components are prime) and mutually exclusive (using "50). So Lemma 5.5 (c) applies (indeed, the special case), and we assume (preparatory to 3-elim. from the result)

FORMAL INTUITIONISTIC ANALYSIS

CH. I

3 if -Seq(a), 1 if Seq(a) & lh(a)=O, (y(a))~*2S+1 if Seq(a) & lh(a)#O & cr((?(a))~*2S+l>=O, (Y(~))~*~x((Y(&))=)+ 1 if Seq(a) & lh(a) #O & cr((p(a))~*2S+1) #O where B is IIiilh(a)flp~a)i and S is (a)lh(a)fll1. If in (p) we use ~ ( 0 ) for a (via V-elim.), the second case applies and gives y(~(0))= y(1) = 1 (using *23.5, X23.1, "B3). If in (p) we use ti(x') for a, then the third or fourth case applies; furthermore using "23.4, *23.2, *23.8 etc., B=E(x), S=a(x), E(x') = a = B.p:+' = B*2S+l=ti(~)*2"(~)+~, so B < a (using *143b, "3.10 etc.) and (y(a))B= y(B) = y(ti(x)) (by "24.2). Now by ind., using (2) in the basis, and (a)to deal with the fourth case of (p) in the ind. step, (Y)

"(Y

(W)) =o.

Let "ay" abbreviate At(y(E(t')))tL1. Now we deduce by induction (8)

",x)=y(E(x)).

BASIS: trivial. IND.STEP.o(,(x') = o(,(x)*2((~(~(~')))"l-l)+' [*23.8, XO. 11 = y ( E ( ~ ) ) * 2 ( ( ~ ( ~ ( ~ ' ) ) )[hyp. ~ ~ ' ) +ind.], ' which (using (y(ti(x))*2*+1), = ( & ( ~ ) * 2 * + l [hyp. )~ ind.] = A+1), if the third case of (p) applies to a=ii(x'), = y(E(~))*2"(~)+' = y(E(x')) {if the fourth case applies, - y(CC(x))*2X(Y("(x)))+1 = Y(~(X')U.- BY (Y) and (a), (E)

a,€".

We also deduce by induction

(t)

o(E(x))=O 3 y(E(x))=E(x).

IND.STEP. Assuming c(E(x'))=O, the third member of (1) gives o(E(x))=O, so by hyp. ind. y(E(x))=E(x), and the third case of (p) applies. - By (8), (C), "23.2 and*6.3, c(E(x'))=O 3 ay(x)=a(x),whence (q)

a E c z) uy=u.

11. Assume also the remaining hyps. (3)-(6) of *26.4a. We shall apply X26.3awith R(y(a)), A(y(a)) as the R(a), A(a). If we can then verify the four hyps. of X26.3a, the concl. of *26.4a will follow using y ( l ) = l (in I). We get the first hyp. by (y) with (3) (using *23.6 to put a=E(x) preparatory to 3-elims.). For the second, by ( E ) and (4)

§6

THE BAR THEOREM

59

3xR(<(x)), whence by (6) 3xR(y(E(x))). We get the third (putting a=E(x)) by (y) with (5). For the fourth, assume Seq(a) &VsA(y(a*28+1)). By (y) with *23.6, cr(y(a))=O. Put x=lh(a). Assuming c(y(a)*28++1) =0, and using *22.8, *22.5, *23.6 to put a*28+l=E(y) (then y=x‘ [*22.8, *20.3, *23.5], a-psxf’ = a*28+1 [X21.1 etc.] = Z(x’) = E(X).~;(~)+’ [*23.8], so s=a(x) [*19.11, *22.2, “19.9, *6.3] and a=E(x) [*133]), the third case of (p) applies to E(x’) and gives y(z(x’))=y(a)*28+1, so VsA(y(a*28+1)) gives A(y(a)*2s+l); thus Vs{o(y(a)*2s+l)=O 3 A(Y(a)*28+1)).BY (611 A(y(a)). 6.10. From his bar theorem Brouwer inferred his “fan theorem” (implicit in 1923a p. 4 (11); 1924 Theorem 2; 1927 Theorem 2; 1954 5 5). A “finite set” or “finitary spread”, most recently called a fan, is a spread in which each choice must be from a finite collection of numbers. Say e.g. that, for t = 0, 1, 2, . . ., the number a(t) must be chosen from among 0, 1, . . .,@ ( E ( t ) ) ; i.e. (t)a(t)
60

FORMAL INTUITIONISTIC ANALYSIS

CH. I

holds for the subfan issuing from any sequence number a = 8(y) securable but not past secured (in the given fan with respect to the given R ) and the predicate ilw R(a*w). The subfan issuing from a sequence number a such that R(a) has 0 as a z for the fan theorem. Consider a sequence number a whose securability follows from that of all a*2s+l for s < ,!l(a); by the hyp. ind., for each s < @(a)the subfan issuing from a*2s+l has a z , call it zs, for the fan theorem. So the subfan issuing from a has 1 +max(zo, . . ., as a z for the fan theorem. This completes the induction. But under the hyp. of the fan theorem, 1 is securable but not past secured. So the conclusion of the fan theorem holds for the subfan issuing from 1 and the predicate ilw R(I*w), i.e. for the given fan and R. This is easily pictured geometrically. Our fan is represented by a tree in which from each vertex, occupied by the sequence number a, finitely many arrows (namely @(a)+l of them) lead to vertices, occupied by a*20+1, . . ., ~ * 2 ~ ( ~ )This + ‘ . is illustrated by Figure 1 in 6.5 for the case (a)[@(a)=2](the binary fan), where now we are not to imagine arrows for s > 1. Again consider a predicate R ( a ) ; and suppose that, for each a, we underline the first &(x) (if any) for which R(C(x)).Figure 1 illustrates a case in which (a)(Ex)R(&(x)). To simplify terminology, let us suppress in each branch all vertices to the right of an underlined C ( x ) ; so in Figure 1 only the part of the tree printed in bold face remains. The hypothesis of the fan theorem then says that all paths are finite. The conclusion says that there is a finite upper bound to their lengths. The proof is by induction, corresponding to the inductive definition of the class of the securable (but not past secured) sequence numbers a (6.5, but now in the fan rather than in the universal spread). The induction proposition is that there is a finite upper bound to the lengths of paths in the subtree issuing from a. As basis of the induction, this upper bound is 1 (the z is 0) for a at the end of any branch. As induction step, in proceeding leftward from all a*28+1 (s = 0, 1 in Figure 1) to a, we graft finitely many subtrees (2 in our Figure 1) with respective finite upper bounds onto a to obtain a subtree with upper bound the maximum of the respective upper bounds increased by one. In formalizing this proof, we first prove a lemma *26.5, in which b, s, z, w are any distinct number variables, and B(s, z) is any formula not containing b, w free in which w is free for z.

§6

61

THE BARTHEOREM

*26.5. VsVzVw[B(s, z) & w 2 z

=I B(s, w)]

1VSS,b3ZB(S,

2)

2 3ZvS,,bB(S,

2).

PROOF.We assume (a) VsVzVw[B(s, z) & w>z 2 B(s, w)], and deduce the rest by ind. on b. IND. STEP. Assume Vs,,,JzB(s, z), whence 3zB(b', z) and Vss,,3zB(s, 2). By hyp. ind., 3zvs,,,B(s, z). Assume for 3-elim., B(b', 21) and Vs,,,B(s, 22). Using (a) and *8.4, B(b', max(z1, 22)) and Vs,,,B(s, max(z1, ZZ)), whence VsSl,,B(s,max(zl, zz)), whence 3zVs,,,B(s, z). *26.6a. k Va[Seq(a) =I R(a) V lR(a)] & VaBca,3xR(E(x)) 3 32VcrB(a)3xx~zR(E(x))

where B(a) is Vta(t)
PROOF OF "26.6a. I. B(a) does indeed restrict a to a non-empty spread. For, we can introduce a function variable r~ so that the following formula (a) holds. Specifically, using #22, #D, #E, etc., the right member of (a) is equivalent to p(a)=O for some term p(a) (with k p(a)
-

Seq(a)

vtttlh(a)(a)t- 1 g P ( n i < t ~ P ) ~ ) I .

(The following also proves *26.6a' in which the "Seq(a)" of *26.6a is replaced by the right side of (a).) By "23.2, *23.4, "23.5 and *6.3, (b) o(E(x))=o

Vt,,,a(t)


Thence

(c) B(a) - a ~ r ~ . Furthermore, the first two hyps (1) Spr(a) (using 0 for the s in the second member) and (2) o(l)=O of *26.4a now hold. 11. We shall apply *26.4a with the present r~ and R taking A(a) as follows. A(a) :

3zVcr[Vta(t)
=I 3x,,,R(a*E(x))]. -

With this A(a), the concl. 3~Va~(~,3x,,,R(E(x))of *26.6a will follow from A(l) by *22.7. So it will suffice, assuming the two hyps. of *26.6a, to deduce the other four hyps. (3)-(6) of "26.4a. The next

62

FORMAL INTUITIONISTIC ANALYSIS

CH. I

three (3)-(5) we quickly obtain (with 0 for the z in (5)). To deduce (6), assume (a) ts(a)=O and ( e ) Vs{a(a*2s+l)=O 2 A(a*28+1)}; we must deduce A(a). Using (d), (a) and *23.6, we can put (for 3-elims.) (f) a=S(y). Then by (d) and (b) : (g) Vt,,,S(t) (t))) [*23.7] = P((6(y)*2a‘0’+1)*2(t)) [*22.9] ; SO Vta’(t) <~((8(y)*2a‘0’+’)*2(t)). So from B(a(O), z), 3~~~~R((8(y)*2~(~)+’)*~(x)). Assume x
=I-

More generally, the choices permitted for m(t) in a fan need not be a non-empty initial segment of the natural numbers. The choice law is then a function (T satisfying the first two hypotheses of: *26.7a. I- Spr(c) & Va[a(a)=O 3 3bVs{a(a*28+1)=0 3 s l b } ] & Va[o(a)=O 2 R(a) V -R(a)] & Vaa,,3xR(~(x)) I’zVaac$xx< zR(k(x))*

PROOF.CASE 1 : c( 1) #O. Then l a ~ c .Use * 10a. (The fan is empty and the theorem holds vacuously.) CASE 2: c(l)=O. Assume the four hyps. (1’)-(4‘) of *26.7a. I . We have the first two hyps. (1) and (2) of *26.4a, so we can introduce x and y as in I of the proof there and (a)-(7) will hold.

§6

T H E BAR THEOREM

63

11. Using ts(a)=OVts(a)#O and (2‘), Va3b[o(a)=O 1Vs{a(a*28+1)=0 I scb}]. Applying *2.2, we may assume for 3-elim.

Va[o(a)=O 3 Vs{o(a*28+1)=0 z, s
X26.8a. Va[Seq(a) 3 R(a) V -.IR(a)]& Va[Seq(a) & R(a) 3A(a)] & Va[Seq(a) & VsA(a*2*+1)3 A(a)] 5 ) {Seq(w)& Va3xR(w*E(x))3 A(w)}. X26.8~. VaVxVy[R(E(x))& R(E(y))5 ) X=Y] & Va[Seq(a) & R(a) z, A(a)] & Va[Seq(a) & VsA(a*26+1) z, A(a)] 3 {Seq(w)& Va3xR(w*E(x))3 A(w)).

64

FORMAL INTUITIONISTIC ANALYSIS

CH. I

x26.8d. VaVx[R(E(x))& VY~<~-R(E(Y)) 2 A(E(x))]& Va[Seq(a) & VsA(a*2S+1) 2 A(a)] 2 { v ~ C 1 ( 2 ; ) = P ( I ) ~ X H ~ Z [ R ( E (& X )vYy<xlR(z(Y))l ) 3 A(P(z))). DERIVATION O F X26.3a FROM X26.8a. Substitute 1 for w, and use Seq(l ) , “22.7 and “23.5. DERIVATION OF X26.8a FROM X26.3a. Assume the five hyps. of X26.8a. Using *22.8, “22.9 and “22.5, the four hyps. of X26.3a follow for R(w*a), A(w*a) as the R(a), A(a). So by X26.3a A(w*l), whence by “22.6 A(w).

6.12. Finally we consider Brouwer’s longer proof of the bar theorem given in the text of 1927, and slightly differently in 1924 (cf. 1924a) and 1954. Brouwer confined his attention (in 1924, 1927) to an R such that ( a ) ( E ! x ) R ( E ( x )of ) , which we now assume the uniqueness part (a)(x)(y)[R(E(x)) & R ( E ( y ) )--f x=y]. We take the case of the universal spread (though Brouwer was considering any spread), since the theorem for arbitrary spreads is a corollary “26.4. In this longer proof, Brouwer begins with the following interpretation. Consider any sequence number w securable (i.e. securableE) but not past secured; i.e. assume Seq(w) & (a)(Ex)R(w*E(x)).(Here we edit Brouwer’s proof slightly; he considered the unsecured sequences, which is a bit less convenient.) That w is securable means intuitionistically that there is a “proof (Beweisfuhrung)” of w’s being securable. Such a proof must rest “ultimately (in letzter Instanz)” (1924, 1927) upon the “atomic” facts in the situation (1954), which are only the truth of R(v) for certain sequence numbers v, and the relationships between sequence numbers and their immediate extensions v*2s+l (s = 0, 1, 2, . . .). So when a proof that w is securable is analyzed into its atomic inferences (“Elementarschlusse”), these will be of three kinds : 7-inferences (from 0 premises) that v is securable because R ( v ) , F-inferences (from K O premises) that v is securable because all v*2s+l (s = 0, 1, 2, . . .) are securable, and p-inferences (from 1 premise) that v*28+1 is securable because v is securable. (Brouwer did not use the term “7-inference”.) Disregardmg considerations of the symbolism, such a proof differs from the proofs in metamathematics (IM § 19 especially p. 83 lines 8-10, end § 24 especially bottom p. 106) only in that one of the present three rules of inference (namely F-inference) has infinitely many

96

THE BAR THEOREM

65

premises. The logical structure of such a proof is represented directly by taking it in tree form (IM p. 106), but now with infinitely many branches concurrent downward at each F-inference. (Brouwer in 1924, 1927 uses instead the sequence form I M p. 106, which brings the inferences or the propositions inferred into a linear ordering, which when F-inferences occur is a transfinite well-ordering. Thus he connects with his theory of well-ordered species 1918-9 I 3 3, 1924-7 111, 1954 3 4, which Vesley plans to formalize. Brouwer says “species (Spezies)” rather than “set”, as he used “set” for what he later called “spread”. - Brouwer 1954, and Heyting 1956 pp. 43-44 dealing with a fan, omit this unnecessary linearization.) Now we can use induction over proofs by q-, f- and (-inferences, i.e. the form of induction corresponding to the inductive definition of ‘provable formula’ (IM p. 83, top p. 260) when used with the present three rules of inference. (Proceeding downward in the trees of I M pp. 106-107 corresponds to proceeding to the left in trees as drawn here, e.g. in 6.5 Figure 1.) Brouwer’s interpretation (just described) of what is entailed in ‘w is securable (but not past secured)’ taken with the principle of induction over proofs, like his 1927 Footnote 7 (6.5 above), introduces a reversal of direction. We start from the hypothesis (a)(x)( y )[I?(@)) & R(E(y))-+ x=y] & Seq(w) & (a)(Ex)R(w*E(x)),with the prefix (a)(Ex) looking forward from w into the tree of diverging paths of growth of a choice sequence a. The interpretation (with the induction principle) then enables us to argue inductively, proceeding backward from many bases at the ends of the branches of a tree along converging paths to a single conclusion. But this time, instead of proceeding in the convergent direction in the original tree of the sequence numbers issuing from w and terminating in the w*E(x)’s for which R(w*&(x)),we are doing so in another tree constituting a proof of w’s being securable. By induction over this latter tree, we readily establish that the (-inferences are eliminable, so we obtain the bar theorem. To formalize this proof of the bar theorem, we shall require a postulate expressing Brouwer’s presupposition,

( 4 ( 4 ( Y ) [ R ( W )W Y ) ) (***)

+

x=Yl

& Seq(w) & (a)(Ex)R(w*&(x)) --f {there is a proof of ‘wis securable’ by q-, F- and (-inferences}.

To formulate this postulate, we must find a way of rendering the

66

FORMAL INTUITIONISTIC ANALYSIS

CH. I

notion of a proof by q-,f-and 5-inferences into the formal symbolism. Agreement on this must necessarily be reached outside the formalism. All the propositions in a proof of securability by q-, f- and 5inferences are of the form that a number 2) is securable. So by replacing (the occurrences of) the propositions ‘z, is securable’ in the prooftree by (occurrences of) the numbers 2) which those propositions are about, we obtain a tree isomorphic to that proof-tree, while escaping any difficulty from the lack of formal metamathematical symbols in the system. (In 1924 Brouwer dealt at once with the well-ordered species of the finite choice sequences in the order in which their securability is established by the proof. It was in 1924a that he began first to speak of the proof as a well-ordered species of inferences.) Now we must find a way to talk about the latter tree of (occurrences of) sequence numbers. But indeed it constitutes simply a mapping of sequence numbers onto the vertices of a certain tree of terminating choice sequences. This, when viewed in the forward (divergent) direction, is precisely a spread, where as in 6.9 7 4 w is to be correlated to the empty sequence of choices, i.e. to the sequence number E(0) = 1. (Of course here we have no interest in the finite sequences constituting the elements of the spread as such.) In specifying this spread by a choice law and a correlation law (6.1, 6.9), instead of sometimes terminating sequences of choices or forbidding some choices (while allowing others), it is simpler to correlate 0, which isn’t a sequence number, to the choices which would thereby be disallowed (cf. 6.9 7 3 (b)). By this device, the choice law u becomes trivial, and a correlation law p will specify the spread. More explicitly, there are two spreads, the one specified directly by p which overlies the universal spread of all the choice sequences and has sequence numbers and 0’s as the correlated objects, and the other which is obtainable from that by disallowing the choices to which 0’s are correlated and is the spread isomorphic to the proof of ‘w is securable’. But we are viewing the latter spread in the backward (convergent) direction, as is essential to Brouwer’s argument. Now we can formulate conditions that p thus represent a proof of securability. When Seq(a) & p(a)>O, then p(u) =z, where ‘v is securable’ is the result of an q-, f- or [-inference as described above; etc. That the proof represented by p is of ‘wis securable’ is expressed by p( 1 ) = w. Altogether, the “local” requirements P ( w , p, R) on p that it represent a proof of ‘wis securable’ are as follows.

§6

67

THE BAR THEOREM

P ( w , p , R ) : p ( l ) = w & (a){Seq(a)&p(a)>O + [R(p(a))& (~)p(a*2"+~)=0] v (s)p(a*28+1)=p(a)*28+1 v [Seq(p(a*2)) & (Es)p(a)=p(a*2)*28+1& (s)p(a*2"2) =O]} & (a){Seq(a)&p(a)=O + (s)p(a*28+fl)=O}. The principle of induction over proofs of 'w is securable' by q-, Fand [-inferences, as we have now represented such proofs, is (A)$(w,p, A ) where s ( w , p, A ) is as follows.

S ( w 4 :(4{Seq(a) &p(a)>O&

(s)[A(p(a+28+1)) v p(a*28+1)=0] + A ( p ( a ) ) } + A ( w ) .

Let P(w, p, R) and g(w, p, A) be the correspondingly constructed formulas, for any formulas R(a) and A(a). The conclusion of (***) is now (Ep){$(w,p, R) & (A)$(w,p, A ) } . Thus we are led to the following axiom schema, expressing (as nearly as we can in the absence of predicate variables) the presupposition for Brouwer's longer proof. VaVxVy[R(E(x)) & R(E(y)) zi x=y] & Seq(w) & Va3xR(w*E(x)) 3 3 d P h P, R) 9(w*PI A)}. With X26.9 as an axiom schema, we can prove each axiom by X26.3c, formalizing Brouwer's longer proof, thus. We take 1 for the w (since in X26.3~we specialized to 1, though Brouwer didn't), and for the A(a) the following formula.

X26.9.

A'(a) : 3yy..Ih(a)R(IIi
Assume (a) Va3!xR(E(x)). If from (a) and the axiom by X26.9 we deduce A'(I), we will have what we want by taking 1 for the c (using lh(l)=O, Seq(l), "22.7). Using (a), we have the three hyps. of that axiom, so our problem reduces to seeing that from (a), (b) p ( l ) = 1, (c) Va{Seq(a) & p(a)>O 3 [R(p(a)) & Vsp(a*28+1)=0] V Vsp(a*28+1)= p(a)*28+1V [Seq(p(a*2)) & 3sp(a)=p(a*2)*28+1 & Vsp(a*28+2) =O]} and (a) Va{Seq(a) & p(a)=O zi Vsp(a*2S+l)=O} we can deduce (A) Va{Seq(a) & p(a)>O & Vs[A'(p(a*28+1)) V p(a*2s+l)=0] 1 A'(p(a))}. By ind. on x, using (b)-(d): (e) p(E(x))>O 2 Seq(p(E(x))).Using (a) in Lemma 5.6: (f) Va[Seq(a) zi R(a) V lR(a)]. Using (for p(a)>O) "150 (p(a))i)

v

with (f)* *23*4 (g) Seq(a) zi 3yyO and (j) Vs[A'(p(a*28+1)) V p(a*28+1)=0], and undertake

68

FORMAL INTUITIONISTIC ANALYSIS

CH. I

to deduce A'(p(a)). Using (8) we have two cases; and using (c) we have three subcases for use in the second case. CASE 2: (k) l~yy0, so by (j): (u) A'(p(a*2)). By (t) with (k) : (v) ~3yy
Then (g) p(a)>O

=I p(a)=w*a.

(For more detail, cf. (A) in 14.1.)

§7

BROUWER’S PRINCIPLE

69

The first and third conjunctive members (h) and (i) of $(w, p, R) are immediate from (f). Toward the second (j), assume Seq(a) & p(a)>O. Using (e), we have two cases. CASE 1: R(w*a). Then by (g) R(p(a)), and by (f) Vsp(a*2s+l)=O. CASE 2: TR(waa). Using this with p(a)>O and (f), Vsp(a*28+1) = w*(a*28+1) = (w*a)*28+1 = p(a)*28+1 [by (€91Toward $(w, p, A), assume (k) Va{Seq(a)& p(a) >O & Vs[A(p(a*28+1)) V p(a*28+1)=0] ZI A(p(a))}. To deduce A(w) by use of X26.3c, it will suffice to deduce the hypotheses (d), (A) and (B) of X26.3~with R(w*a), A(w*a) & p(a)>O as the R(a), A(a). Using *149a: (1) p(E(x))=O ZI 3yy,,[p(E(y))=0 & Vz,,,p(E(z>) >O]. DEDUCTION OF (A). Assume Seq(a) & R(w*a). Put a=E(x). By (f), Vsp(a*28+1)=0. To apply (k), we still need (m) p(a)>O. Assume p(a)=O. From ( I ) , assume (n) y(x & p(E(y))=O & Vz,,,p(E(z))>O. By p(E(y))=O with (f) (and w>O from (b)), y>O and R(w*E(yLl)) V p(E(yLl))=O. But p(E(y-l))=O is excluded by (n) with y>O. Thus R(w*E(y:l)) & R(w*E(x)) & y~ 1 O. DEDUCTION OF (B). Assume Seq(a) & V s ( A ( ~ * ( a + 2 ~ & + ~p(a*28+1)>0). )) By (i) : ( 0 ) p(a)>O. Using (g), VsA(p(a*28+l)).So by (k), A(p(a)),whence by (0)and (g), A(w*a) & p(a)>O. REMARK 6.3. In Brouwer’s “bar theorem” 1954 p. 14, he comes out with a (linear) well-ordering, but this entails an induction principle similar to that in our X26.3 etc. At no other place we know in Brouwer’s writings is a similar proposition stated explicitly as a theorem, except in 1924 $ 1 (not cited from 1954), where the hypothesis of 1954 p. 14 or X26.3 is replaced by one entailing it via Brouwer’s principle 3 7 below (making the theorem false classically). However, from 1954 p. 14 he cites the passage pp. 63-65 of 1927 proving that version. We do not consider a proposition to be a version of the bar theorem unless, from an hypothesis giving (or entailing) securabilityE or barrednesss, it concludes securability1 or barredness1 or a wellorderedness property or an induction principle; for, that is what the labor in Brouwer’s proof in 1954 or 1927 is devoted to obtaining.

Q 7. Postulate on correlation of functions to choice sequences (Brouwer’s principle). 7.1. Suppose that to each choice sequence a(O),a(1),a(2), . . . a natural number b is correlated (“zugeordnet”). Since intuitionistically choice sequences are considered as continually

70

FORMAL INTUITIONISTIC ANALYSIS

CH. I

growing by new choices rather than as completed, this correlation can subsist intuitionistically only in such a manner that at some (finite) stage in the growth of the sequence a(O),a(l), 4 2 ) , . . . the correlated number b will be determined (effectively). That is, intuitionistically the b must be determined effectively by the first y choices a(O),. . . , a(y- 1) of CL for some y (depending in general on those choices). We call this “Brouwer’s principle (for numbers)”. Brouwer maintained it in 1924 § 1, 1927 3 2 and 1954 p. 15, in the course of proving his version of the fan theorem. He also maintained it in 1918-9 I end 1 and 1924-7 I end 5, in arguing (without using Cantor’s diagonal method, I M 3 2 bottom p. 7) that the universal spread C cannot be mapped 1-1 onto a subset of the natural numbers A ; taking this with the usual definition of > for cardinal numbers (IM p. lo), and an obvious 1-1 mapping of A onto a subset of C , it is a rather trivial theorem intuitionistically that the cardinal number of C is greater than the cardinal number of A . Brouwer’s principle must be made explicit for our formal development. One might do so by incorporating it explicitly into each statement that a b is correlated to each a. Then the simple statement without the added explanation would not be used. In any case, Brouwer elected rather to consider the determination of the correlated number b by an initial segment a(O), . . ., a(y- 1) of a as implicit in his use of the mode of expression ‘a b is correlated to each a’.Then formulas expressing his principle, or something entailing it, must be postulated for the intuitionistic formal system. For the principle is false classically, e.g. in the case of the classically-admissiblecorrelation of 0 to the sequence consisting of all 0’s and 1 to all other sequences. So (with 9.2) the principle is independent of the other intuitionistic postulates, which are all true classically. Brouwer’s intention that the determination of the b be effective is attested by his phraseology “the algorithm of the correlation law (dem Algorithmus des Zuordnungsgesetzes)” (1924 1, 1927 3 2, italics his). Toward formulating the necessary postulate, consider first what the algorithm must do. I t must decide for each initial segment a(O),. . ., a(y-1) of a choice sequence a whether from that segment it will produce the correlated number b. When this decision is affirmative (as it must be for some y), it must produce the b. It will be convenient,

$7

71

BROUWER’S PRINCIPLE

as in $ 6 , to represent each initial segment a(O),. . ., a(y - 1) of a choice sequence 01 by the sequence number Z(y). Now we can combine the two operations the algorithm must perform into one function z, which operates on sequence numbers G(y), thus. As the initial segment a(O), . . ., a(y-1) of a choice sequence 01 grows (y increasing), z(&(y)) remains 0 so long as the algorithm does not accept a(O),. . ., a(y- 1) or G(y) as a basis for producing the b ; but when it first does, then z[E(y)) = b+ 1. (This representation was selected in January 1956 in the course of the present study. It has meanwhile been used in Kleene 1959a. Using the terminology of that paper intuitionistically, Brouwer’s principle for numbers can be stated: Each type-2 functional is countable.) We are not saying that, as y increases, z(Z(y)) remains 0 only as long as 01(0), . . .,a(y-1) or G(y) does not “determine” the b in the sense that the same b is correlated to all a’s having a(O), . ., a(y- 1) as initial segment. Intuitionistically, the correlation is first established by the algorithm itself, i.e. z(E(y)) changes from 0 when G(y) determines the b effectively by the algorithm. This does not exclude the possibility that we may be able to prove about the algorithm that, in some case, it puts off producing the b to beyond the first y at which, for all ways of extending the choices a(O),. . ., a(y- l ) , it will ultimately produce the same b. (Cf. Brouwer 1924 $ 1 end T[ 1, 1927 3 2 end T[ 1 .) We give an example (from .Kleene 1959a* p. 83) in which such postponement by the algorithm is essential, if z to express an algorithm must be general recursive (Church’s thesis). Let a b be correlated to each a by the rule that b = 0 if p1(01(0), 01(0), 01(1)) and b = a ( l ) + l otherwise (IM p. 281). If the production of the 6 is always put off until y = 2, a primitive recursive z suffices, namely

.

1 if F~((u)o;l, ( u ) 0 ~ 1( ,u ) l ~ l &lh(u)=2, ) (u)l+l if 7’1((u)o~l, (u)o‘l, ( u ) l ~ l &lh(u)=2, ) 0 otherwise.

But if the b were always produced as soon as classicallyit is determined, then we would have (y)F,(x, x , y ) = t(Yfl)= 1, so by I M p. 283 t would not be general recursive. It will be convenient to take z(&(y)) = 0 after the first y for which z(G(y)) > 0, so that we will have (a)(E!y)z(E(y))>O. Had we merely a z with (a)(Ey)z(E(y))>0, we could get one with (a)(E!y)z(Z(y))>O,

72 call it

FORMAL INTUITIONISTIC ANALYSIS 71,

CH. I

by putting

The formulas by which we shall presently state Brouwer’s principle will thus be interderivable with ones expressing it without taking the y to be unique, using Lemma 5.5 (a), *149a (with “159) and *174b. Next, consider the hypothesis that a natural number b is correlated to each choice sequence a. This asserts the existence of a function from a’s to b’s, i.e. a type-2 functional F. We might add functional variables to our formal symbolism to formalize the hypothesis of Brouwer’s principle. But we wish to keep the symbolism as simple as possible. At least as far as we go in this monograph (including Vesley’s Chapter 111), all we need is that, for each a , b = F(a) bear some given relation A(a, b) to a. Then we can take the hypothesis to be simply (a)(Eb)A(a,b). As we understand intuitionism, (a)(Eb)A(a, 6) means exactly that there is a process F by which, given any a , one can find a particular b such that A ( a , b). Thus an axiom of choice (a)(Eb)A(a,b) + (EF)(a)A(a,F(a)) holds intuitionistically. (Other axioms of choice, corresponding to the prefixes (x)(Ey) and (%)(Ear),were introduced above as *2.2 and X2.1.) Brouwer’s principle says that any F can be represented by a z in the manner just described. Then the t can replace the F. So the F has only a transitory role, between the initial hypothesis (a)(Eb)A(a,b) and the appearance of the t. By combining the principle of choice implicit in the intuitionistic meaning of the prefix (a)(Eb)with Brouwer’s principle for numbers as he directly stated it, we obtain a version of Brouwer’s principle, (each instance of) which can be expressed in our formal symbolism as it stands. Thus we arrive at *27.2 as our formalization of Brouwer’s principle for numbers. (In 1957* (4) we already stated in like manner a consequence of Brouwer’s principle.) As another case of Brouwer’s principle (the case for functions), if to each choice sequence a(O),a( l), a(2), . . . a function /? is correlated, then intuitionistically each value B(t) of /? must be determined effectively by t and some initial segment a(O), . . ., a(y-1) of a. This can be handled as the correlation of a number b = j3(t) to each of the choice sequences t , a(O),a ( l ) , a(2), . . .. We originally supposed that this would be required in formalizing Brouwer’s proof of his uniform continuity theorem (1923a Theorem 3,

§7

73

BROUWER’S PRINCIPLE

1924 Theorem 3,1927 Theorem 3,1952 p. 145, 1954 5 6). But Vesley in

3

15 of Chapter I11 has managed using only Brouwer’s principle for numbers for each value B(t) of j3 separately. Still, it seems to us that the intuitionistic reasons for accepting the principle for numbers apply equally to the principle for functions (even though we do not know of an explicit affirmation of it in Brouwer’s writings). So we elect to postulate the latter, as X27.1 (for the intuitionistic, but not the basic, system) ; cf. 7.15.

7.2. For x27.1, u, (4 and 7 are any distinct function variables, t and y are any distinct number variables, and A(u, p) is any formiila not containing T free. x27.1. Va3PA(u, p)

ZI 37Va{Vt3!y~(2t+l*ti(y)) >O & VP[Vt3~~(2t+l*E(y))=p(t)+1 3 A(K,p)]}.

Thence we can specialize to Brouwer’s principle for numbers. >O & *27.2. I- Va3bA(u, b) 5) 3~Vct3y{~(E(y)) VX[T(~(X)) >O 5) Y=X] & A(a, z(E(y))2 1)). PROOF. Assume Vu3bA(a, b). By *0.6, Vu3pA(a, p(0)). Applying X27.1, omitting 37 preparatory to 3-elim., and using V- and &-elim.,

(1) (2)

Vt3!y7(2t+l*ti(y))>o, vp[vt3yT(2t+l*a(y))=P(t)+ 1 3 A(% P(0))l.

Applying *2.2 to (1) (unabbreviating 3!y by IM p. 199), and omitting 3u from the result (preparatory to 3-elim.), (3)

Vt[7(2t+l*ti(u(t)))>o & Vx[7(2t+l*ti(x))>o 3 .(t)=x]].

By V-elim. from (2) (using At7(2t+l*E(u(t)))1 1 for (4)

p) and XO. 1,

vt3yT(2t’1*a(y))=(T(2tf1*&(U(t))) L 1) + 1 3 A(u, T(~*E(u(O)))L 1).

+

By V-elim. from (3) and *6.7, 7(2t+l*E(u(t)))=(7(2t+l*E(u(t)))2 1) 1, whence Vt3y~(2t+l*E(y)) =(7(2t+l*a(u(t)))A 1) 1. Thence by (4), A(a, 7(2*E(u(O)))2 I), and by XO. 1,

+

(5)

A(a, {At7(2*t)}(E(u(O))) 1).

Also by V-elim. from (3) (with 0 as the t) and xO.1, (6)

{At7(2*t)}(ti(u(O))) >O & VX[{/?~T(~*~)}(E(X)) >O ZI u(0) =XI.

74

FORMAL INTUITIONISTIC ANALYSIS

CH. I

The formula of "27.2 follows from (6) and (5) by &- and 3y-introd., the h-elim., Vu- and 37-introd., the 37-elim., and 3-introd. A third case of Brouwer's principle (the case for decisions) applies when A and B are classes such that each choice sequence a belongs to A or to B . Then intuitionistically one of A and B to which a belongs must be determinable from a sufficient initial segment a(O),. . .,a(y- 1) of a.

*27.3. k Va(A(a) V B(a)) 3 37vCr3y{vX[T(E(X))>O 2 Y=X] & WW & M Y ) ) = 1) v ( B ( 4 & .MY)> = 2 N -

PROOF.Assume Va(A(a) V B(a)). By V-elim., A(a) V B(a). CASE1 : A(a). Using *loo, A(a) & O=O, whence by V-introd. (A(a) & O=O) V (B(a) & O= l ) , whence by 3-introd. 3b{(A(a) & b=O) V (B(cr) & b= 1)). CASE 2: B(a). Similarly. - By V-introd., Va3b{(A(a) & b=O) V (B(a) & b= I)}. By "27.2, 3~Va3y{~(E(y)) >O & VX[T(E(X)) >O 3 Y=X] & {(A(a)& T ( ~ ( Y ) ) 1 =0) V (B(a) & ~ ( h ( y ) ) 1 = 1))). Via 3-, V-, &and V-elims. and introds. and "6.7 in proper sequence, and 3-introd., we obtain the formula of *27.3. Using *2.2 and *6.7, the converse of x27.1 can be proved in the basic system. So in X27.1 the outermost 3 can be strengthened to N (analogously to x2.1); call it then *27.la. Likewise (usually more simply), "27.2-*27.10, "27.15 can be strengthened to *27.2a-*27.10a, *27.15a ( B x A , C & B I A , D & C & B 3 A becoming B-A, C 3 ( B - A ) , D&C2(B-A)). 7.3. In 7.2 we introduced Brouwer's principle for the universal spread. It applies to other spreads. "27.4. k Spr(o)& Va,,,3pA(a, p) 2 37Vaa,,(Vt3!y~(2tf1*Ft(y))>O & Vp[Vt3y.c(2t+l*E(y))=P(t)+ 1 xAA(a,p)]}. *27.5. t- Spr(a) & Va,,,3bA(a, b) 3 3~Va,,,3y{~(E(y)) >O & VX[T(E(X)) >O 3 Y=X] & A(a, .(E(y)) A 1)). "27.6. k Spr(o) & Va,,,(A(a) V B(a)) 3 3~Va,,,3y{Vx[~(E(x))>O 3 Y = X l &{(A(a)&.(.(Y))=1) v (B(4 WE(Y))=2)1).

PROOFS. *27.4. CASE 1 : o(l)#O. Then

IREG. Use *10a. CASE 2: Assuming also Spr(o), we can introduce x and y as in I of the proof of *26.4a and (a)-(3) will hold. Assume Vcra,,3pA(a, p). By V-elim. ayes 2 3pA(cry,p), whence by (E) and V-introd. Va3pA(ay,p).

o(l)=O.

$7

75

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Using x27.1 for A(a,, p) as the A(a, p), and omitting 37 preparatory to 3-elim., Va{Vt3!y~(2t+l+~(y)) >O & VP[Vt3y~(2t+l+ti(y)) = p(t)+ 1 2 A(a,, p)]}. Thence, assuming a~cs and using V-elim. and (q), Vt3 !~~(2t+l*ti(y)) >O & Vp[Vt3~~(2t+l*Ei(y))=P(t)+ 1 3 A(a, p)]. "27.5, *27.6. Similarly from "27.2 and "27.3, respectively, or successively from *27.4 as *27.2 and "27.3 from X27.1. 7.4. Brouwer combined his principle (for numbers) with what we already have in *26.6 (or *26.7) t o state his "fan theorem" thus: If to each element 6 of a fan a natural number ba is correlated, then a natural number z can be determined such that, for each 6 , the correlated number ba is completely determined by the first z choices of the choice sequence a generating 6. We establish this now. First we deal directly with the fan of the choice sequences a (cf. 6.10). *27.7. k VaB,,,3bA(a, b) 3 3zVaB(a)3bVyB(y){VxX
=a(x) 2 A(y, b)}

where B(a) is Vxa(x)
PROOFS. *27.7. As in I of the proof of *26.6a, we introduce cs so that (a)-(c), (1) and (2) hold. Assume (d) Va,,,,3bA(a, b). Applying *27.5 with (1) and (c), and omitting the 37 for 3-elim., (e)

V ~ B ( , ) ~ Y { ~ >O ( ~ (& YVX[+(X)) )) >O 2 Y=X] & A(a, .(E(y))

2

1)).

We have the first hyp. of *26.6a for .(a) >O as the R(a) by * 159 or by "15.1 and *158, and the second using (e). So, applying *26.6a and omitting 32 for 3-elim.,

(f) VaB(a)'xxO* Toward the concl. of "27.7, assume B(a). Using (e), assume for (&- and) 3-elim. (h,) tlx['~(E(x))>O3 y1=x]. Assume B(y). Using (e), assume (g2) .(y(Yz))>O, (hi?)Vx[.(W)) >o 3 yz=x1 and (4) A(y, .(Y(yz)) 1); and using (f), assume (j) xO. Assume Vx,<,y(x)=a(x). Thence using (j) with *B19 and X23.1 y(x)=ti(x). By (k) and (hz) y2=x, so Y(y2) = y(x) = ii(x), whence by (gz) and 011) y1=x, so y(yz)=E(yl). Hence by (4,A(y, ~(E(y1))2 1). Now we V- and 3-introds. and &- and 3-elims. in proper can carry out the 3-, order, with ~(E(yl))-l as the b.

76

FORMAL INTUITIONISTIC ANALYSIS

CH. I

"27.8. Similarly from *26.7a. Now consider the fan of elements 6 determined by a choice law (I (satisfying the first two hypotheses of *27.8) and a correlation law p (cf. 6.1, 6.9). That a b (such that A ( 6 , b ) ) is correlated to each element 6 can be expressed in the formal symbolism by V6{3aa,,Vt8(t)= p(E(t)) 2 3bA(6, b)}. Via "96, X O . l and Lemma 4.2, this is equivalent to Vaa,,3bA(Itp(~(t)),b). The fan theorem for such a fan is expressed by "27.8 with A(iltp(E(t)),b) as the A(%, b). 7.5. The postulates and results thus far obtained place us in a favorable position for following Brouwer's development of his analysis. This is done up to a point by Vesley in Chapter I11 below. We continue here with investigations concerned more with foundations. Using Brouwer's principle, we can dispense with the hypothesis Va[Seq(a) I) R(a) V -1R(a)1(Va[o(a)=O 2 R(a) V yR(a)]) in our previous version *26.6a ("26.7a) of the fan theorem.

*27.9.

1 V O I ~ ( ~ ) ~ X R2 ( E3zVa,~,,3xx,,R(E(x)) (X))

where B(a) is Vta(t)
2 3bVs(a(a*2S+1)=0 2 sib)] & Vaa,,3xR(E(x)) I) 3~Va,,,3x,,,R(iii(x)).

1 Spr(a) & Va[o(a)=O

PROOFS. "27.9. This will follow from *27.11 as *26.6a from "26.4a; the following proof is more direct. We proceed as in the proof of *27.7 down through (f), for R(E(b)) as the A(a, b). For 3-elim. from Lemma 5.3 (b), assume

(g)

Y(o)=

8~vk[y(k')=y(k)*(pk exp

+maxt5,(~)P(t))1

(cf. Kleene 1956 Footnote 8). By ind. on k as follows,

(h)

B(a) & x < k 3 E(x)ly(k).

IND.STEP. Assume B(u) & x l k ' . CASE 1 : xO. Thence assume (j) xO; and using (e), assume (k) Vx[~(Ei(x)) >O 2 y=x] and ( I ) R(E(~(ti(y)) A 1)). By (j)

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77

and (k) y=x, so by ( I ) : (m) R(k(~(E(x)):l)). By *HJ and (h) with (i) and (j): (n) T(E(X))LI <max,,,(,,.c(s)~ 1. Using (n) and (m), we obtain the concl. of "27.9 by &-, 3-, 3- and V-introds. and &- and 3-elims. in proper sequence, with ~(E(x))-l as the x and maxs,y(z)T(s)- 1 as the z. "27.10. From "27.9 as *26.7a from "26.6a. We can similarly weaken the hyp. of the bar theorem *26.4a in the case the spread is a fan. These results, with those in 6.7-6.9, 7.6 and 7.14 give a survey by proof-theoretic methods of intuitionistic vs. classical forms of the bar theorem. & "27.1 1. I- VaB(a)3xR(?i(x)) VaVx[Vtt<+(t)
where B(a) is Vta(t)
PROOFS. "27.1 1. We proceed as in the proof of "27.9 through (h). Let R'(a) be 3u,,.(,)[o(u)=o&1h(a)+ =T(u) 8L vii~min(lh(a),lh(u))(a)i=(U)il~

Now assume the remaining two hyps. ( 0 ) and (p) of "27.1 1. We shall apply "26.4a with R'(a) as the R(a). Of the remaining hyps. (3')-(6') of *26.4a, we have (3') by Remark 4.1 (or #D, #E etc.). Toward (4'), assume QEG, whence (c) gives (i) B(a), so we may further assume (j). Using (h)-(j), (b), *6.7, "23.2 etc. we get R'(ii(~(C(x)):l)) with E(x) as the u. To deduce (5') from (0)and (a), (b) etc., it will suffice (using (a), "23.6) to deduce o(<(x~))=O& R'(g(x1)) 3 R(G(x1)). So assume ~ ( K ( x 1 ) ) = & 0 R'&(xl)), and for 3-elim. (simplifying): (q) umax(xl, lh(u)) 3 a(i)=O. So using first "B19 and X23.1, then also *22.3, "6.7 with "22.1, etc.: ( s ) E(x1)=<(XI) & ii(lh(u))=u. Thence using also o&(xl)) =O and o(u)=O with (b), (c) and (r): (i) B(a). So as in the proof of "27.9 we can assume (k) and ( I ) . By (s) with (4): (t) xl+l=~(E(lh(u))).So by

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FORMAL INTUITIONISTIC ANALYSIS

CH. I

(k) y=lh(u), whence by (1) R(E(-r(E(lh(u)))Al)),whence by (t) and "6.3 with ( s ) , R(iG(x1)). - Finally (6') follows from (p) and (b). "27.12. Assuming the six hyps. (1")-(6") of "27.12, we take over I and 11 of the proof of *26.7a Case 2. 111. We shall apply "27.11 for the p of (0) with R(y(a)) as the R(a) and o(a)=O 2 A(y(a)) as the A(a). B. The first hyp. of *27.11 (= the second of *26.6a) is verified as before. For the second, assume R(y(E(x))).By (y) and (5") A(y(E(x))),whence o(E(x))=O 3 A(y(E(x))). For the third it will suffice, assuming (1)

Vs{s
to deduce o(E(x))=O 2 A(y(E(x))). Assume (K) o(E(x))=O. By (y) and (6") it will suffice, assuming (A) o(y(ii(x))*2s+1)=0, to deduce A(y(~(x))*28+1). From (A) by (x) and (C): (p.) o(Z(x)*28+1)=0. By (6) with (x)and (p) : (v) s
7.6. Using (X27.1 via) *27.2, we can establish a fifth intuitionistic version of the bar theorem (cf. X26.3a326.3d). *27.13. t VaVx[R(E(x))2 Vy,,,R(E(y))] & Vcr3xR(E(x))& Va[Seq(a) & R(a) 2 A(a)] & Va[Seq(a) & VsA(a*2s+l) ZI A(a)] ZI A( 1).

PROOF.Assume the four hyps. (1)-(4) "27.2, and omitting 3-r for 3-elim., (a)

of *27.13. Using (2) with

Vdy{-r(~(y)) >O & Vx[-r(E(x)) >O 2 y=x] & R(E(-r(E(y))A 1))).

Let R'(a) be Ih(a)=<(lh(a))o,(lh(a))l) 8~7(ni<(lh(a))op~a)1)=(lh(a))l+ 1, so

(b)

R'(E(x)) -x=<(x)o, (x)i>&-r(E((x)o))=(x)i+1

[*23.4 with * 19.2 etc.]. We shall apply X26.3a with R'(a) as the R(a). We have the first hyp. of X26.3a by Remark 4.1 or #D. The second with (y, -r(E(y))2 1) as the x we obtain from (a) and *25.1, *6.7. Toward the third, assume Seq(a) & R'(a), and a=E(x), so we have the right side (c) of (b). Using (a), assume for 3-elim. (d) Vx[-r(E(x))>O y=x] and ( e ) R(E(-r(E(y)):l)). By (c) with (d) y = ( x ) ~ , so ~ ( E ( y ) ) 2= 1 -r(E((x)o))'l = ( ( ~ ) 1 + 1 ) 2 1 = (X)I [*6.3] 5 x [*19.2,

=I

97

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BROUWER'S PRINCIPLE

*19.6]. Hence by (e) and (l), R(E(x)), i.e. R(a). So by (3), A(a). The fourth is (4). DERIVATION OF X26.3a FROM "27.13 (as postulate in place of X26.3, without X27.1). Assume the four hyps. (1)-(4) of X26.3a. Pick R'(a), A'(a) so that via "23.4 (a) (b)

R'(E(x)) A'(Z(x))

--

3t,,,R(h(t))> A(&(X)) V 3tt<xR(E(t)).

We shall apply "27.13 with R'(a), A'(a) as the R(a), A(a). The first hyp. of *27.13 is immediate. The second follows from (2). Toward the third, assume Seq(a) & R'(a), and put a=h(x). Using (l), R(a) V yR(a). CASE 1: R(a). Then A(a) by (3), whence A'(a) by (b). CASE 2: l R ( a ) . Then by R'(a) with (a) 3tt<,R(E(t)), whence A'(a) by (b). - Toward the fourth, assume Seq(a) & VsA'(a*2*+1), and put a=E(x). By (1) with *150, 3tt<,.R(E(t)) V -,3tt<,
=

3

7.7. Brouwer's principle for numbers can be stated (similarly to the fan theorem in "27.7) without mentioning the algorithm z explicitly. "27.15. t Va3bA(a, b) 2 Va3y3bVy{VxX<,y(x)=a(x)

A(y, b)}.

This formula appears in Kreisel 1962 Remark 9, where it is misnamed "the bar theorem"; cf. Remark 6.3. (Kreisel seems to be suggesting it as a postulate to replace the postulate "27.7 of Kleene 1957; but by Corollary 9.9 below, "27.7 would not then be provable lacking X26.3.) I t is Heyting 1g3oa 12.22, allowing for considerable differences in the symbolism. In the hypothesis Heyting explicitly uses a function from choice sequences to natural numbers, which our symbolism doesn't provide. In "27.15 our practice (initiated in 1957) is followed under which the existence of the correlation of b to 01 is implied by the simple prefix V d b , or for a fan VcrB,,,3b etc. (cf. 7.1).<- The proof of *27.15

80

FORMAL INTUITIONISTIC ANALYSIS

CH. I

from X27.1 via "27.2 (without X26.3) is straightforward by reasoning used in the proof of "27.7. We do not see how "27.2 could be proved from "27.15 as a postulate replacing X27.1. 7.8. In 7.1 we explained Brouwer's principle on the basis of the intuitionistic conception of choice sequences as continually growing rather than as completed objects. This applies in talking about the totality of choice sequences a making up the universal spread or any other spread. It is true that the property ~ E Cof being a choice sequence a of a spread involves all the values of ar (cf. 6.9). But it involves them only in such a way that, whenever at a given stage G(x) in the growth of 01 the property a ~ is o satisfied "thus far", i.e. o(E(x))=O, the growth can continue so that the property will be satisfied at every subsequent finite stage, and thus so that ~ E itself O will be satisfied. But a species (not a spread) of choice sequences a can be characterized by a property C(a) involving all the values of a not merely in that manner (cf. 6.12 T[ 3). In reasoning about the members a of such a species, C(a) functions as a non-constructive hypothesis, which so to speak augments what the intuitionist can do "by himself". (This theme will be developed in Chapter 11, especially 8.6.) We now show that Brouwer's principle as formulated in "27.4-*27.6 for any spread (characterized by a ~ o cannot ) be extended consistently with the basic system to arbitrary species (characterized by C(a)). Indeed, using only the intuitionistic Postulate Groups A-D : "27.16. I- l[Vac(,,(A(a) V B(a)) 3 ~TV~,(,,~Y{VX[T(E(X)) >O 3 y=x] & {(A(a)& T M Y ) ) = 1) v ( B ( 4 i?L .MY)) =2)})1 when A(a) is Vxcc(x)=O, B(a) is lVxa(x)=O, and C(a) is A ( a ) V B(a). PROOF.Vac~(,)(A(a)V B(a)) holds by the principle of identity * 1. So via 3-elim. it will suffice to deduce a contradiction from

Vac(,,3y{Vx[~(E(x)) >O 2 y=x] & {(A(a)& T ( ~ ( Y ) ) =1) V ( B( 4 & +(Y)) =2)H. Assume Vxal(x)=O (Lemma 5.3 (a)), so A(al), hence C(a1). Applying (a) and omitting 3yl for 3-elim., (a)

(b) G l ( Y 1 ) ) = 1 . Now assume Vxa2(x)=x'-y1, (c)

x c y 1 3 ~z(x)=o,

so (by "6.11 etc.)

(d)

) x>yi 3 a 2 ( ~>O.

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81

By (d), az(yl)#O, so ~ V x a ~ ( x ) = O i.e. , B(az), hence C(a2). Applying (a) and omitting 3y2 for 3-elim., (e) VX[T(G(X))>O 3 yz=xl, (f) 7(G(y2))=2. But by (c) (and *B19, X23.1), a2(yl)=<(yl), so by (b):

(g) 4 d Y l ) ) = 1. By (g) and (e) yZ=y1, so by (f) and (g) 2=1, contradicting 2 f l . Thus, with a particular choice of A(a), B(oc), C(a), we have refuted the modification of *27.6 obtained by suppressing Spr(a) as hyp. and changing KEG to C(a). The corresponding modifications of *27.4 and h27.5 are likewise refutable, since that of *27.6 is deducible from each of them in the way that *27.3 was deduced from *27.2 and thence from X27.1.

7.9. Now (through subsection 7.14), we shall explore consequences of Brouwer’s principle which contradict classical results. We begin in the next subsection with refutations of laws of classical logic. Demonstrations that various laws of classical logic are not provable in intuitionistic logic have been given in a number of ways. For the propositional calculus, cf. Godel 1932, Gentzen 1934-5 (or IM pp. 479-486), JaSkowski 1936 (and Pil’Cak 1952,G. F. Rose 1953), Wajsberg 1938, Stone 1937-8, Tarski 1938, McKinsey-Tarski 1946, 1948 (which uses ideas that appeared in Skolem 1919 ; cf. Skolem 1958, Scott 1960),Scott 1957, Kreisel-Putnam 1957, Schmidt 1958, Harrop 1956, 1960, Kleene 1962a. For the predicate calculus, the methods of Gentzen 1934-5 were applied to this end by Curry I950 and Kleene 1948 and I M 580. Mostowski 1948 used a topological interpretation of the predicate calculus (extending Stone 1937-8, Tarski 1938). Kleene 1945 used unrealizable number-theoretic formulas constituting (free-)substitution instances of the predicate letter formulas, from which the unprovability of the latter formulas follows by Nelson’s theorem 1947 that all formulas provable in the predicate calculus are realizable (also in I M 9 82). An elegant new method of Beth 1956, related to Gentzen 1934-5, is available on the basis of an outline in Kreisel 1958b p. 381 of how to correct errors in Beth’s proof (cf. the reviews Kleene 1957a and Kreisel 1960) and a report Dyson-Kreisel 1961 carrying out these corrections. Other criteria are in Rasiowa 1954, 1954a, Harrop 1960, Kleene 1962a.

82

FORMAL INTUITIONISTIC ANALYSIS

CH. I

Brouwer in 1924-7 I, 1925, 1927, 1928 and Heyting 1930 p. 50, 1g3oa p. 65 used contradicting propositions in intuitionistic analysis to refute formulas of the classical propositional and predicate calculi. We shall return to this earliest method here. But we shall be using explicitly given formation and transformation rules, while Brouwer’s and Heyting’s examples were given informally. (Heyting had such a formalism later in 1g3oa, but he didn’t restate those examples in it.) Also their examples presuppose that intuitionistic analysis provides a model for intuitionistic predicate calculus, which hardly had been demonstrated explicitly then. Likewise our exhibiting a contradicting forniula in our system of intuitionistic analysis will not show unprovability in the intuitionistic predicate calculus, until we have given a consistency proof for the system. Such a proof, employing a realizability notion for intuitionistic analysis, will be given in Chapter 11. By Godel’s second theorem (1931;I M pp. 210-213), no proof of the consistency can be elementary. The only demonstrations of the intuitionistic unprovability of classically provable formulas of the predicate (not merely the propositional) calculus which seem to us really elementary are those based on Gentzen’s Hauptsatz 1934-5 (IM p. 453), including Beth’s demonstrations. Simply to demonstrate unprovability in the intuitionistic predicate calculus, there would be slight point to the examples here. But having for other reasons gone through the work of reaching the position in which we now stand, and granting what we will also do in Chapter 11, the examples are simple and sweeping. Moreover, as we shall see in a moment, the existence of a contradicting formula in intuitionistic analysis is a stronger result than unprovability in the intuitionistic predicate calculus. Also, it rules out provability in any extension of the intuitionistic predicate calculus compatible with intuitionism. The examples are also applied in analysis. There is no effective method or “decision procedure” (IM $$30,60, 61) to decide in general whether a predicate letter formula A (IM p. 143) already known to be provable in the classical predicate calculus is provable in the intuitionistic predicate calculus, as the following argument (due to Kleene, and published in Beth 1955a p. 341) shows. Let C be a fixed (say closed) predicate letter formula provable in the classical predicate calculus, but unprovable in the intuitionistic predicate calculus (examples in the literature or below). Let A be any (closed) predicate letter formula. Then A V C is prov-

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83

able in the classical predicate calculus. But by Gentzen 1934-5 p. 407 (or IM Theorem 57 (a) p. 486 but for the predicate calculus), A V C is provable in the intuitionistic predicate calculus exactly if A or C is so provable, i.e. since C is not, exactly if A is. So, were there a decision procedure for the intuitionistic provability of classically provable formulas, there would be one for the intuitionistic provability of arbitrary formulas. This is contrary to the version for the intuitionistic predicate calculus of a theorem of Church 1936a and Turing 1936-7 (IM Theorem 54 p. 432). We should not seek to “refute” formulas of the predicate calculus in intuitionistic analysis simply by proving the negations of substitution instances. Thus J V 12is unprovable in the intuitionistic propositional (and predicate) calculus. But no formula of the form l(A V -A) is provable in intuitionistic analysis, if that is consistent. For l l ( 2V 12) is provable in the intuitionistic propositional calculus (*5la), so -,l(A V -A) is in intuitionistic analysis. In general, we must seek rather to prove negations of closures of (free-)substitution instances (cf. Kleene 1945 $9 10, 16). For example, we will “refute” J V 1J by proving in intuitionistic analysis lVa(A(a) V lA(a)) for a suitable A(a) (*27.17). This of course does show J V 1J unprovable in intuitionistic propositional (and predicate) calculus, if intuitionistic analysis is consistent, since its provability in intuitionistic propositional (or predicate) calculus would entail that of A(a)V y A ( a ) , and by V-introd. Va(A(a) V lA(a)), in intuitionistic analysis. Not every predicate letter formula provable in the classical predicate calculus but not in the intuitionistic can be refuted in this way, using our formal system of intuitionistic analysis or any other given formalism F consistent with the intuitionistic predicate calculus. For if all could be, we would have a decision procedure for the provability in the intuitionistic predicate calculus of a predicate letter formula A provable in the classical predicate calculus thus: search through an enumeration of the provable formulas of the intuitionistic predicate calculus for A itself, and simultaneously through an enumeration of the provable formulas of F for the negation of the closure of a substitution instance of A. 7.10. However, with one possible exception, all the predicatk letter formulas mentioned in I M as being classicallyprovable but intuitionistically unprovable can be thus refuted in our intuitionistic analysis.

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FORMAL INTUITIONISTIC ANALYSIS

CH. I

*27.17. t- -IVa(Vxa(x)=OV lVxa(x)=O). PROOF. Assume Va(Vxa(x)=O V -IVxa(x)=O). Writing A(a) for Vxa(x)=O and B(a) for -IVxa(x)=O, and applying "27.3,

(4

V~~Y{VX[T(E(X)) >O 3 Y=X] & {(A(a)& T(E(Y)) = 1) V (B(4 8l .(E(Y)) =2)1)*

We continue as in the proof of "27.16, except that C(a1) and C(a2) aren't required for using (a). "27.18. I- -1Va(-13xa(x)#O V -I-I3Xa(X)#O).

PROOF.Vxa(x)=O V -tVxa(x)=O comes from -I3xa(x) #O V -,13xa(x)#0 by *86, "158 and *49c. In "27.17 we directly refute (a closure of a substitution instance of) GI V Vx(J(x) V l J ( x ) )and l l V x ( J ( x ) V 1 2 ( x ) ) (cf. *49b), etc. to this list. and "27.18 adds 1J V -1li2' There must be a like refutation of any predicate letter formula B from a (free-)substitution instance B1 of which a formula A thus refutable is deducible in the intuitionistic predicate calculus. PROOF. Given: in intuitionistic analysis I- lVA* where A* is a substitution instance of A, and in intuitionistic predicate calculus B1 I- A. By V-elim. and 3-introd. t VB1 3 A, whence by substitution (IM Theorem 15 p. 159) and "69 (and perhaps "75) t- VB1* 3 VA* in intuitionistic analysis, whence by contraposition * 12 !- lVA* 3 lVB1*. By 3-elim. t- lVB1". But B1* is a substitution instance of B. For example, we can hence refute also 1-12 3 J (by I M Remark 1 p. 120), l l ( V x ~ l J ( x ) 3 -~-rVxJ(x))(which is -I-I(ICI3 Ib) I M pp. 166, 491), l V x l J ( x ) 3 3 x J ( x ) (since 1-12 3 J follows after substituting ZP for ZP(x)), etc. Altogether, using these examples and deductions noted in I M pp. 486 and 491, we can from *27.17 and "27.18 thus refute all the examples in I M for the propositional calculus (Example 4 p. 485, and those listed in Theorem 57 (b) p. 486), and all of those for the predicate calculus (listed in Theorem 58 p. 487) except (b) (i) (or *92), (b) (iii), IIc ZIIIb, IIIb z, IIIa and *97. We now refute (b) (i) (or "92). "27.19. I- iVa{Vx(a(x)=O V lVxa(x)=O) 3 VXC~(X)=OV-.lVxa(x)=O}.

PROOF.By * 158, a(x)=O V - I ~ ( x=O. ) Thence by cases (using *85a in the second case) and V-introd., (a) Vx(a(x)=O V -rVxa(x)=O).

§7

85

BROUWER'S PRINCIPLE

Assume Va{Vx(a(x)=O V lVxa(x)=O) 3 Vxa(x)=O V lVxa(x)=O}. Thence by (a) and "41 Va(Vxa(x)=O V lVxa(x)=O), contradicting "27.17. Next we refute *97. "27.20. I- lVa{(3xa(x)=O 3 3xa(x)=O) 3 3x(3xa(x)=O 3 a(x)=O)}. PROOF. Assume Va{(3xa(x)=O 3 3xa(x) =0) 3 3x(3xa(x)=O 3 a(x)=O)}. Using * 1 and "41, Vdx(3xa(x)=O 3 a(.) =O). Applying "27.2, and omitting 37 for 3-elim.,

(a)

>O 3 y=x] & Va3y{.r(ii(y))>O & VX[T(Z(X)) [3xa(x)=o 3 .(.(E(y)) A 1) =O]}.

Assume Vxal(x)=l. Using V-elim. from (a), and omitting 3yl for 3-elim.,

Assume Vxaz(x)=sg(max(yl, ~ ( Z ( y 1 )AX), ) ) so (c) (d)

X<max(yl, +i(yl))) 3 a 2 ( 4 = 1, xrmax(y1, ~(Z(y1))) zi a2(4=0.

Applying (a), and omitting 3y2 for 3-elim., (e) (g)

'6(Yd)>'J

(f)

VX[7(z(X))

>o

3 y2=xl,

3xaz(x)=O 3 a2(7(z(y2)) 1) =O.

By (c): (h) czZ(y~)=orl(yl).So by (b) and (f): (i) y2=y1. By (c) with . (d) 3xa~(x)=O, so by (g): (h) and (b): (j) a 2 ( ~ ( i $ y 1 ) ) ~ l ) = l By (k) a2(7(tC2(y2))-1)=0. By (j), (k) and (i) 1=0, contradicting I#O. By "158 and *49c a(x)=O lla(x)=O, so we also refute (2'3 3xB(x))3) 3 x ( 3 3 llB(x)). But this we can deduce from a substitution instance of IIc 3 IIb, thus. Assume (a) L2' 3 3xB(x) and (b) V x l ( 2 ' 3 B ( x ) ) . From (b) by V-elim. and *60d, 112' and lB(x); from the latter by V-introd. and "86, 13xB(x), whence from (a) by "12, l J , contrad. 1-2'. So by l-introd. (discharging the assumption (b)), l V x l ( J 3 B(x)).Thence by a substitution instance of IIc3 3 IIb, 3x11(2' 3 B(x)), and by "60g, h 3 x ( J zi l l B ( x ) ) . By 3-introd. (discharging (a)), (2'3 3xB(x)) 3 3 x ( J 3 -.nB(x)). Finally, IIc2 3 IIb is a substitution instance of IIIbl 3 IIIa, which for the predicate calculus leaves only (b) (iii) unrefuted.

-

86

FORMAL INTUITIONISTIC ANALYSIS

CH. I

7.11. We refute the closure of an instance of the least number principle "149 (whence "148 is refutable, like IM p. 513 (vii)). "27.21. t -1Va{3xC(a,x) 2 3y[C(a, y) & Vz(z
7.12. We refute the classically-provable duals of "2. la, *2.2a and "25.9, beginning with the last. (In each of these duals, one implication is provable by the predicate calculus simply.) "27.22. I- qVa{Vx3ii
Va{Vx3ii
Assume Vx(A(a, x) V B(cr)). By V-elim., A(a, ( x ) ~V) B(a). CASE 1 : A(a, (x)o). Then A(a, ( x ) ~& ) O=O, hence A(a, 0, ( x ) ~ )hence , 0<2 & A(a, 0, ( x ) ~ )hence , 3ii
=I

7.13. We take one illustration from Brouwer's theory of species. In 1924-7 I p. 246 7 9 he gives a pair of species M and N which are "congruent" but not "identical". As a newcomer to this field in 1941, the present author found this example hard to decipher from Brouwer's austere text. (A different example is in Brouwer 1954 p. 6 7 4.) We

§7

87

BROUWER’S PRINCIPLE

now give the 1924-7 example rearranged and simplified. Let abbreviate Vxa(x)#O V lVxa(x) #O, ‘‘aEN” abbreviate Vxa(x)=O V yVxa(x)=O.

“aEM”

That the species M is “congruent” to the species N is expressed by (i)

-3a(aEM & -raEN) & -3a(a€N & 1aEM).

This formula is provable, since by *51a 11aEN and 1 1 a E M are both provable. That M and N are not “identical” is expressed, after a simplification permissible because the elements of M and N are already choice sequences (and not merely spread elements overlying choice sequences by a correlation law), by -.I{Va(aEM3 aEN) & Va(aEN

2

aEM)}.

Both lVa(aEM 3 aEN) and -rVa(aEN 3 aEM), a fortiori (ii), are provable. To prove the first, assume Va[aEM 2 EN], i.e. (a)

V a ~ x a ( x#O ) V lVxa(x) #O 3 Vxa(k)=O V lVxa(x)=O].

Assume Vxal(x)=sg(x)-a(xA l), so al(O)=O, al(x‘)=a(x). Then lVxal(x) #O, hence (b) Vxar(x) #O V lVxal(x) #O. Also (c) Vxal(x)=O -Vxa(x)=O. From (a) using V-elim. (with a1 for a), (b), (c),3al-elim. and V-introd. we deduce Va(Vxa(x)=OVlVxa(x)=O), contradicting *27.17. 7.14. Finally, we refute the classical version *26.1 of the bar theorem.

*27.23. I- iVP{Va3xR(P, ii(x)) & Va[Seq(a) & R(P, a) 3 A@,a)] & Va[Seq(a) & VsA(P, a*28+1) 2 A@,a)] 3 A(P. I)} when R(P, a) is (a= 1 & lVxP(x)=O) V (lh(a)=l & P((a)o- 1)=0) and A@, a) is R(p, a) V VxR(P, a*2~+1). PROOF.Call the formula lVpB(p). Assume VpB(p). I. We establish the three hypotheses of the implication B(P). (A) For the first, by *I58 P(a(O))=O V p(a(0))#O. CASE 1 : p(a(O))=O. Then by *23.5, *23.2 and *6.3, lh(ii(l))= 1 & P ( ( ~ ( 1 ) ) o - 1)=0. So R(p, ii(l)), hence 3xR(P, ii(x)). CASE 2: p(a(0))#O. Then 3xP(x)#O, whence by *85a ~Vxp(x)=O. Also ii(O)= 1. So R(p, ii(O)), hence 3xR(P, &(x)). Completing the case argument (V-elim.), and using V-introd., Va3xR(P, ii(x)). (B) The second hyp. of B(P) is immediate,

88

FORMAL INTUITIONISTIC ANALYSIS

CH. I

because R(P, a) is a disjunctive member of A@, a). (C) Assume Seq(a) and VsA(p, a*28+1),i.e. Vs[R(P, a*28+1) V VxR(p, (a*2s+l)*2X+l)].But lVxR(P, (a*29+1)*2X+1); for by "22.8 with "22.5, lh((a*28+1)+2X+1)= lh(a)+2, which with lh( 1)=0 contradicts both disjunctive members of R(P, (a*2S+l)*2X+l).So VsR(p, a*28+1), whence A@, a). 11. Now by V- and 3-elim. from VpB(p) we infer A@, l), whence (using "22.7 with "22.5) R(p, 1) V VxR(p, 2x+1). CASE 1 : R(p, 1). Then since lh( 1) =0, lVxp(x) =0, whence Vxp(x)=O V lVxp(x) =O. CASE 2: VxR(P, 2x+1), i.e. Vx[(2x+l=l & lVxp(x)=O) V (lh(2~+1)=1& p((2"+l)o~l)=O)]. But by "3.10, 2X+1#1. SO V~P((2x+l)o'l)=O, whence by "19.9 and "6.3 Vxp(x)=O, whence Vxp(x)=O V ,Vxp(x)=O. Completing the case argument and using V-introd., we contradict "27.17. 7.15. In 1957 Kleene proposed "2.2 as a postulate. Subsequently he thought there would be need for the apparently stronger X2.1, which likewise is acceptable intuitionistically (as well as classically). Then Kleene (in February 1963) obtained a result *R15.1 bypassing the only direct use (not via "2.2) of x2.1 in Vesley's Chapter 111; Joan Rand (in March 1963) eliminated Kleene's direct use of X2.1 in deriving X26.9 from X26.3; and finally Vesley (in July 1963) obtained the following: DERIVATION OF X2.1 FROM "2.2. Assume Vx3pA(x, p). By "0.5, Va3pA(or(O),p). By X27.1 (omitting 3~ prior to %elim.), Va{Vt3!y >O & VP[Vt3y~(2t+l*E(y)) =p(t) 1 3 A(a(O), p)]}. Using ~(2t+l*~(y)) V-elim. with ilzx for or, XO. 1 and V-introd. : (a) VxVt3y~(2t+l*lG(y)) >0, (b) VxVp[Vt3y~(2~+~*ilZX()=p(t.)+1 3 A(x, p)]. Using (a) in *25.7 (cf. Remark 5.4), assume (omitting 36 prior to 3-elim.): (c) V X V ~ T ( ~ ~ + ~ * A G (t>))) ~ ( <>O. X , Assume (prior to 3-elim. from Lemma (w)~)))) 2 1. Now 5.3 (a)): (d) VW~(W)=T(~(~)~+~*~Z(W)~(~(<(W)~,

+

( i l t ~ ( 2 ~ - 3 ~ ) )1( t= ) +y(2".3t)+ 1 = (T(2t+1*ilE(8(<X,t>))) 1)+ 1 [(d), "25.11 = ~(2t+l*l&(6(<x,t>))) [*6.7 with (c)]. By 3- and V-introd., Vt3y~(2t+l*h.x(y))=(Aty(2X-3~))(t)+ 1. Thence using (b), A(x, hty(2".3t)), whence 3yVxA(x, hty(2x-3t)). This derivation uses X27.1, directly (not via "27.2). So replacing ~ 2 . 1as postulate by "2.2 would leave it uncertain whether x2.1 and "25.8 (our sole remaining result other than "2.1 a using x2.1 directly) would hold in the basic system (or even in the present classical

§7

BROUWER'S PRINCIPLE

89

system); cf. end 5 2. Also X27.1 is in a similar status to X2.1, as a postulate that is acceptable, but could for our essential purposes be replaced by its specialization from functions to numbers; it is used directly only here (and for *27.4). The modification *27.1' of X27.1 with Va3P replaced by Va3!P (where 3!P is like 3!x I M p. 199, using the = for functors 4.5) is derivable from "27.2.HINT:Va3 ! PA(a, P) I- Va3bA'(a, b) where A'(a, b) is YP{P(a(O))=b 8~A(Ata(t+ I), P) & Vy[A(lta(t+ I), y) 2 P=yI).

CHAPTER I1

VARIOUS NOTIONS OF REALIZABILITY by S. C. KLEENE

Q 8. Definition of realizability. 8.1. In the introductory 3 1 we proposed to relate intuitionistic analysis and the theory of general recursive functions. But in setting up the formal system in Chapter I we did not use the general recursive functions. In particular, we did not carry out the early proposal by Beth and ourselves to take the “laws” or “algorithms” in Brouwer’s definition of a spread to be general recursive functions; those laws, and also the one in Brouwer’s principle, we expressed simply by function variables, the c in *26.4, the p in X26.9 and the T in x27.1. (Cf. Remark 6.1.) A non-classical meaning of the prefixes Va3p and V d b is incorporated into the intuitionistic system through the postulate X 2 7 . 1 expressing Brouwer’s principle. But can one give a special meaning to just these prefixes, without having to consider the effect in the presence of the predicate calculus on all other forms of composition of formulas by the logical connectives? Initiates to intuitionism may ask for an explicit interpretation that applies to all formulas while satisfying Brouwer’s principle for those beginning with Va3p or Va3b. Indeed, a classical mathematician might question the consistency of the intuitionistic system with Brouwer’s principle. He can be assured of its consistency by the interpretation we now give, which is to be based on only principles acceptable classically as well as intuitionistically. If our semantical (or model-theoretic) arguments using this interpretation should be formalized in the basic formal system (end 32), a metamathematical consistency proof for the intuitionistic system relative to the basic system would result. We shall discuss later the possibility of such a formalization (cf. 9.2 7 5).

§8

DEFINITION OF REALIZABILITY

91

8.2. We begin by recasting some results in the theory of general and partial recursive functions of number and (one-place) function variables (end 3.2) into a form convenient for the present application. Let !P be a list of variables, number or function or both. We write e.g. q [ Y ] ,with square brackets instead of parentheses, for a function of the variables Y with (partial or total) one-place number-theoretic functions as values. We say q [ Y ] is primitive (general) [partial] recursive, absolutely or in 0, if q[y7 = I t q(Y,t) where the function q(Y,t) with natural numbers as values is such. As in 9 7 where we formalized Brouwer’s principle, a certain kind of functional which correlates a one-place number-theoretic function to a one-place number-theoretic function a (in fact, a “countable” functional Kleene 1959a 3 5) can be represented by a one-place numbertheoretic function z such that, for each t and a, z(2t+l*C?(y))> 0 for exactly one y , and z(2t+l*&(y))= B(t) 1 for that y . We now write {z}[a]for the function j3. In order to be able to construe (z}[a]as a partial recursive function of z and a, we define it in general by

+

(8.1)

{ 4 [ a=~at t(2t+1*or(yt))1.1

where yt II ,uyz(2t+l*C?(y))>O. But we shall say that {z}[a]is properly defined if (t)(E!y)t(2t+l*&(y)) >O. LEMMA 8.1. To each partial recursive function q[0, a ] , there is a primitive recursive function y [ 0 ] such that, for each 0, a : {y[O]][a]= q[0,a],and if v[0, a] i s completely defined then { y [ O ] } [ a ]i s properly defined. (Proof follows.) We shall write

do,

(8.2)

011

= y[@I,

i.e. A a q [ 0 ,a] shall be a notation for some primitive recursive function y [ 0 ] with the properties in the lemma, so that for each 0,a : (8.3)

{A.

.I>bl

= q[@, 011,

and {Aa q[0,a ] } [ a ]is properly defined if completely defined. If ~ [ z0, , a] is partial recursive and y[ z, 01 = A a q [ z , 0, a ] , and a] = q[e, 0, a] for a fixed e, then y [ e , 01 = Aa q [ e , 0, a] we put q[0, is a Aa q[0, a ] , i.e. it has the properties. PROOFOF LEMMA 8.1. Consider e.g. q [ b , p, a] with b, p as the 0. Write q [ b , p, a] = At ~ ( bj3,, a , t) where the latter q is partial recursive. By the normal form theorem I M pp. 292, 330, there is a number e

92

REALIZABILITY

CH. I1

such that, for each b, p, a,t : p(b, p> a,t ) = ~ ( P y w ( p ( y q) y, ) , e, b, t , y ) ) , (ii) T:,l(&y), & ( y ) ,e, b, t, y ) for at most one y . Let y [ b , 83 = I s y(b, 8, s) where

(4

where the y , t , 01 on the right are to be expressed in terms of s by

y = lh(s)Ll, t = ,l'o)s( a ( i ) = (s)g+l:l (i < y ) . We shall generalize these notations to allow other lists of arguments a l , . . ., uk, al, . . ., al in place of a single function a (the case (k, I ) = (0, 1)). First, we write PT-. . :p$ as (= 1 when m = - 1) ; I t as l , omitting the superscript 1 when the context makes it clear that I > 0 or that the whole is a function; I t (a(t))gas (a)$,and ((a)i)jas (a)i,jetc. (cf. 5.7, Kleene 1959 5 2). Now we define (8.W (8.lb) (8.lc)

(4.1 {.>

= {.>[At a ] ,

= {z}[O] = { r ) [ I t 01, {~>[ai, , ak,011, , a23 = {+
.. .

.. .

-

--

* , ak, a1, .,az>l (h+l > 1) and say the results are properly defined if the expressions under (8.1) to which they reduce are properly defined. In conjunction with this notation, we shall avoid using curly brackets as simply marks of inclusion. Furthermore, we write

(8.2a) Aa y [ O , a] = Aa p[O, a(O)], (8.2b) A p[O] = AU p[@] = Aa v [ O ] , (8.2~) A u ~ ... ~ k a l .. .a1 p[O, ~ 1 .,. ., al, . . . , at] = Aa (a(O))o, * ( a ( 0 ) ) k - L ( 4 k , . * ,( 4 k + l - l ] ( k + J Now f

(8.34 {nu do,4>[4 = p[@, a ] , (8.3b) { A d @ I } = d@I, ( 8 . 3 ~ ) { A u ~. .~kal. . . .WJ p [ @ , ~ .l ., ., ~ =

-

a ,

do,a1, . . .

k al, , . . ., al]}[al,. f

Uk, a19

. . .,a23

each being properly defined when completely defined.

> 1).

..,~ k ,

. ., a21

(k+l

> I),

011,.

§8

DEFINITION OF REALIZABILITY

93

The notations of this subsection are analogous to those of I M pp. 341-342, 344, which we shall also use occasionally but writing the functions on the line as in 1957, namely: We shall write (z}(Y) for the ( Z } ~ ~of( Y p. ~ 341 ) where Y1 are the functions, and Y2 the numbers, in order among Y . We write A Y ~ ( 0 Y),for the S ~ * ’ - l ( e ,0) of p. 342 or A1-1!P2 ~ ( 0 !P),of p. 344 when 0 consists of numbers only (m of them), where e is a Godel number of A0Yz p(0, Y ) uniform in as many functions as there are Y1 (Yz being n numbers, and superscripts 1) ; thus, for each 0, A Y ~ ( 0 Y ),is a Godel number of AY2 ~ ( 0 Y),uniform in Y1, or briefly a Gddel number of ilY ~ ( 0 Y). , The T-predicates of I M pp. 291-292 can be reformulated to use & instead of 8, etc.; e.g. Ti>’(/?(y),Z(y), e, a ) = T:,’(B(y), 8 ( y ) , e, a, y) when we put T:*’(.u,v , e, a) = T:,l(ni
8.3. In our former interpretation of intuitionistic number theory 1945 (found in 1941) or IM $82, we began with the idea that intuitionistically each statement, except of the most elementary kind, constitutes an incomplete communication of information, asserting that further information could be given effectively to complete it. For example, an existential statement “ ( E x ) A(x)” is an incomplete communication, asserting that an x could be given such that A(%) together with information which would complete the statement “ A(x)” for that x. A generality statement “ ( x ) A(x)” is an incomplete communication, asserting that an effective general process could be given by which, to each x , information which would complete the statement “A(x)” for that x could be found. A statement containing no logical connective is a value of one of the fundamental predicates,

94

REALIZABILITY

CH. I1

and requires no completion, assuming those predicates have been given effectively. Here “ ( E x ) A(x)” and “ @ ) A(x)” to be “statements” must either have no “free” variables, or each free variable in them must be understood as having a specified value; then in turn we talk about “A(x)” for a specified value of x as a statement. In applying these ideas to the number-theoretic formalism it was convenient, instead of specifying natural numbers as values of the free variables, to substitute for the variables the numerals expressing those values. Thereby the formal counterpart of “statements” became closed formulas. Furthermore, the information by which incomplete communications can be completed admits of being codified in natural numbers, using for effective processes Godel numbers of recursive functions. We said these numbers “realize” the respective closed formulas. Thus we said a natural number e realizes a closed formula 3xA(x), if e = 2X3a where a (= (e)l) realizes the (likewise closed) formula A(x), where x is the numeral for the natural number x (= (e)o).A number e realizes a closed formula VxA(x), if e is the Godel number of a general recursive function p such that, for each x, p(x) realizes A(x). A number e realizes a closed prime formula P, if e = 0 and P is true under the usual interpretation of its individual, function and predicate symbols. (For the other four clauses of that definition of ‘realizes’, see Kleene 1945 or I M 9 82. Some minor improvements in details appear in 1960 § 5.) Finally we said a closed formula A is realizable if some number realizes it. An open formula A(y1, . . ., ym) containing free only the distinct variables yl, . . . , ym is realizable, if its closure is realizable (1g45), or equivalently (1948 and I M 9 82) if there is a general recursive function y such that, for each y1, . . ., ym, the number p(y1, . . ., Ym) realizes A(y1, . . .,y m ) . 8.4. We shall use these ideas with some modifications to obtain an interpretation of intuitionistic analysis. We now say a statement “ ( E p ) A(p)” is an incomplete communication asserting that a /? could be given such that A ( / ? )together with information which would complete the statement “A(/?)”for that /?.Here from the constructive point of view the /? should be a general recursive function, unless “ ( E p ) A(/?)” contains free function variables having non-recursive

§8

DEFINITION OF REALIZABILITY

95

functions as their values. The latter situation arises e.g. in considering (a, B)” ; say this contains no free the interpretation of “(a)(Ep)A variables. This is to be completed by giving an effective process by which, to each a, information which completes “(Ep)A(a, B)” for that a could be found. Here a ranges over all one-place number-theoretic functions. (By Kleene 1g5oa 9 3 or Lemma 9.8 below, the fan theorem would not hold if the functions were restricted to be general recursive.) The to complete the communication “(Ep)A(or,p)” for a given a will in general depend on what that a is, and we should not restrict the p to be a general recursive function. In the simple example that “A(a,p)” is “(x)a(x)=,!?(x)”,the p must be a itself. In general, the p to complete “ ( E p ) A(B)”, when “ ( E p ) A(B)” has free function variables having specified functions as values, should be a function general recursive in those functions, unless “(Ep)A(p)” is contingent (cf. 8.6). Proceeding now to the formal symbolism, we cannot in general avoid specifying functions as values of the free function variables by substituting for those variables functors expressing their values. For the formal system does not have a functor to express each particular function; no formal system (with only countably many symbols) can. This being the case, we might as well specify the values of function and number variables alike. (In our 1957 definition of realizability, found in 1951, we specified values of the function variables, while substituting numerals for the number variables.) We find it advantageous now to codify the information, by which communications are to be completed, not in natural numbers but in one-place number-theoretic functions. Before any functions have been introduced, as to evaluate function variables, the “realizing” functions will be general recursive; later they will be general recursive in the functions which have been introduced. Actually functions primitive recursive absolutely or in those functions will suffice, in consequence of our using via the notation { }[ 3 a form of the normal form theorem (cf. 8.8). (In 1957, we still used numbers, while adding function arguments to t.he recursive functions represented by Godel numbers. The definition in 8.5 is equivalent to the 1957 one, but is more manageable in the proofs.) 8.5. Consider any formula E, and let Y be a list of distinct variables of both types including all which occur free in E. We define when a one-place number-theoretic function E ‘realizes’ E for a given assign-

96

REALIZABILITY

CH. I1

ment of numbers and functions Y as values of Y', or in brief when 'realizes-Y' E. The definition, as before (1945,I M § 82), is by induction on the number of (occurrences of) logical symbols in E. We have 9 cases according to the form of E. 1. E realizes-Y a prime formula P, if P is true-Y, i.e. if P is true when Y' have the respective values Y. 2. E realizes-Y A & B, if ( E ) O realizes-Y A and (E)I yealizes-Y B. 3 . E realizes-!P A V B, if ( ~ ( 0 ) )= o 0 and (E)I realizes-Y A, or (~(0))# o 0 and (E)I realizes-!?' B. In Clause 4 (next) it is to be understood that, when realizes-Y A, { ~ } [ c c ] is properly defined; and similarly in Clauses 6, 8 and 9. 4. E realizes-Y A ZI B, if, for each a, if a realizes-!?' A then {€}[a] realizes- Y B . 5. E realizes-Y TA, if, for each a, not a realizes-Y A. (Equivalently, since by Clause 1 a realizes-Y 1 =O for no cc, Y : E realizes-Y -A, if E realizes-Y A ZI 1 =O under Clause 4.) In Clause 6 it is to be understood that by first changing bound variables if necessary it has been arranged that x not occur in the list Y', and that !P, x are the values of \r, x, respectively. Similarly in Clauses 7, 8, 9 (in Clause 7, ( ~ ( 0 ) )is o the value of x ; etc.). 6. E realizes-Y VxA, if, for each x, { E } [ x ]realizes-Y, x A. 7 . E realizes-Y 3xA, if (€)I realizes-Y, ( ~ ( 0 ) ) A. o 8. E realizes-!P VaA, if, for each a, {~}[a] realizes-Y, a A. 9. E realizes-!P3uA, if (€)I realizes-Y, {(E)o} A. We say a closed formula E is realizable, if a general recursive function E realizes E ; an open formula, if its closure is. Realizability can be relativized to a class (or list) T of (total) number-theoretic functions, thus. A closed formula E is realizablelT, if a function E general recursive in (some finitely-many functions of) T realizes E ; an open formula, if its closure is. Consider, now in the formal symbolism,the example Va3pVxa(x)=a(). discussed in 8.4. This formula is provable by elementary logical reasoning. It is realizable, if there is a general recursive function E which realizes it, i.e. by Clause 8, such that, for each a, { ~ } [ arealizes-a ] . Clause 9, (01 must realize-a., 3pVxcr(x)=p(x). Write 5 = { ~ } [ a ]By {(c)~}Vxa(x)=p(x). The obvious choice for is a ;then by Clause 6, for each x, {(T)l}[x]must realize-a, a,x a(x)=p(x). But, for each x, a(x)=p(x) is true-a, a,x ; so {(c)l}[x] can be any function, e.g. At 0. Using Lemma 8.1 etc., it suffices to take C =
{(c)~}

§a

DEFINITION OF REALIZABILITY

97

( A a,Ax ilt 0). That is, this E is general (actually, primitive) recursive, and realizes Va3pVxa(x)=p(x). As a second illustration, take VaVx(a(x)=O V a(.) #O) (cf. Remark 6.1). To realize it, E must be general recursive and, for each 01, {E}[o~] = where, for each x, {C}[x] = 11 where q realizes-a, x a(x)=O V a(%)#O. We can take 11 =
= Aa

8.6. The new notion of realizability, besides departing from the earlier one 1945 in obvious respects connected with the presence of the function variables, alters the treatment of implication and (thence under the alternative Clause 5) of negation. Consider a closed implication A 2 B, taken by itself. We said in 1945: e realizes A 2 B, if, for each a, if a realizes A then {e}(a) realizes B. Now we say: E realizes A 2 B, if, for each a,if a realizes A then {e}[a]realizes B. The range of 01 is all number-theoretic functions, not just the general recursive ones. So the new interpretation of implication requires in the general recursive E a process by which, from even a highly non-constructive completion a of the incomplete communication expressed by A, one can get constructively a completion of the communication expressed by B. In other words, the new realizability interpretation treats A 2 B as “true constructively” whenever, for A “true” but not necessarily “constructively” so, B will be likewise “true” with “degree of non-constructiveness” not greater than that of A. This enforces the intuitionistic demand for constructiveness in a less drastic form than before; we now allow non-trivial “contrary-to-fact” conditionals, instead of placing all B’s on a par as consequences of a non-(intuitionistically-true)antecedent A. Under the 1945 notion, if A was unrealizable, then A 2 B was always realizable. That would leave no place in intuitionistic mathematics for the theory of relative recursiveness. (The change in question, which enters rather unobtrusively here with functions used as the realizing objects, was effected in the 1957 version of the new notion of realizability by departing from what in that notation is the direct generalization of the 1945 definition.)

98

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We were forced to make this change, since without it we failed to extend Nelson’s 1947 Theorem 1 (IM Theorem 62 (a) p. 504) from intuitionistic one-sorted predicate calculus with number variables to the intuitionistic two-sorted predicate calculus with number and function variables. But the change alters the notion of realizability for number-theoretic formulas, i.e. those not containing function variables or A. Consider a closed such formula A. Formerly, -A was realizable if and only if no number a realized A, i.e. if and only if A was unrealizable. Now -A is realizable (and realizablelq for any given q), if and only if no function a realizes A, i.e. if and only if A is unrealizable/8 for every 8. In I M Theorem 63 (ii) p. 51 1 we gave an example A of a closed formula such that A was unrealizable, so -A was realizable, so 1-A was unrealizable. The formula A was Vx(A(x) V lA(x)) where A(x) is 3zA(x, z) where A(x, z) numeralwise expresses TI(%, x , z) ; A was unrealizable because, if a number a realized it, ({a}(x))o would be a general recursive representing function of the predicate (Ez)Tl(x,x , z), which is not general recursive (IM Theorem V p. 283). Now similarly no general recursive function a realizes A (details in a moment), so A is unrealizable; but a function a primitive recursive in the representing function t of (Ez)Tl(x,x , z) can be defined classically which realizes A, so A is realizablelt. Hence now -A is also unrealizable, and unrealizablelq for every q, and --A is realizable. To simplify the details (which otherwise should follow Corollary 9.6) we may either suppose the list of function symbols (introduced as described in 5.1) extended sufficiently to include one f for the representing function of TI(%, x , z ) and let A(x, z) be f(x, z)=O, or without extending the symbolism we may choose the A(x,z) to numeralwise express TI(%, x,z) by the method of Lemma 8.5 below. In either case (using Lemma 8.4a in the second), (EE)(E realizes-x, z A(x, z)} + T I ( %x,, z ) , and there is a primitive recursive function E ~ ( ~ (arbitrary in the first case) such that TI(%, x , z) + ( E ~ ( ~ realizes-x, , ~ ) z A(x, 2)). The proof that A is realized in the former sense by no number a is as before (IM p. 512 (i), after applying Clause 7 p. 503). Similarly, A is realized in the present sense by no general recursive function a, because then the general recursive function sg(({a}[x](O))o)would represent (Ex)Tl(x,x , 2). Let y [ t , x ] = <Sg E z T ~ (x, xz), , < ~ z T l ( x ,, z ) , E ~ ( ~ , ~ ) (cf. > ) IM p. 317). Then y [ z , x] is general, a fortiori partial,

, ~ )

98

DEFINITION OF REALIZABILITY

99

recursive; so using Lemma 8.1 via (8.2a), and 8.2 next to last 7 , the function 01 = Ax p[z, x] is primitive recursive in z, and realizes A (using (x)Fl(x, x, 0) and cases from (Ez)Tl(x, x,z ) V (Ez)Tl(x, x,2)). Although we now consider that the new notion af realizability gives a more faithful interpretation of intuitionistic number theory, the old one remains of interest. It is simpler for establishing unprovability results for intuitionistic formal number theory as in I M Theorem 63 p. 5 1 1. Also it lends itself to investigations of intuitionistic number theory which depend on the realizability of each formula being expressible by another formula of the same system or the system inessentially extended (cf. I M top p. 406), as in Kleene 1945 $9 12-16 and Nelson 1947 $5 2, 1 1, 12. This property of realizability is lost under the new notion as applied to the formal system of number theory, though we have it again for the intuitionistic formal system of analysis. Kleene in 1945 used this property of 1945-realizability for intuitionistic number theory to set up a system of number theory corresponding to the starker form of constructiveness which that realizability interpretation represents, by adjoining to intuitionistic formal number theory certain realizable but classically false formulas (also cf. I M p. 514). Such a form of constructivism has been favored by Markov and Sanin, if we correctly understand their position. A differentappearing interpretation by Sanin 1958,1958a is shown in Kleene 1960 to be equivalent to 1945-realizability. 8.7. LEMMA 8.2. Let Yl be a list of those of the variables Y which actually occw free in E, not necessarily in the same order as in Y ;and let Y1 be the list of the corresponding ones of the nwnbers and functions Y . T h e n E realizes-YI E, if and only if E realizes-Y E. PROOF.By induction on the number of logical symbols in E. Equivalently to the definition in 8.5 (as we show in a moment), a formula E containing free only Y is realizable (realizable/T), if there is a function p, general recursive (general recursive in T ) , such that, for each Y , p[Y] realizes-Y E. Any function p with the latter property we call a realization function for E (in the list Y); and any function E which realizes-Y E, e.g. p[y7, we call a realizing-Y function for E (in the list Y).This notion of realizability is independent of the choice of the list Y ,as we see thus (similarly for realizability/T). Suppose p is a general recursive realization function for E in a given list Y ,and let ipl[Y1] = p[Y*] where Y1 is a minimal list (as in

100

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Lemma 8.2) and Y* comes from Y by replacing each number (function) correlated to a variable in Y not in Y1 by 0 (by I t 0). Then by Lemma 3.2 (with 8.2 7 2) q1 is general recursive, and by Lemma 8.2 is a realization function for E in the list Yl. Inversely, if y1 is a general recursive realization function for E in YI, and y [ Y ] = yl[Yl], then y is a general recursive realization function for E in Y . To prove the equivalence to the definition in 8.5, take Y to be the free variables of E in order of first free occurrence. For E closed, Y is empty; and if E is general recursive q [ Y ] = E is, and inversely. For E open, say e.g. its closure is VaVaE. If q is a general recursive realization function for E in u, a, then using Clauses 8 and 6 and Lemma 8.1 Aa A a q [ a , a] is a primitive recursive realizing function for VaVaE. (If q is a realization function for E general recursive in T, then using end 8.2 q[a, a] = q ~ [ aa,, Z]for some partial recursive function q1 and list 2 of one-place functions primitive recursive in T, and A d a q1[a, a, 21 is a realizing function for VuVaE primitive recursive in T.) Inversely, if E is a general recursive realizing function for VaVaE, then, using Clauses 8 and 6, ilaa {{~}[a]}[a] is a general recursive realization function for E. In the foregoing definition of ‘E is realizable/T’ the functions in the class or list T are not assigned as values to respective function variables. Now say as before that E contains free at most the variables Y (of either type), suppose Y is (@, a), and let !P = (a,0)be values of Y. We say E is realizable-@/T if there is a function cp general recursive in (functions of) 0, T such that, for each @, q[@] realizesY E ; or equivalently, if a function E general recursive in @, T realizes-@ the closure VQE of E with respect to 0.Similarly, without the T (cf. Kleene 1957). The notion ‘E is realizable-@IT’ differs from ‘E is realizable/T’ in that 0 are assigned as values to 0, and the universal quantification ‘for each @’ (or V@) applies only to the rest of Y (or Y). In view of Lemma 8.2, functions among 0 correlated to function variables among 0 not actually occurring free in E can equivalently be considered as part of the T . For the same reason, one-place functions of T can equivalently be included among the 0 by correlating them to function variables not free in E ; and so only in the case T is an infinite class of functions is the notion ‘E is realizable-@/T’ more general that ‘E is realizable-@’. The reader may elect to start 3 9 next (filling in later).

§8

DEFINITION O F REALIZABILITY

101

8.8. From a realization function y general recursive (general recursive in T ) ,we obtain realizing-Y functions v[!PI general recursive in Y (in Y , T ) . However, by the next lemma, with 0 = Y, then TE is another such realization function with realizing-Y functions v E [ y ] primitive recursive in Y (in Y , T ) . LEMMA 8.3. To a n y formula E containing free only Y, a n y 0 C T, and a n y function x general recursive absolutely (in T ) , there i s another such function XE such that: For each Y, XE[@] is firimitive recursive in 0 (in 0 , T ) , and if x[@] realizes-Y E so does XE[@] (where 0 are those of Y which correspond to 0). PROOF,by induction on the number of logical symbols in E. CASES 1 AND 5: E is prime or E is -A. Let XE[@] = At 0. CASE 2: E is A & B. Let XE[@] = <(X[@])OA, (X[@])lB> where (X[@])OA is (A@ (X[@])O)A[@] etc. CASE 3 : E is A V B. Let XE[@] = <(~[0](0))0, ). It %((X[@I (0))0 )'(X[@I) lA(t) %((X [@I (0))0 ) '(X[@I) CASES AND 8: A 1 B or VuA. Aa(x[@]}[a]. (For x general recursive in T , Aa (XI[@, Z]}[a]; cf. 8.7 7 4.) CASE 6: VxA. Ax (x[@]}[x]. CASE 7: 3xA. < ( ~ [ @ ] ( O ) ) O(x[@])lA>. , CASE 9: .

+

8.9. Brief indications of the following results were given in Kleene 1960 (in the last sentence of 9 1 with Lemma 2.la and 2.lb and Footnote 9, and Footnote 1). That E in Lemma 8.4a is true-Y is simply

the proposition expressed upon translating E directly into the informal language under the usual reading of the symbols with Y as the values of Y. (In Clause 1 of 8.5 we did not have any logical symbols to translate. On I M p. 499 we had no free variables, while on p. 500 we gave the free variables the generality interpretation p. 149 instead of values under the predicate interpretation p. 146.) LEMMA 8.4a. To each formula E containing free only the variables P ' and not containing V or 3, there is a primitive recursive function EE such that, for each Y:

(i) (ii)

If (EE)[E realizes-Y El, then E i s true-Y. If E is t r u e - y , then E E redizes-Y E.

PROOF,by induction on the number of logical symbols in E. According as E is of the form P (a prime formula), A & B, A 1 B, -A,

102

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CH. I1

VxA or VuA, let EE = I t 0, <€A, EB), A ~ E BI t ,0, AX EA or A ~ E A , respectively. One case will suffice to illustrate the reasoning. CASE 4: E is A 3 B. Then EE = Aa EB. (i) Suppose E realizes-Y A 3 B. Suppose A is true-Y; then by hyp. ind. (ii), EA realizes-Y A, so { E } [ E A ] realizes-Y B, and by hyp. ind. (i), B is true-Y. (ii). Suppose A 3 B is true-Y. Suppose a realizes-Y A ; then by hyp. ind. (i), A is true-Y, so B is true-Y, and by hyp. ind. (ii), EB realizes-Y B. Thus Aa EB realizes-Y A ZIB. LEMMA 8.4b. To each formula E containing free only the variables Y, and containing no 3, other than in parts of the form 3xP(x) with P(x) prime or of the form 3aP(a) with P(u) prime, and no V, there i s a partial recursive function E E [ Ysuch ] that, for each Y : (i) (ii)

If ( E E[)E realizes-!€' El, then E i s trwe-Y. If E i s true-Y, then EE[Y] (= I t EE(Y,t ) i s completely defined and) realizes-Y E.

PROOF, by induction. According as E is of the form P, A & B , A 2 B, qA , VxA(x), 3xP(x) or VccA(a), let ~ ~ [ l=f Ilt 0, <€A[Y],E B [ ~ ) , Aa E ~ [ YI t] 0, , Ax E * ( ~ ) [XI, Y ,< p x P ( Y ,x),I t 0 ) (where P ( Y , x) is the primitive recursive predicate expressed by P(x)) or Aa E * ( ~ ) [aY ]., CASE 9: E is 3uP(u). Say e.g. Y ! is a, p. Now P(u) expresses a primitive recursive predicate P(a, fi, a ) . Using I M Theorem VI* (a) with its proof from Theorem IV* (6) (IM pp. 292 etc.) with p, C-U instead of /?, a (cf. end 8.2), there is a number e such that (Ea)P(a,fi, a) = ( E ~ ) ( E Y ) T : ~ ~ ( PE(y), ( Y ) e, , a ) = (Es)[Seq(s)8~T:*l(L@(s)),s, e, 41 and ~ So we take EE[a, fi] Seq(s) & T:>'(B(lh(s)),s, e, a ) + P(a, p, I t ( s ) t1). =
LEMMA 8.5. T o each general recursive function v(Y) [predicate P ( Y ) ] , there i s a formula P(Y, w) [ P ( Y ) ] not containing V or 3 which numeralwise represents [expresses] i t (5.3 [5.4]) in the intuitionistic formal system of analysis (or a n y subsystem including Postulate Groups A and B and A x i o m x l . l ) , besides representing [expressing] it under the usual interpretation of the symbolism. PROOF.We modify the proofs of I M Theorems 27 p. 243 and 32 (a) p. 295 and Corollaries as extended on p. 298 (lines 13-16), and material contributing thereto, as follows. In Case (IV) for Theorem I p. 241, we change to p(x1, . . ., xn, a1, . . ., az)=w = ( y l ) . . .( y m ) [ x l ( x l ,. . ., x n , 011, . . .,al)=y1& . . . xm(x1, . . .,x n , a l , . . ., a ~ ) = ~ + m ~ ( Y I. ., .,

§8

103

DEFINITION OF REALIZABILITY

ym, q , . . ., at)=w]. In Case (Vb) for Theorem 27 (and similarly in Case (Va)), we make several changes. First (also on top p. 296), we take a < b to be an abbreviation for Vca#b+c (not 3cc'+a=b) or to be prime (cf. above preceding "15.1). Then, instead of using B(c, d, i, w) as defined on p. 203, we take it to abbreviate lVvc#(i'.d)'.v+w & w<(i'.d)' (cf. *180b p. 204). Finally, the displayed formula middle p. 243 becomes VcVd{Vu[B(c, d, 0, U) 3 Q ( x ~., . .,Xn, a1, . . ., ui,u)] & Vi,,,VuVv [B(c,d,i',u)&B(c,d,i,v)3R(i,v,xz,.. . , X ~ , Q I., .,az,u)l3B(c,d,y,w)}, .

..

abbreviated P(y, x2, ., xn, al, . . ., al, w). Instead of proving directly that this works, we can establish its equivalence to the previous formula, as we shall do in Lemma 8.7. REMARK 8.6. I M pp. 244-245 Remark 1, concerning the formula P(Y, w) given by the method of proof of I M Theorem 27 to numeralwise represent a primitive recursive function cp(Y),extends to the case Y may include function variables. LEMMA 8.7. For each primitive (general) recursive function q.(Y), the forwiulas P(Y, w) and P1(Y, w) given by the proofs of I M Theorem 27 (32 (a)) and the present Lemma 8.5 respectively to numeralwise represent v(Y) are eqwivalent (even in the subsystem of Lemma 8.5), i.e. t- P p r , w) Pl(Y, w). PROOF, for v(Y) primitive recursive, by induction on the length k of a given primitive recursive description of cp(Y).In Cases (Va) and (Vb), the proposition will follow from I, 11, IV below and the hyp. ind. ; in these cases, we write (y, 0) for Y, and in (Va) Q(w) is w=q. I. k a < b a
-

-

-

N

--

-

N

-

t- 3c3d{3u[B(c, d, 0, u) & Q(@,u)] & Vii,,3u3v[B(c, d, i', u) & B(c, d, i, v) & R(i, v, 0, u)] & B(c, d, y, w)} VcVd{3u[B(c, d, 0, u) & Q(0, u)] & Vii,,3u3v[B(c, d, i', u) & B(c, d, i, v) & R(i, v, 0, u)] 3 B(c, d, y, w)}.

-

104

CH. I1

REALIZABILITY

IIIa. Assume, preparatory to 3-elims., (a)

3u[B(co,do, 0, u) & Q(@,u)] & Vii,$u3v[B(co, do, i', u) & B(co,do, i, v) 8~R(i, v, 0 , u ) l 8~B(co, do, y, 4.

Assume, preparatory to 2- and V-introds., (b)

3u[B(c, d, 0, u) & Q(0, u)] & Vii,,3u3v[B(c, d, i', u) 8~ B(c, d, i, v) & R(i, v, 0, u)].

We aim to deduce B(c, d, y, w). But first we shall deduce, by ind. on i, (c)

i c y I> VuoVu[B(co, do, i, UO) & B(c, d, i, u) I> uo=u].

BASIS. Similar to the: IND.STEP. Assume (1) i'O (or CASE 2: y=O). From (b), assume B(c, d, y, u). By (c), w=u. So B(c, d, y, w). IIIb. Write the converse (to be proved) as VcVd{A(c, d) 2 B(c, d, w)} 3 3c3d{A(c, d) & B(c, d, w)}. Assume (a) VcVd{A(c, d) 2 B(c, d, w)}. Using I M p. 245 (3), 3w3c3d{A(c, d) & B(c, d, w)). Thence assume (b) A(c, d) & B(c, d, WO). By (a), B(c, d, w). So 3c3d{A(c, d) & B(c, d, w)}. IV. I n Cases (Va) and (Vb) for I M Theorem 27, tP(y,O,w)-VcVd{Vu[B(c,d,O,u) 3Q(@,u)]&Vi,,,VuVv[B(c,d,i',u) & B(c, d, i, v) & R(i, v, 0, u)] 3 B(c, d, y, w)}. Use successively: 111; "91; "181 p. 408 with "180~;*95; *4, "5. LEMMA 8.8. Let P(Y, w) [P(Y)] be picked by the method of $roof of Lemma 8.5 (or of I M Theorem 27 [Corollary Theorem 271) to numeralwise represent [express] a primitive recursive function q~('u) [pedicate P ( Y ) ] .T h e n (even in the subsystem)

c- P(Y, w) v lP(Y, w)

[t P(Y) v lP(Y)].

99

REALIZABILITY UNDER DEDUCTION

105

PROOF, for Lemma 8.5. Then P(Y) is P(Y, 0), where by I M p. 245 (3) (with Remark 8.6 above) and Lemma 8.7 (with P(Y, w) as the Pl(Y, w)), t 3!wP(Y, w). Hence by Lemma5.6, t P(Y, w) V -IP(Y,w), whence by substitution, t P(Y) V l P ( Y ) .

Q 9. Realizability under deduction in the intuitionistic formal system. 9.1. LEMMA9.1. (a) Let Y be a list of distinct variables not including x ; let A(x) be a formula containing free only Y, x ; and let t be a term containing free only Y, x, free for x in A(x), and (for given values Y ,x of Y, x) expressing the number t ( Y , x). Then E realizes-Y, t(Y,x ) A(x) if and only if E realizes-Y, x A(t). (b) W i t h similar stipulations (u a functor), E realizes-Y, u [ Y , 011 A(a) if and only if E realizesY,or A(u). With this we may combine uses of Lemma 8.2, hereafter tacit. For example, if A(a) contains free only a (distinct from x), and a and x are free for a in A(a): { E realizes-+) A(a)} = { E realizes-or, x , E ( x ) A(a)} [Lemma 8.21 = { E realizes-a, x, a A(E(x))} [Lemma 9.1 (a), with a,x as the Y and a as the x] = {~realizes-a,x A(E(x))) [Lemma 8.21. LEMMA 9.2. E realizes-Y E if and only if E realizes-Y the result of replacing each part of E of the form -A where A i s a formula by A 2 1 =O. THEOREM 9.3. (a) If I? t E in the intuitionistic formal system of analysis, and the formulas I? are realizable, then E i s realizable. (b) Similarly, reading “realizablelT” in place of “realizable”. (c) If I?, E contain free only Y = (a,a), and in the intuitionistic formal system of analysis I? t E with 0 held constant, and I? are realizable-@ (end 8.7), then E is realizable-@. (d) Similarly, reading “realizable-@IT’’ in $lace of “realizable-@”. PROOF.(a) (Cf. the proof of IM Theorem 62 (a).) AXIOMS(except by X26.3, X27.1). For each particular axiom E (For most of the axiom schemata), we give a particular primitive recursive function E such that, for (for any axiom E by the schema,) any list Y of variables including all that occur free in E, and any assignment Y of values to Y, the function E realizes-Y E ; so taking y [ Y ] = E (i.e. y [ Y ] = At y ( Y , t) where p(Y, t ) = E ( t ) ) , y is a primitive recursive realization function for E (cf. 8.7). For the rest of these axiom schemata, the E may depend on some of the Y as parameters (e.g. for Axiom Schema 13 on x). la. A 2 (B 1 A) is realized-!P by A a Aj3 a. For, suppose (1)

106

REALIZABILITY

CH. I1

a realizes-Y A ; by Clause 4 in 8.5, we must infer from (1) that {Aa Ap a}[a] realizes-Y B 2 A. But by (8.3), {Aa Aj3 a}[a]= Aj3 a,

Suppose (2) p realizes-Y B ; we must infer from (1) and (2) that {Ag .}[PI realizes-!€‘A. By (8.3), {A@a}[!?]= a ; so what we need is (1). lb. (A 2 B) 2 ((A 2 (B 2 C)) 3 (A 2 C)). Also 7, via Lemma 9.2. A n AP ~~P~[~l~[~~X~Il. 3. A 3 (B 3 A & B). Aa A@( a , @>. 4a. A & B 2 A . A y ( y ) ~ . 4b. A & B 3 B . A y ( y ) l . 5a. A A V B. Aa (0, a>. 5b. B 2 A V B. A@< I , p). 6. (A 2 C) 2 ((B 3 C) 2 (A V B 2 C)) . An Ap Aa It sg(( 4 0 ) )0) ((4 [(411) (4 sg((40))0) ((PI“411)(4. 81. 1 A 3 (A 3 B). AnIt 0. Suppose n realizes-Y 1 A . Then no function a realizes-!€‘A. So any function, e.g. At 0, realizes-Y A 2 B. ION. VxA(x) 3 A ( t ) , where A(x), t are as in Lemma 9.1 (a), so the free variables of the axiom are only ly, x. An {n}[t(Y, x ) ] . For, suppose n realizes-Y, x VxA(x). Then by Lemma 8.2, n realizes-?P VxA(x). So {n}[t(!€’, x)] realizes-Y, t(Y,x) A(x), whence by Lemma 9.1 (a) {n}[t(Y,x)] realizes-Y, x A(t). 10F. VaA(a) 3A(u). An{n}[u[Y, a]]. 1 IN. A(t) 2 3xA(x). A n (t(Y,x),,n). 11F. A(u) ~ 1 3 a A ( a )An .
+

-

-

d o , a1 = ( 4 0 ,

dx’,a1 = {{(411[~1>b[% all. Writing p [ x , a] = It p(x, a, t ) , this takes the form P(0, 0 f

1 9 4

(x’,a , 4

= y(a, t ) , x(x, a,At p(x, % t ) ,t)

=

where y is primitive, and x is partial, recursive. To prove this p partial recursive, we apply the recursion theorem IM p. 353 for a as the Y (I = 1) with uniformity t o solve for z the equation y(a, t ) if x = 0, , a , At { z } ( x 1, ~ a, t),t) if x

# 0.

Call the solution e, and put p(x, a, t) = {e}(x, a,t). (Cf. Lemma 3.2, Kleene 1956 5 4, 1959 XXIV.)

99

REALIZABILITY UNDER DEDUCTION

107

14, 17, x l . l : AnltO. 16: AnApltO. 15, and all prime axioms (namely, 18-21, XO.1, the axioms of Group D): It 0. ~ 2I . . Vx3aA(x, a) 3 3aVxA(x, Iya(<x,y>)). An .Suppose n realizes-YVx3aA(x, a). Then, for each x , ({n}[x])1 realizes-Y, x , {({n}[x])o}A(x, a). Hence by Lemma 9.1 (b), ({4[x1)1 realizes-y, x , At {({n}[(t)01)0}((t)1)A(x, l y a(<x,y>)). RULESOF INFERENCE. 2. A, A 3 B / B . Noting 8.7, we choose Y to include all variables free in A 3 B. By hyp. ind., there are general recursive functions a and y such that, for each Y, a[Y] realizes-Y A and y [ Y ] realizes-Y A 3 B. Let a,[Y] = {y[!Pl}[a[Yl]. Then a, is partial recursive, and, for each Y, q[Y] realizes-Y B; hence a, is general recursive. 9N. CIA(X) / C 3VxA(x). Say, for each Y and x , y[Y,x] realizes-Y, x C 5 ) A(x). Then, for each Y, Ay Ax { y [ Y ,x]}[y] realizesY C 3 VxA(x). 9F. Ay Aa { y [ Y , a]}[y]. 12F. A(a) 2) C / 3aA(a) 3 C. An { y [ Y , {(n)o}])[(n)1] is a realization function for the conclusion, if y [ Y , a] is one for the premise. 12N. An { Y [ Y (n(O))oI~[(n)11. AXIOMSCHEMA X26.3~.Vcc3 !xR(E(x))& Va[Seq(a) & R(a) 3 A(a)] & Va[Seq(a) & V~A(a*2~+l) 3A(a)] 3 A( 1). Assume that n realizes-Y' the antecedent of the main implication of an axiom by this schema containing free only 'Y; all the definitions and inferences below are under this assumption until the final step, except that, when we say a predicate or function with n as a variable is partial recursive, n ranges over all functions. Now (n)o,o realizes-Y Va3 !xR(E(x)), i.e. Va3x[R(z(x)) & Vy(R(E(y))3 x=y)] ; (n)o,1realizes-Y Va[Seq(a) & R(a) 3 A(a)]; and (n)l realizes-Y Va[Seq(a) & VsA(a*2s+l) 3 A(a)]. Hence: (1) For realizes-Y, a,x R(E(x))& Vy(R(a(y))I x=y) each a, ({(~)o,o}[cL])~ for x = ({(n)o,o}[a](O))o. (2) For each a, P O , p l , if PO realizes-a Seq(a) and p l realizes-Y, a R(a), then ({(n)o,l}[a]}[po,pl] realizes-Y, a A(a) (cf. (8.1~)). (3) For each a, po, pl, if po realizes-a Seq(a), and, for each s, (pl}[s] realizes-lv, a, s A(a*28+1), then {{(n)1}[a])[po,p ~ ] realizes-Y, a A(a). Furthermore: (4) For each 0,a,y , if 0 realizes-Y, a,y R(ti(y)),then y = x for the x of (1). For, by (1) { { ( { ( n ) 0 , 0 } [ a ] ) 1 , 1 } [ ]realizes-x, }[~~] y x=y, so x=y is true-x, y .

108

REALIZABILITY

CH. I1

By ( l ) , {(n)o,o}[a](O) is defined for each a. Let R(n, 4

EE

(Ea)[a= &(x) for x = ({(~)0,0}[~1(0))01.

Clearly ( 5 ) (8)(Ex)R(n,B(x)). We define a partial recursive predicate R1 thus.

Rl(T a)

= [a = G(x1) for

a1 =

At

@It-

1 , x1

=

({(~)0,0}[~11(0))01.

We show that (6) R(n, a ) = Rl(n, a ) . Assume R(n, a ) , and put 4%) with x = (((~)o,o}[~l(O))o. By (11, ({(n)o,o}[aI)1,0realizesR(E(x)), whence by Lemma 9.1 (a) it realizes-!$,' &(x)R(a). Let a1 = At ( a ) t l l . Then a1 agrees with a in its first x values, so <(x) = &(x), and by Lemma 9.1 (a) (((n)o,o}[a])1,0realizes-Y, al, x R(E(y)). By (4)with al, x and x1 = ({(z)o,~}[a~](O))~ as the a,y and x , x = XI. So a = <(XI). Conversely, a = <(XI) implies R(n, a ) . We shall find a partial recursive function q with the following property. Let S; be the set of the sequence numbers barred with respect to l a K l ( n , a) (cf. 6.3, 6.5, 6.6). (7) a E S; + {q[n,a] realizesY , a A(a)}. To prove this, we use an intuitive application of the bar theorem, i.e. we use an induction over S; (the informal analog of X26.8a in 6.1 1, with the recursiveness of l a Rl(n, a ) providing the first hypothesis). We begin by giving the basis and induction step. In each we derive a specification for q[n,a] that will suffice there. Afterwards we show that a partial recursive q can be chosen to satisfy both specifications. BASIS: R&, a). Then a = <(XI) etc. By ( l ) , (((n)o,o)[a1])1,0 realizes-Y, al, x1 R(E(x)), whence by Lemma 9.1 (a) it realizes-Y, a R(a). Also Seq(a), so I t 0 realizes-a Seq(a). So using (2), q[n,a] will realize-Y', a A(a) if q[n,a1 = {{(~)0,1}[~1}[~~ 0, ({(~)0,0)[~11)1,01. IND.STEP: Seq(a) & (s)[u*2'+l E ST]. By hyp. ind., for each s, ~ [ na*2s+l] , realizes-Y, a*2s+l A(a), whence using Lemma 9.1 (a) it realizes-Y, a, s A(a*28+1). So, using ( 3 ) ,q[n,a] will reazile-!&,' a A(a) , if q[n,a] = {{(n)l}[a])[lt0, As ~ [ na*2s+l]]. DEFINITIONOF 7. It will suffice to have q[n,a] = lu q(n,a, u ) where

a

= Y , a,x

V(n, a , .u) 2:

{

{{(~)0,1}[~1}[~~ 0, ({(n)o,o)[lt ( 4 t - lI)l,Ol(.u) if R1(n, a ) , {{(n)l}[a]}[lt 0, As ltq(n,a*2S+1, t)](u) otherwise.

Upon replacing q by {z}, this equation assumes the form {z)(n,a, u) II z is given by

y(z, z,a, u ) with a partial recursive y . A solution e for

39

109

REALIZABILITY UNDER DEDUCTION

the recursion theorem I M p. 353 for 7c as the Y with uniformity. We take ~ ( na,,u ) N {e}(n,a, u); as remarked after Lemma 8.1 (with (8.2a)), the specialization of z to e under the operation As is valid. By (5) and ( 6 ) :(8) 1 E S;. Hence by (7), ~ [ z11 , realizes-Y, 1 A(a), whence by Lemma 9.1 (a): (9) ~ [ nI] , realizes-Y A(1). Finally, Anq[7c,11 realizes-Y the axiom. AXIOMSCHEMA x27. 1. Va3PA(a, p) 3 37Va(vt3!y~(2~+l*E(y)) >O & Vp[Vt3y~(2t+l*E(y)) =p(t) 1 2 A(Q,p)]}. Assume (outside the definitions of the recursive functions and the final step) that 7c realizes-Y Va3PA(a, p). Consider any a. Now (1) for each a, ({7c}[a])1realizes-!P, a, A(a, P) for B i = {({7c}[aI)o).Let t = AaBi = Aa {({7c}[011)0).By ( I ) and >O, Lemma 8.1, {t}[a]is properly defined, i.e. (2) (t)(E!y)t(2t+1*&(y)) and {z}[a]= B1, whence by (8.1): (3) (t)z(2t+1*&(yt)) =Bl(t) 1 where yt = pyz(2t+l*or(y))>o. Now we seek a function po to realize-z, a Vt3 !yT(2t'1*i?(y)) >O, i.e. Vt3y[~(2t+l*ii(y)) >O & V~(7(2t+~*ii(z)) >O 3) y=z)], taking the inequality as a prime formula (cf. preceding "15.1 in 5.5). Consider any t. Using (3),~(2~+l*ii(y)) >O is true-z, a,t, yt and hence is realizedz, a, t, yt by I s 0. If (T realizes-t, a, t, z 7(2t+l*E(z))>O, then ~(2~+l*E(z))>O is true-t, a,t, z, hence by (2) z = yt, and hence Is 0 realizes-z, yt z=y. Combining these results, Vt3 !yT(2t+1*E(y))>O is realized-z, a by po = At (,uyz(2t+l*&(y)) >O, (As 0, Az Ao As 0)). Next we seek a function p l to realize-Y, t,a vp[Vt3~~(2t+l*E((y))=P(t)+ 1 2 A(a, p)]. Consider any B. Suppose (T realizes-z, a, Vt3~~(2~+l*i?(y)) =P(t) 1. Then, for each t, ({(~}[t])l realizes-z, a, B, y t ~(2~+l*E(y))=B(t)+lfor yt = ( { o } [ t ] ( O ) ) ~thus ; (t)t(2t+h&(yt))=B(t)+1. Hence by (2) and (3), B = B1, so by (1) ({7c}[a])1 realizes-Y, a,B A(a, P). So we take p l = A! d o ({z}[a])1. Altogether, the axiom in question is realized-Y by An ( A z, Aa
+

+

+

110

REALIZABILITY

CH. I1

(c) By an application of I M Lemma 8b p. 104 and changes of bound variables, we can replace the given deduction of E from I? by one of B from in which 0 do not occur as bound variables, where f‘, E are congruent to I?, E. Then also are realizable-@, and E is such if f? is. Now we can reconstrue 0 to be individual and function symbols rather than number and function variables; in the resulting formal system, contrary to Lemma 3.3, terms express under the intended interpretation functions primitive recursive in 0 instead of absolutely. So now we adapt the proof of (b) to construct realization functions AYi cpi[Yi] of the form AYt cpg[Yi,01 with AY@ cpt[Ui,01 partial recursive. (d) The realization functions are of the form AYt cpi[Yg,0,Z]. 9.2. COROLLARY 9.4. If A i s realizable, and B i s unrealizable, then A 3 B i s unrealizable. If A i s realizable or realizable17 for a n y 7,then -,A i s unrealizable and unrealizable18 for every 8. Similarly reading “realizable-@” in place of “realizable” ; etc. COROLLARY9.5. T h e intuitionistic formal system of analysis i s simply consistent, i.e. for no f o r m d a A are both A and -A provable. PROOF.For no A are both A and -.IA realizable, by Corollary 9.4. This consistency proof uses informally only methods corresponding to the common portion of the classical and intuitionistic formal systems. It is of course not a metamathematical consistency proof, as a non-elementary interpretation is used. 6 But presumably it can be formalized in the basic formal system (i.e. intuitionistic analysis minus X27.1) to give a strictly finitary metamathematical consistency proof for the intuitionistic formal system of analysis relative to the basic system, just as in Nelson 1947 with Kleene 1945 5 14 the proof by the old realizability interpretation of the consistency of a certain non-classical extension of intuitionistic number theory was formalized to provide a metamathematical consistency proof for the extended system relative to the unextended one. Such a formalization must be quite laborious, as it must begin with a formalization of the presupposed theory of partial recursive functionals. Our confidence that it can be carried out is based on a careful review of everything which went into the above interpretative consistency proof, and on a part 6 The “N” of IM pp. 500 ff. applies to the theorems, corollaries, lemmas and remarks here (not already labelled “C”) which use realizability notions.

99

REALIZABILITY UNDER DEDUCTION

111

of the work of formalization already carried out. (We shall perhaps say more in a later publication.) COROLLARY 9.6. If P(a1, . . .,ak, al, . . ., az) numeralwise ex#resses a general recursive predicate P(a1, . ., ak, 011, . . ., mi) in the intuitionistic formal system of analysis (or any subsystem thereof), then, for each al, . . ., a*, al, . . ., q ,P(a1, . . ., ak, a1, . . ., at)i s realizable-al, . . ., ak, al, . . ., at if and only if P(a1, . . ., ak, orl, . . ., az). PROOF, adapting that of I M Lemma 47 p. 512. Suppose P(a1, . . .,ak, 011, . . ., q).Then by I M 3 41 (i) p. 195 with p. 298 (cf. 5.4), E;:::::: I- P(a1, . . ., ak, al, . . ., ai) with a1, . . ., a1 held constant. But EZ:::;: are prime and true-al, . . ., 011 and hence realizable-orl, . . .,011. So by (c) of the theorem, P(a1, . . .,ak, air . . .,at) is realizable-q, . . ., at, and hence by Lemma 9.1 (a), P(a1, . . ., ak, a1, . . ., az) is realizable-al, . . ., ak, a1, . . ., q. Conversely, suppose P(a1. . . ., ak, a1, . . ., az) is realizable-al, . . ., ak, al, . . ., at. Values of 011, . . ., a1 being available as required, P(a1, . . ., ak, al, . . ., al) v P(a1, . . ., ak, 011, . . ., az); etc. as before. Corollary 9.6 may be used in conjunction with I M Corollaries to Theorems 27 and 32. A different approach is provided by Lemmas 8.5 and 8.4a.

.

9.3. In the definitions of ‘realizes-Y’ and ‘realizable’, let the range of the informal function variables E , Y ,a be confined to the functions belonging to a class C closed under general recursiveness (i.e., whenever Z E C and p is general recursive in Z, then cp E C), or general recursive in a function or in section -d (Kleene 1963 p. 133), and hence containing all general recursive functions. There result notions Clrealizes-Y and Clrealizable, or tlrealizes-Y etc. We can simultaneously use recursiveness in T instead of recursiveness in the definition of ‘realizable’, obtaining for a T C C a notion ClrealizablelT (and similarly for 0,T C C , Clrealizable-@IT;etc.). The C acts as a ceiling on all the functions considered, the T as a threshold below which constructivity is not demanded. THEOREM 9.7. For any class C of functions closed under general recursiveness (e.g. the general recursive functions): If I? I- E in the intuitionistic formal system of analysis without Axiom Schema X26.3 (the bar theorem), and the formulas I’ are Clrealizable [ClrealizablelT, where T C C ] , then E i s Clrealizable [C/realizable/T] (cf. Theorem 9.3 (a), (b)). With corresponding changes, Lemmas 8.2, 8.3, 8.4a, 8.4b,

112

REALIZABILITY

CH. I1

9.1, 9.2, Theorem 9.3 (c) and (d), and Corollaries 9.4 and 9.6 hold (call them LEMMA C/8.2 etc.). PI~OOF. We reexamine the former proofs, omitting the case of X26.3 in that of Theorem 9.3, to verify that the reasoning holds good when the universe of functions is C. LEMMA9.8, toward Corollary 9.9. (Kleene 1g5oa 5 3 . ) There i s a primitive recursive predicate R ( a ) such that, writing u E 0 = {a i s general recursive} (Kleene-Post 1954) and B(a) = (t)a(t)5 1 :

(4

(.),~O&N(,,(Ex)K(.I(x)) ;

(b)

(.)(E.)~~o&H(d)(x)z,,R(ol(x)))

whence

(4

( E 4(4&O&B(,)(E~)Z&(“.x)),

whence by the f a n theorem (the informal analog of *26.6a) (d)

(a),,,,(Ex)R(W.

PROOF.Using the W O ,W1 of 181 p. 308, let

R ( 4 = (Et)t
Then, for each a with B(a), (1)

R(+)) = (Et)t
(a) Consider any general recursive I V p. 281,

(2) (3)

.@I= .(t)=O

1

f

f

a

with B(a). Using IM Theorem

(EY)Tl(fO,t, y ) = ( E Y ) T l ( ( f ) Ot,, y ) , ( E Y ) T l ( f l ,t , Y ) = ( E Y ) T l ( ( f ) lt,, Y ) ,

for suitable numbers fo,

f l ,f

= .

CASE 1 : a(f) = 1. Then

(EY)Tl((f)O, f , y ) ; and (Ey)Tl((f)l> f >Y ) , whence (.)%((f)l?f , 4. so ( E y ) W l ( f ,y ) , i.e. ( E Y ) W , ( ~ y, () ~. ,CASE 2: a(/)= 0. Similarly. - By (1) (with x = f+y+ 1 , t = f ) , (Ex)R(E(x)). (b) Consider any z . Let 1 if t
99

REALIZABILITY UNDER DEDUCTION

113

Then ~ E O& B(a). Consider any x < z. Suppose I?(%(%)). By ( l ) , there is a t < x < z and a y < x - t < z 2 t such that W,(t,(t, y ) . Thence we obtain a contradiction, by cases. CASE 1 : a(t) = 1. Then W l ( t , y), and by Case B of the definition of a, a(t) = 0. CASE 2: a(t) = 0. Similarly. COROLLARY 9.9. T h e bar theorem X26.3 and the f a n theorem *26.6 or "27.7 do not hold in the intuitionistic system without the former as axiom schema, i.e. some formulas of the forms X26.3, *26.6, "27.7 (and via deducibility relationshi@, *26.4, "26.7326.9, "27.8-*27.14) are unprovable i n it. Also, by Corollary 9.5, the negation of no instance of X26.3 etc. is unprovable. So X26.3 etc. are "independent" of the other postulates of the intuitionistic system. PROOF OF COROLLARY 9.9. Taking C = {the general recursive functions} = 0 in Theorem 9.7, all formulas provable in the system in question are O/realizable. X26.3, *26.6. We shall show that the following substitution instance of the fan theorem *26.6a (deducible in this system from an instance of X26.3a) is not O/realizable: Va[Seq(a) 2 R(a) V -R(a)] & VaB(,)3xR(ii(x))3 3zVa,(,,3x,,,R(ii(x)), where B(a) is Vta(t) < 1 (At1 being substituted for p in *26.6), R(a) is a formula numeralwise expressing the primitive recursive predicate R ( a ) of Lemma 9.8 obtained by the method of proof of Lemma 8.5, and for simplicity x VaB(,,3xR(ii(x)).Hence q = ( { E } [ [ o , 5'1])1 O/realizes-zVaB(,~3x,<,R(E(x)) for z = ( { E } [ [ o , [1](0))0. Now consider any general recursive a such that B(a). By Lemma 0/8.4a (ii), B(a) is O/realized-a by E ~ ( ~ )so; x
114

CH. I1

REALIZABILITY

we obtain 7 and z such that 7 O/realizes-z Va,,,,3bVyB(y)[Vxxcz y(x)=cr(x) 3 R(y(b))], whence we infer that

6)

(4asO&B(

).

(Eb)(Y)reO&i?(r) [(x),
(4=“(4

-+

R(.i;@)I.

Let S be the finite set of the numbers a such that Seq(a) & lh(a)=-z & (t)[(a)t<2], and let z1 be the maximum of the b’s given by (i) for a = At 1 for a E S . Then ( . ) a E o & B ( . , ( E b ) b ~ z , ~ (contradicting ~(b)), Lemma 9.8 (c). 9.4C. In this and the next subsection we use classical reasoning. The “jump” operation ’ of Kleene-Post 1954 takes a predicate ila A ( a ) into the predicate ila (Ex)T;’(a,a, x ) , or more generally a function ila .(a) into (the representing function of) the predicate ila (Ex)T,“(a, a, x ) (cf. I M p. 292). A (number-theoretic) function y(a1, . . ., a,) is arithmetical if its representing predicate y(a1, . . .,a,)=w is arithmetical IM pp. 239,285. LEMMA9.1OC. T h e arithmetical functions constitute the least class of functions closed under general recursiveness and the jump operation ’. (Kleene-Post 1954 with IM; or Kleene 1955b $9 2, 3.) PROOF, from results in IM. First, y and its representing predicate are general recursive each in the other, by ##14, C pp. 227-228 (with Theorem I1 p. 275), and Theorem I11 p. 279 with y(a1, . .., a,) = pw[y(al, . . ., a,)=w]. Now suppose y is general recursive in arithmetical functions y1, . . .,yr. By Theorem VII (d) p. 285, the representing predicates Q1, . . ., QZ of y1, . . ., yr are expressible each in one of the forms of Theorem V Part I1 p. 283, say with K1, . . ., kz quantifiers, respectively. Then, by introducing redundant quantifiers if necessary, all are expressible with k = max(k1, . . ., kr) quantifiers. So by Post’s theorem Theorem XI p. 293 (and Theorem VII (b)), the representing predicate of y is arithmetical. Thus the arithmetical functions are closed under general recursiveness. As T;(a, a, x ) is primitive recursive in a (p. 292), hence by Theorem I1 general recursive in a, and by definition the arithmetical predicates are closed under number-quantification, the arithmetical functions are closed also under ’. To show conversely that each arithmetical function is definable using only general recursive operations and ‘, it suffices by Theorem VII (d) to show it for predicates expressible in the forms of Theorem V. We do this by induction on the number k of the quantifiers. For k = 0, it is immediate. Say P(a1, . . ., a,) is ex-

§9

REALIZABILITY UNDER DEDUCTION

115

pressible using k+ 1 quantifiers. If an existential quantifier (Ex) is outermost (otherwise we first consider P(al, . . ., an)), then (using ai = (
(a)B(a)(Ex)'

('(x))

+

('2)

(01)B(u)

EX),^1('

)

where B(a) is (t)a(t) p(E(t)). This is equivalent via contraposition etc. to (b)

(2)(E"')B(,,(x>,I,B(;;(.))

+

(EO1)Bcu,(x)~(w),

which upon being expressed geometrically will be Konig's lemma. For the geometrical version, after representing the fan by a tree as in 6.10 7 3, we now print in bold face along each path 01 all the vertices up to the first one (if any) inclusive (underlined) at which R(&(x)),i.e. those occupied by sequence numbers not past secured 6.3. Again we suppress the part of the tree which is not in bold face. Now the hypothesis (Z)(EO~)~(~)(X),.~~(E(X)) of (b) says that in the tree remaining there are arbitrarily long finite paths (not illustrated by Figure 1 in 6.5). The conclusion (E01)~(,&)&i(x)) says there is an infinite path. Indeed, we trace an infinite path by the following rule (in general, not effective). Under the rule, we start of course at the initial vertex marked [ 3. Suppose we have traced the path as far as any vertex a (either [ ] or a later vertex), and that a has the property of belonging to arbitrarily long finite paths (as [ 3 does by hypothesis). Then one of the next vertices a*28+1 (s = 0, . . ., p(a)) will have the same property; for, if all paths through a*28+1 were of length 5 z,+ 1 (s = 0, . . .,B(a)), all through a itself would be of length max(zo+ 1, . . .,zSca,+ l), contradicting our supposition that a has the property. The rule says to pick as the next vertex on the path one of the a*2S+1 which has the property, say the one with least s. Thus, starting from

116

REALIZABILITY

CH. I1

[ 1, we are able successively to pick a next vertex, always with the property, ad infinitum. LEMMA 9.12C. For a n y class C of functions closed under general recursiveness and the jump operation ' (e.g. by L e m m a 9.10, the arithmetical functions) : T h e f a n theorem holds informally in the version (a) of Remark 9.11, when j3 and the representing function p of R belong to C , and the function variable a ranges over C. PKOOF.By the proof in Remark 9.11, it will suffice to show that the OL represented by the infinite path determined by the rule in that proof belongs to C. We analyze that rule by using a sequence number g, subject to appropriate conditions, to represent a finite path of length z + l through a. Thus g = p(z) for some y such that B ( y ) , i.e. ( t ) y ( t )2 /l(y(t)). It follows that g 2 y(j3, z ) where y(B, 0 ) = 1, y(P, t') = y(@,t).ptl+msxs~y(B,t)B(s)

(Kleene 1956 Footnote 8). Thus the question satisfies the equation

(4 4%) =iu~s~p(u(,),(z){~>x

+

cy

represented by the path in

(Eg),,y(B,z,{Seq(g) & lh(g)=z & n,,,P?(=w 8z.

( t ) t , , [ ( g ) t l 1 +P(ni
(g),=s+ 1 & ( t ) , , z m < t P l " " ) l > , which defines it by induction. Let (T = (j3, p), so 0 is primitive recursive in /l, p, and /l and p are each primitive recursive in (T.Now the right z )zR) where ~ ( < ; R" ;(X is )primi, side of (c) is of the form ~ . s ~ ~ ~ ( ~ ( , ~ ) ) (s>, tive recursive in (T. Applying the jump operation to u, let S ( a ) = ( E z ) T ; ( a ,a , 2 ) . Since j3, ~ E C so , does (T, and hence so does (the representing function of) S . By I M p. 343 Example 2 for 1 = 1, (Ez)R"(a,z ) = S ( y ( a ) ) for a primitive recursive y . Letting %(a)= p s r p c a , g ( y ( < as, ) ) ) , (c) becomes

(4 44 = x(S(4). By I N #G p. 231 adapted to use .I(%) instead of a(%),01 is primitive recursive in x, which is primitive recursive in j3, S , which E C ; so 01 E C. THEOKEM 9.13C. For a n y class C of functions closed under general recursiveness and the jump operation ' (e.g. the arithmetical functions) : If I? t E in the intuitionistic formal system of analysis with the fan theorem "26.6 (or via deducibility relationships, "26.7, *27.7-*27.10)

99

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117

replacing the bar theorem X26.3 as axiom schema, and the formulas I' are Clrealizable [ClrealizablelT, where T C C ] , then E i s Clrealizable [C/realizable/T]. PROOF.To the proof of Theorem 9.7, we add a treatment of the fan theorem with the present C. "26.6~. VaB(,)3!xR(C(x))3 3zVaB(,,3xx,,R(C(x)), where B(a) is Vta(t)
R(n,a) = ( E 4 B ( a , [ a= "4 for x

= ({{7C)[O;I}[EB(G()l(0))01.

Clearly (5) ( C X ) ~ ( ~ ) ( Eti(%)). ~ ) R (Define Z, a partial recursive R1 by &(n, a) = [a = crl(x1)& (t)tq,,w(t)G & ( j ) ) for 011 = At ( a ) t L l and X I = ({{~}[~l]}[~g(~)l(0))0]Using (1) and (4): (6) R(n, a) = Rl(n, a ) . Also, if B(a)& R1(n, & ( x ) ) , then putting a = &(x) and a1 = ilt (a)t- 1, B(a1) holds, and as in the basis for Axiom Schema X26.3~( { { n } [ a 1 ] } [ E B ( , ) ] ) l , O C-realizes-Y, a R(a), and so C/realizes-!P, a,x R(E(x)).Thus (10) B(a) & R&, & ( x ) ) -+ ({{n}[At( & ( x ) ) t - 1 ] } [ E B ( , ) ] ) l , 0 Clrealizes-Y, a, x R(E(x)). But ila Rl(n,a) is general recursive in p, n, which E C, so l a Rl(n, a ) E C ; and the range of 01 in (5) is C. So by Lemma 9.12 we can apply the fan theorem +)). informally to (5) with (6) to obtain (11) (E~)(or),,,)(Ex),~,Rl(n, Using also (0) and (lo), 3zVaB(a)3~,s,R(E(~)) is Clrealized-fl, Y by A p <WRl(Z,+)), ). To avoid the function quantifier in this expression, we can introduce a number quantifier (g), subjected to conditions which effect that g = E(z) for some a such that B(a), as in the proof of Lemma 9.12; thus (12) ( ~ ) B ( a ) ( E ~ ) x s z ~=l ((g),,,(p,z){SeS(s) ~~ & W)=z ZZ (t),,,[(g)t< 1 +B(n,,,fiY') & (Ex),,,Rl(n, lLZfiig)')}. Using (121, our

W)

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expression Clrealizing-8, Y 3zVa,~,$x,,,R(ii(x)) takes the form ~ [ pn], with a partial recursive V. Now An y[B, n] Clrealizes-8, Y * 26.6~. 9.5C. A predicate is analytic Kleene 1955 9 2, if it is expressible in terms of general recursive predicates of number and function variables by the operations of the predicate calculus. Two number-theoretic predicates are of the same degree Kleene-Post 1954, if each is general recursive in the other. THEOREM 9.14C. We can correlate to each arithmetical [analytic] (analytic} predicate P(a) a system S between the intuitionistic formal system of number theory T and the classical one [a system S between the basic formal system T artd the present classical formal system of analysis] {a consistent supersystem S of the intuitionistic formal system of analysis T } so that, when Pl(a) and P4a) are of distinct degrees, the correlated systems S1 and S2 are distinct, i.e. have different classes of provable formulas. Each system S arises from T by adjoining an axiom of the form P(a) V l P ( a ) (or equivalently by IM p. 120 Remark 1, 1 1 ( P ( a )V l P ( a ) ) 3 P(a) V l P ( a ) ) . PROOF.A given expression for P(a) is equivalent to a prenex expression (by the informal counterpart of I M Theorem 19 p. 167) with a general recursive scope (by IM #D p. 228), say e.g. (x)(Ey)(z)R(a,x,y, z). By Lemma 8.5, we can express R(a, x, y, z) by a formula R(a, x, y, z) not containing V or 3. Let P(a) be VxlVylVzR{a, x, y, z), and form S as stated. Using cases (P(a) v P(a)),Lemma 8.4a (ii) and Clauses 3 and 5 in 8.5, for each a P(a) V i P ( a ) is realized-a by where n(a) is the representing function of P(a). Thus P(a) V l P ( a ) is realizable/P. So by Theorem 9.3 (b) each provable formula of S is realizable/P, and by Corollary 9.4 S is (simply) consistent. When S1 and S 2 are thus constructed for PI(,) and P2(a) of different degrees, with say P1 not recursive in P 2 , the axiom Pl(a)V l P l ( a ) of S1 is not provable in S2. For if it were, it would be realizablelP2, say with the realization function g~ general recursive in Pz.Then PI(,) = (p[a](0))0=0, since by Lemma 8.4a (i) (q~[a])l realizes-a PI(,) (lPl(a)) only if Pl(a) (i"l(a)). So P1 would be recursive in ilat p[a](t) and thence in Pz, contrary to hyp. REMARK 9.15C. In the system S formed similarly by adjoining as axiom Pk(a) V lPk(a) (or any formula realizablelpk) for each of a list

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of predicates Po(a),Pl(a), Pz(a), . . ., or Po(a), . . ., P,(a) (then let E $‘,(a)), each provable formula is realizablelAaR P&). REMARK 9.16C. Were V not excluded in Lemmas 8.4a and 8.4b, 3 would be a classical *27.17 [P(a)V l P ( a ) for P(a) = (Ex)Tl(a, a, 4 counterexample to (i) [(ii)]. Counterexamples for unrestricted 3 follow by t- A V B 3x((x=O 3 A) & (xfO 2 B)) and *0.6.

Pm+&)

-

Q 10. Special realizability. 10.1. ’ Early in the investigations of 1945-realizability (since 194 l), formulas were encountered whose realizability was only proved classically (e.g. Kleene 1945 p. 114 (h) and (1); G. F. Rose 1953 p. 11). In such a case, the realizability interpretation fails to exclude the formula’s being provable intuitionistically, but on the other hand, we lack adequate grounds for affirming that it should hold intuitionistically, though we know classically that it can consistently be adjoined to the intuitionistic system as a postulate. The situation is the same with the present notion of realizability. One such formula (by Theorem 11.7 (a) below with *25.3) is MI:

VzVx[+’ylTl(z, x, y) 3 3yTl(z, x, y)1

where Tl(z, x, y) is a formula numeralwise expressing x , y) (IM p. 281) chosen by (the method of proof of) Lemma 8.5. In view of IM $3 62, 63, this formalizes Markov’s principle 1954a (introduced in lectures in 1952-53), known to us from the statement of it in Sanin 1958a, “If the process of application of the algorithm Ic1 to an initially given P does not continue infinitely, then 52 shall apply to P.” The like formulas M n with Tn(z, XI, . . ., xnry) for n > 1 in place of Tl(z, x, y) follow under the interpretation from M1 by contraction of XI, . . ., xn (IM (18) p. 285) and I M Theorem IV p. 281. Especially in connection with the author’s investigations of 1957realizability (since 1951), it became apparent to him (before he learned of Markov’s 1954a) that considerable interest attaches to the result of adjoining this principle to intuitionistic systems, because of a variety of results which it then (or only then) becomes possible to obtain. An example recently come to light is Godel’s that, if “strong completeness” of the predicate calculus is provable intuitionistically, so is each instance (particularizing z, x to numerals) of Markov’s principle. This result appears in Kreisel 1958a Remark 2.1 and 1959a, with sketch of proof in 1962b, and as Theorem 2 of 1962. Godel seems

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not to have published it himself. Kreisel states the principle as “l(x)A (x)--f ( E x ) l A(x)for each primitive recursive A” (or similarly) without mentioning Markov. The bearing of the principle on Brouwer’s theory of the continuum will be discussed in Chapter IV Remark 18.6. Kreisel in 1959a with 1959 para. 3.52 shows that this principle is not provable in intuitionistic number theory or in the formal system of Kleene 1957. Kreisel obtains this result by applying an interpretation which he describes as “closely related to Kleene’s realizability” (1945 or IM 9 82), but which uses ideas from an interpretation of Godel’s (Kreisel1959 paras. 3.1-3.3 and Godel 1958), from which it also differs. Kreisel deals further with these matters in 1962. The formal system of Kleene 1957 is not as strong as the present formal system (letter from Kreisel, 22 November 1963). We now give, in this and the next section, a proof of Kreisel’s result for the present formal system. We arrived at this proof by attacking the problem directly with the help of some inspirations gained from Kreisel1959a and Godel 1958, and we have not determined the precise relationship of the interpretation used in it (‘,realizability’) to the one Kreisel used (1959 para. 3.52). 10.2. In realizability as treated above ($3 8, 9), an implication A XIB is realized-Y by E, if { E } [ o ~ ] is any partial recursive function q~[01] such that, for each 01 which realizes-!PA, q ~ [ 0 1 ] is completely defined and realizes-Y B ; for an 01 which does not realize-!P A, q[a] need not be completely defined. In the ‘special realizability’ (briefly, ‘,realizability’) to be introduced in this section, we shall use instead of {E}[o~] an analogous operation on E which will produce a function q ~ [ 0 1 ] (termed ‘special recursive’) which is partial recursive but such that p[a] is completely defined for every 01 of the appropriate sort or ‘order’ (determined by A) whether or not 01 ‘,realizes-!P’ A. To carry this out, we first assign ‘orders’ to the formulas, and t o the one-place (total) number-theoretic functions. A function to realize a formula must have the same order as the formula. We begin with an inductive definition of the ‘orders’ to be used. ( 1 ) 1 is an order. (2) If a is an order, so is a + 1. (3) If a0 and a1 are orders, so is (ao, al). The only orders are those given by these three clauses. Orders differently generated by use of these three clauses are different. An order given by Clause (2) is a successor order (s-order); by (3),

9

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a pair order (p-order). The orders 1+1, ( l + l ) + l , ( ( l + l ) + l ) + l , ... we write 2, 3, 4, . . . simply. If a is an s-order, a- 1 is the order b such that a = b+ 1. The variables 801, ap, . . ., aa1, "012, . . . will be used for functions of order a (as specified next). For any one-place number-theoretic functions 01 and p , we define

and say a@) is properly defined when (E!y)a(b(y))>O. Now we specify which one-place number-theoretic functions are 'of a given order'. (1) All such functions are of order 1. (2) To each order a, the functions &+lo1 of order a+ 1 are those such that, to each (function of order a) 801, there exists a unique y such that a+la("G(y)) > 0 (so a+1a(&01) is properly defined, and, for that y, a+la(%(y))= a+la(tLa)+1). (3) To each pair of orders a0 and al, the functions a01 of order a = (ao, al) are those such that pa)o is of order ao, and is of order al. An object a+l01of an s-order has a dual role; it is a number-theoretic function Is a+la(s),and it serves via (10.1) as an operator 1% a+l01(%). The operators of orders 3,4, 5, . . . are not simply countable functionals of the types 3 , 4 , 5 , . . . (Kleene 1959a), since e.g. 3a(201) will depend in general on As %(s) and not simply on I l a %(la). (An interpretation using functionals here seemed to work for all postulates except Brouwer's principle X27.1.) (We could extend our theory of orders to include the natural numbers as objects of order 0.) Any function of an order a # 1 is also of order 1, and there are other possibilities for functions to be of more than one order. By 10 we mean I s 0. For any order a, a+10 is defined by a+10(1) = 1, a+lO(s) = 0 for s f 1.

For any orders a0 and al, if a = (ao, al), "0 = <"", each order a, "0 is primitive recursive and of order a.

&lo>.Now,

for

10.3. By a special recursive function (a function special recursive in T ) we mean a function ~ ( a )where , a is a list of zero or more distinct number variables, and of zero or more distinct one-place numbertheoretic function variables for each of which a respective order is specified, such that : (a) when the ranges of the function variables among a are restricted to their specified orders, q(a) is completely

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defined, and (b) when the ranges of the function variables among a are unrestricted, p(a) is partial recursive (partial recursive in T ) . The notions extend to function-valued functions y[b] = As y(b, s). A primitive or general recursive function y(a) or y[b] is a fortiori special recursive. By (10.1) with the definition of 'function of order a+ l', a+la(%) = p(a+la, "a) with p special recursive. l function y(b, "a),there is a LEMMA 10.1. T o each f ~ ~ r t i arecursive @imitive recursive function y[b] = A% y(b, "a) szlch that, for each b for which p(b, aa) is defined for all aa of order a: A% y(b, "a) is of order a+ 1, and for each aa of order a (10.2)

{A% p(b, &a)}(%) '= p(b, aa).

Ordinarily our p(b, a a ) will be special recursive, so the conclusions will apply whenever (the functions among) b are of the specified orders. Here for brevity we have introduced the A-notation right in the lemma; cf. Lemma 8.1. If y(z, b, aa) is partial recursive and y[z, b] = A% p(z, b , % ) , and we put p(b, &a)= y(e, b, &a) for a fixed e, then y[e, b] = Aaa p(e, '6, a a ) is a A% y(b, aa). PROOF OF LEMMA 10.1. Say e.g. b is (a, bp). By the normal form theorem I M pp. 292, 330, but using ol, B instead of 6,B (end 8.2), for any Godel number e of p, for each a, bp, aa:

(4 (ii)

= U(pYT:*'(bg(y),".l(y), e, a)), T : . l ( b g ( y ) ,aol(y), e, a) for at most one y.

p(a, bp, aa)

So let y [ a , b/?] = As y(a, b/3, s) where y(a, bB>

=

U(lh(s))+ 1 if T:?'(bB(lh(s)), s, e, a ) , 0 otherwise.

10.4. Let 1, =1 $9 3 la = $9 3 (x)[h(x) = 1p(x)]. When a is an s-order, let aa =a "/3 3 P-ly)[aa(a-ly) = 9!I(8-ly)]. When a is a porder (ao, al), let aa =a ap 3 (aa)o =ao ("p)o & (aa)l =al ("/?)I. Now a, = "/I --f "a =& a/?, but (for a # 1) not in general conversely. For any two choices Alas p(b, "a) and A& p(b, "a) of A% y(b, aa), by (10.2) A1aa p(b, aa) =a+l Azaa y(b, "a) (for each b as supposed for (10.2)). For each two orders a and b, we now define an order a*b; and for e = a*b we choose a special recursive function le@a b(ee}[aa] so that, for each e ~ , a aof the specified orders, b(ee)[aa] is of order b. The

9

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definitions are by recursion on b, corresponding to the inductive definition of 'order' (cf. I M p. 260). When b = 1, e = a*b = (a, I ) + 1 and b(ee}[aa] = I s ee(<%, It s)). When b is an s-order, e = a*b = (a, b- 1)+ 1 and (using Lemma 10.1) b(ee}[%] = Ab-lp %(<"a,b-lp}), for some choice of the latter. When b is a p-order (bo, bl), e = a*b = (a*bo, a*bl) and b(ee}[%~] = (where bo{(ee)o}[aa] = (Ia*boe%bo(&*boe}[%])[(ee)~, %I, etc.). By b(ee}[a] we mean bPe}[la] for la = a. For different choices Aeeaa b(ee}[%]l and Ie@a b(ee}[%]2 of Ieeaa b(ee}[%], by irduction on b b(ee)[aa]l = b b(ee}[%]2 (for each ee, "a of the specified orders). Moreover : LEMMA10.2. I f eel =e e ~ 2 , then b(eq}["a] = b b(ee2}[aa] (where e = a*b). PROOF, by ind. on b. CASE 1 : b = 1 . Then, for all x , b("q)[%](x) = It x ) ) [def. etc.] = ee2(<%, I t % ) ) [hyp.] = b(ee2)[aa](x); i.e. b{eel}[sa] = b b(ee2}[aa]. CASE 2: b is an s-order. Then, for all b-1/?, b(eel}[%](b-l/?) = eel(<%, b-lp)) [def., (10.2)] = eez([def.] = b (bo, bl). Then b(e~l}[sa]= [hyp. ind.] = b(ee2}[aa]. LEMMA 10.3. To each partial recursive function v[b, "a], there is a primitive recursive function y[b] = :A% v[b, %] such that, ,for each b for which v[b, "a] is (completely defined and) of order b for all "a of order a : :Aa. v[b, "a] is of order e = a*b, and for each "a of order a (10.3)

b{tAaa v[b, %]}["a]

v[b, "a]. Ordinarily our v[b, "a] will be special recursive with values of order b for all b, &aof the specified orders, so the conclusions will apply whenever b are of the specified orders. We shall understand :Aaa v[b, "a] to be constructed by the method of the proof (below). Again, by the properties: If ~ [ zb,, "a] is partial recursive and y[z, b] = :Aaa ~ [ z b,, "011, and we put v[b, "a] = v[e,b, "a] for a fixed e , then y [ e , b] = :A% v [ e , b, "a] is an ;A% v[b, "a]. The subscript b in ;A% v[b, "a] may be omitted when the context makes it clear that the intended order of the operand y[b, "a] for the operation by eAaa is b. By :Aa v[b, a] we mean CAla v[b, la]. PROOF OF LEMMA 10.3, by ind. on the order b. CASE 1 : b = 1 . Put e = a*b = (a, 1 ) + 1 . Let y[b] = Ae-lp v(b, (e-lp)~,(("-+!?(O))l). Then by Lemma 10.1, for b, &a of the specified orders, y[b] is of order =b

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e, and, for all s, b{y[b]}[aa](s) = (Is ~ [ b ] ( < ~Ita s)))(s) , = y[b](<"a, I t s ) ) = q(b, "a, s) [(10.2) etc.] = q[b, "a](s), i.e. b{y[b]}[%] = b v[b, "a]. CASE 2: b is an s-order. Put e = a*b = (a, b-l)+l. Let y[b] = Ae-lp q[b, (e---lp)o]((e-1,!?)1).Then by Lemma 10.1, y[b] is of order e, and, for all b-1j3, b{y[b]}[%](b-lp) = {Ab-lp y[b](<%, b-l,!?>)}(b-lB) = y[b](<"a, b-lp>) = q[b, aal(b-lp), i.e. b{y[b]}[aa] = b q [ b , "a]. CASE 3: b is a p-order (bo, bl). Put e = a*b = (a*bo, a*bl). By hyp. ind. we can find yo[b], yl[b] with values (for each b as supposed) of orders a*bo, a*bl such that for all %, bo{yo[b]}[%] =bo (q[b, aal)O, bi { ~ i [ b l } [ ~=abll ( d b , &~I)I.. Let y[b] = , which has values of order (a*bo, a*bl) = e. Now b{y[b]}[%] =
b{,"Ax q[b, xl}[x] = b d b , XI. Ordinarily our q[b, x] will be special recursive with values of order b for all b, x (b of the specified orders), so the conclusions will apply whenever b are of the specified orders. We define 1 { 2 ~ }= I s %(It s) (= x[%] with a special recursive x). For q [ b ] a partial recursive function, we write 2A q[b] = Alaq(b, la(0)). Then, for b such that p[b] is completely defined: 2A q[b] is of order 2, and (using (10.2)) (10.3b) 1{2Aq[b]} = q[b]. Ordinarily our q[b] will be special recursive, so the conclusions will apply whenever b are of the specified orders. Also (8.lc)-(8.3c) for k+Z = 2 have analogs; but all we shall use is b(e.z}["op~,* l p l ] as abbreviation for b(ec}[
10.5. Now we shall assign to each formula E an order e (= 'order E'). Simultaneously, for each list Y of variables including all which occur free in E, we shall define when a function ec of order e 'specially realizes' E for a given assignment Y of numbers and functions as the

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values of Y, or briefly when e~ ‘,realizes-Y’ E ; only a function e~ of order e = order E can ,realize-Y E. 1. A prime formula P is of order 1. 1e ,reaZizes-Y P, if P is true-!P. 2. e = order A & B = (a, b) where a = order A and b = order B. e~ ,realizes-Y A & B, if (%)o ,realizes-Y A and ( % ) I ,realizes-Y B. 3 . e = order A V B = (1, (a, b)) where a = order A and b = order B. ee ,realizes-Y A V B, if (ea(O))o=O and (ee)l,o ,realizes-Y A, or (ee(0))ofO and ( e ~ ) l ,,realizes-!P l B. 4. e = order A 2 B = a*b where a = order A and b = order B. ee ,realizes-Y A 2 B, if, for each a a , if aa ,realizes-Y A then b(ea][aa] ,realizes-!P B. 5. e = order -,A = a*l where a = order A. e~ ,realizes-Y -,A, if, for each aa, not &a ,realizes-!€’ A. (Equivalently, e~ ,realizes-Y -,A, if e~ ,realizes-Y A 2 1 =O under Clause 4.) 6 . e = order VxA = l*a where a = order A. e~ ,realizes-Y VxA, if, for each x , a(ea}[x] ,realizes-Y, x A. 7. e = order 3xA = (1, a) where a = order A. e~ ,realizes-Y 3xA, if (%)I ,realizes-Y, (ec(0))o A. 8. e = order VaA = l*a where a = order A. e~ ,realizes-Y VaA, if, for each a, &(ee)[a],realizes-Y, a A. 9. e = order 3aA = (2, a) where a = order A. ee ,realizes-Y 3aA, if ( e e ) ~,realizes-Y, l{(ee)o} A. LEMMA10.4. Let Yl be a list of those of the variables Y which occur free in E, and let Y1 be the list of the corresfionding ones of the numbers and functions Y . T h e n e~ ,realizes-Yl E if and only if e~ ,realizes-Y E. LEMMA 10.5. If eel ,realizes-Y E, and eel = e e ~ 2 then , e q ,realizes!P E. PROOF,by induction, using Lemma 10.2 in Cases 4, 6, 8. We say a closed formula E is ,realizable, if a general recursive function ee ,realizes E ; an open formula, if its closure is (FIRST DEFINITION).

Equivalently (proof below), E is ,realizable, if there is a general recursive function 9, (a general recursive ,realization fwnction for E in Y) such that, for each Y , p[Y] srealizes-Y E (SECOND DEFINITION). This notion of ,realizability is independent of the choice of the list Y, as before (cf. 8.7). To illustrate the equivalence of the two definitions when E is open, say e.g. the closure of E is VaVxE. Write e, f , g for the orders of E, VxE, VaVxE (f = l*e, g = l*f).

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First, suppose q~ is general recursive, and that (1) for each a and x , rp[a,x] (is of order e and) ,realizes-a, x E. Using Lemma 10.3, gAa fAx y[a, x] is a primitive recursive function ga, which we shall show ,realizes VaVxE. By Clause 8, we need that, for each 01, f{ga}[a] ,realizes-a VxE. Consider any a. By (10.3), f{ga}[a] = f f A x y [ a ,x ] ; so by Lemma 10.5 it will suffice to show that f A x ~ [ ax], ,realizes-a VxE, i.e. by Clause 6 that, for each x, e{fAxp[a, x]}[x] ,realizes-a, x E. This follows from (1) by Lemma 10.5, since by (10.3a) “(‘Ax ~ l [ axI)[xI , =e q ~ [ aXI. > Inversely, if g~ is a general recursive function ,realizing VuVxE, then, using Clauses 8 and 6, l a x e{f{g~}[a]}[x] is a general recursive ,realization function for E in a, x. The IT, -0 and C / modifications of ,realizability (e.g. when 0 , T C C , C/,realizable-@/T),and C/,realizes-Y, can be formulated as before (cf. 8.5, 8.7, 9.3).

10.6. LEMMA10.6. Lemma 8.3 holds reading “,realizes-” for “realizes-”. PROOF.Cases 2 and 7 read exactly as before. CASES 1 and 5: prime or -A. e0 (end 10.2) where e = order E. CASE 3: A v B- < ( X [ @ ] ( o ) ) O , <(X[@])l,OA,(X[@])l,lB>>. CASE 4: A 2 B. eA% b{~[@]}[*a] where a, b, e are the orders of A, B, A 3 B. The modifications in CASES 6, 8 and 9 are similar. 10.7. LEMMA 10.7. To each formula E of order e containing free only the variables Y and not containing V or 3 , there is a +rimitive recursive function ~ E Eof order e such that, for each Y : (i) (ii)

If (Eer)[“E ,realizes-Y El, then E is true-Y. If E is true-Y, then ~ E E,realzzes-!P E.

PROOF,by induction. According as E is of the form P (a prime formula), A & B, A 2 B, YA, VxA, or VuA, let ~ C Ebe as follows, where a = order A and b = order B: A t 0, <%A, be^>, eAaa be^, e0, ~ A x ~ Eor A ,eAaa~A.- Cf. Remark 11.9.

Q 11. Special realizability under deduction in the intuitionistic formal system. 11.1. LEMMAS1 1.1, 1 1.2. Lemmas 9.1, 9.2 hold reading “ec ,realizes” in place of “E realizes”.

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127

THEOREM 11.3. (a) If I’ t E in the intzlitionistic formal system of analysis, and the formzllas I’ are ,realizable, then E i s ,realizable. (b) Similarly, reading “,realizable/T” in place of “,realizable”. (c), (d) Theorem 9.3 (c), (d) hold reading “,realizable” in place of “realizable”. PROOF.(Cf. the proof of Theorem 9.3.) (a) We shall understand that A (A(x), A(a), A(a, P)), B, C, R(a) (when present) have orders a, b, c, r ; the prime subformulas have order 1. We show the orders of other subformulas as superior prefixes on the formal operators (IM p. 73). la. A d 3 (B 9 A) is ,realized-Y by dAaa cAb/? &a. For, this is of the required order d = a*c where c = b a , by Lemma 10.3. Suppose (1) &a ,realizes-Y A; by Clause 4 in 10.5 we must infer from ( I ) that C{dAaor cAb/? %}[aa] ,realizes-Y B 3 A. But by (10.3), c{dAaa cAb/? %}[%I =c CAbP aa. So by Lemma 10.5 it will suffice to infer from (1) that c l i b p a a ,realizes-Y B 3 A. By Lemma 10.3 cAb/? 801 is of the required order c = b*a. Suppose (2) b/3 ,realizes-Y B; we must infer from (1) and (2) that a(cAb/3 aa}[bP] ,realizes-Y A. By (10.3), a(“Abb aa}[bP] =a “a; so by Lemma 10.5 what we need follows from (1). lb, 7, 3,4a, 4b, ION, 10F, 1 lF, 1 I N are treated as before, supplying subscripts ,and order superscripts, and using Lemma 10.5. 5a. A d~ A C V B. dAaol> (end 10.2). 5b. B d~ A C V B. dAbp < 1, >. 6. (A d 3 C) 5 ((B e 3 C) h 3 (A fV B g 3 C ) ) . ‘Adz hAep gAfu At sg((‘40))0)’ (” {d4 [(‘4LO]) (4 sg((‘40))0 )* (“(“PI Nf41,lI 1(t)* 81. C q A 9 (A d 3 B). eAcn do. 13. A(0) d& CVx(A(x)b 3 A(x’)) 9 A(x). :Ada p[x, da], where p is defined by

+

p[O, p [x’,

Writing p [ x , da]

= (da)o, = “{b{ (“01)

= At p(x, d a ,

dall.

11 [XI} [ p [x,

t), this takes the form

p(0, d% t) = Y ( d % t) x(x, d% At p ( x 9 da,4, t ) p(x’, 4

9

=

where y is primitive recursive, x is partial recursive, and, for d a , a/? of the specified orders, I t y(da, t) and At ~ ( xd ,a , &PIt ) are (completely defined and) of order a. So, by induction on x, for d a of the specified order, At p ( x , d a , t) is of order a. That this p(x, d a , t ) is partial, and hence special, recursive is seen as before.

128

REALIZABILITY

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14, 17, X1.1, 16, prime axioms, X2.1 as before. 15: 1*10. RULESOF INFERENCE. 2. A, A =I B / B. Noting 10.5, we choose Y to include all variables free in A 3 B. By the hyp. ind., there are general recursive functions 01 and p such that, for each Y , a[Y] ,realizes-Y A and y [ Y ] ,realizes-!P A 2 B. Let y [ Y ] = b { y [ Y ] } [ a [ Y ] ] . For each !P, .[!PI is of order a and y [ Y ] is of order e = a*b, so b{y[Y])[a[Y]] is (defined and) of order b ; so y[Y] (= ils y [ Y , s]) is general recursive. For each Y , y[Y] ,realizes-Y B. 9N, 9F, 12F, 12N as before. AXIOMSCHEMA x26.3c. fVaElx[R(~(x)) d& CVy(R(E(y))b~ x=y)] J& iVa[Seq(a) g& R(a) h~ A(a)] P& Wa[Seq(a) l& kVsA(a*2s+l) m~ A(a)] A( 1). We proceed as before down through (8),now supplying subscripts and order superscripts. Moreover, we now observe that, for Pn of order p, R(pn, a) and R1(Pn, a) are always defined, i.e. whether or not pn ,realizes- !P the antecedent. Furthermore, the definition of q(Pn, a, u) coincides for a E Sp;C with a recursion of form corresponding to the inductive definition of Sy (cf. I M p. 260), call it a bar recursion. By the corresponding form of proof (IM p. 259), always: (9a) a E S y -+ { ~ [ p na] , is (completely defined and) of order a). But we don’t know that q[Pn, 11 is always of order a, which we now would require to complete the proof as before, since the proof of (8) uses the assumption that Pn ,realizes the antecedent. In the following this assumption is not made, except as indicated. We define a second special recursive predicate Rz(Pn, a) thus.

Rz(Pn, a) N [a = <(x) for

011 =

At ( a ) t ~ l x, 2 (e{(Pn)~,o)[~~](0))~].

Clearly: (10) Rl(Pn, a) --f R@n, a). We now show that: (11) (a)(Ex)Rz(Pn,G(x)). Consider any 01. Let xo = (e{(Pn)o,o}[a](O))0.For a fixed pn of order p, since il01 xo is partial recursive in Pn, the computation of xo uses only the first yo values of a , for some yo (cf. I M pp. 330, 292). Put x = max(x0, yo). Since x 2 yo, xo = (e{(Pn)0,0}[a1](0))0 for a1 = ilt ( G ( x ) ) t ~ l Now . Z(x) = al(x) with x 2 xo. So Rz(Pn, Z(x)). Also: (12) R2(Pn, & ( y ) )& Rl(pn, G(y)) --f (Ex),,,R(pn, E(x)). For, ) G(x)for 011 = ilt ( & ( y ) ) t1,~ x = y > (~{(~n)0,0)[~11(0))0 = XI. put ~ ( y= Then G(x1) = <(XI), whence R(pn, &(XI)). Let Sy be the set of the sequence numbers barred with respect to AU Rz(pn, a).

9

11

SPECIAL REALIZABILITY UNDER DEDUCTION

129

Similarly to q[Pn, a ] , we find a partial recursive q2[Pn, a] = I u qz(Pn, a, u) satisfying " ~ ( z t ) if R@n, a) & &(pn, a ) , qz(Pn, a, u) II as before if R1(Pn, a ) , as before otherwise.

I

Using (lo), similarly to (9a): (13) a E S? + {qz[Pn,a] is of order a}. By (1 1) : (14) 1 E ST.Hence qz[Pn, 11 is of order a. To show that qllPnqz[Pn, 11 ,realizes-Y the axiom, it remains for us to show that qz[Pn, 11 ,realizes-Y A(l) when Pn ,realizes-!&' the antecedent. So, with that assumption in force again, consider the set S"" of the sequence numbers securable but not past secured with respect to Aa Rl(pn, a ) . (Using (lo), S"" C SD; C SF.) By (12) with (6), a E S"" + R2(Pn, a) & Rl(pn, a). Therefore on S"", q2(pn, a, u) satisfies the same bar recursion as q(Pn, a, u); hence it is the same function to within possible differences in the results of the operations by k l l s in the "otherwise" case. Hence for the same reasons as (7): (15) a E S"" + {q~[Pn, a] ,realizes-Y, a A(a)). But using (8): (16) 1 E S"". By (15), (16) and Lemma 11.1 (a), qz[pn, 11 ,realizes-Y/ A(1). AXIOMSCHEMA X27.1. CVab3pA(a, p) q 2 ~37nVa{hVts3y[2(2t+l*E(y))>O f& ~ V Z ( T ( ~ + ~ * Z >O ( Z ) )d 2 y=z)] m& Wp[Wti3~~(2t+l*E(y))=P(t)+ 1 k~ A(a, p)]}. Assume that Cn ,realizes-Y Va3PA(a, p). Then (1) for each a, (b{%}[a])1 ,realizes-Y, a, @I A(a, p) for @1= l{(b{cn}[a])~}. Let zl[Cn,t] = I s z ~ ( ~t,ns), = Ala @l(t)= Ala (b(cn}[la])o(Axt). By Lemma 10.1, for each t, zl[cn, t] is of order 2, so, for each a, (2a) (t)(E!y)zl(cn,t, g(y))>O, and @y (10.2)) {TI"% tI}(a) = @I@), whence by (10.1): (3a) (t)zl(cn,t, E(yt))=@l(t)+l where yt = , u y t ~ ( ~t,n , E(y))>O. Let z = I s zl(cn, ( s ) o ~ lIIiO, (3) (t)z(2t+l*E(yt)) =@&)1 where yt = ,uyz(2t+l*E(y))>O. We continue as before. So putting hpo = hllt <,uyz(2t+l*E(y))>O,
+

11.2. COROLLARIES11.4-1 1.6. Corollaries 9.4, 9.5 (with different proof) and 9.6 hold reading ",realizable" in place of "realizable". 11.3. THEOREM 11.7. (a)C Let A(x, y) be picked by Lemma 8.5 to numeralwise express a general recursive predicate A (x,y). Classically,

130

REALIZABILITY

CH. I1

Vx[lVylA(x, y) 2 3yA(x, y)] i s realizable. @) Let T(x, y) be Tl((x)o, ( x ) ~y) , where Tl(z, x, y) i s picked by (the method of f~roof of) L e m m a 8.5 to numeralwise express Tl(z, x , y ) (with x free for z, and z free for y). T h e n Vx[lVy-rT(x, y) 2 3yT(x, y)] i s un,realizable. By #19 and Lemmas 5.2 and 4.2, T(x, y) numeralwise expresses Tl((40, (41,y ) . (c)C Let Ao(x, y) and Al(x, y) by picked by L e m m a 8.5 to numeralwise express general recursive predicates Ao(x, y ) and Al(x, y ) , respectively. Classically, Vx[l(VyAo(x, y) & VyAl(x, y)) 2 lVyAo(x, y) V -,VyAl(x, y)] i s realizable. (d) Let Wo(x, y) be Tl((x)l,x, y) & Vzz,,lT1((x)o, x, z), and Wl(& y) be Tl((X)O, x, y) ~ ~ z ~ y l ~ l x, z), with z l y prime. T h e n Vx[l(VylWo(x, y) & VylWl(x, y)) 2 lVylWo(x, y) V l V y l W l ( x , y)] i s un,realizabZe. By I M p. 202 (C) and (E), WO(X,y) and Wl(x, y) numeralwise express Wo(x,y ) and W l ( x ,y ) , respectively, I M p. 308. PROOF. (a) It is realized by Ax Aa < p y A ( x ,y ) , using Lemma 8.4a. For, consider any x . Suppose a realizes-x l V y l A ( x , y). -_ Then by Lemma 8.4a (i), ( y ) A ( x ,y ) , whence classically ( Ey ) A( x ,y), so A(x, y) is true-x, ,uyA(x, y ) , so by Lemma 8.4a (ii) realizes-x, PYA (XI Y ) A(% y), so Y ) , Ea(x,y)> m3lizes-x 3yA(x, y). (b) Suppose a general recursive function ,realizes fVx"+Vya-.ltT(x, y) e 2 d3ytT(x, y)]. Then, for all x : (1) e{fe}[x] ,realizes-%l V y l T ( x , y) 2 3yT(x, y). By Lemma 10.7 (ii) (and proof) : -_ (2) ( y ) T l ( ( x ) o ,( x ) y~) +. (CO ,realizes-x l V y l T ( x , y)}. Now if (Ey)Tl((x)o,(41, Y ) , then @%((x)o, (%)I,y ) , whence by (2) and (1) d(e{fe}[x]}[cO] ,realizes-%3yT(x, y), whence (d(e{fe}[x]}[cO])l,realizes-x, w(x) T(x, y) for w(x) = (d(e{fe}[x]}[CO](O))~, whence by Lemma 10.7 (i) Tl((x)o, ( x ) ~w(x)). , The converse is immediate. Thus: (3) (Ey)Tl((x)o, (%)I, y ) = Tl((x)o, (%)I, w(x)) for the general recursive function w(x) just introduced. Substituting <x, x> for x : (4) (E y)Tl (x,x , y ) = TI(%,x,v(<x,x ) ) ) , contradicting that (E y )Tl (x,x , y ) is not general recursive (IM p. 283). (c) Ax Aa < p ( x ) ,At 0 ) where 0 if do(% PY[JO(% y ) v Al(% y)l), P(4 1 if Ao(% PY [Jo(x, Y)v &(x, Y ) l ) . (d) Suppose a general recursive function ,realizes hVX[dl(bvyalwWo(X,y) C& bVyal"Wl(x, y)) g 3 elbvyalwWo(X, y) fV e+VyalwWl(x,y)].Then,for allx: (1) g{he}[x] ,realizes-xl(VylWo(x,y) f~

-1

( ( ~ )

9

11

SPECIAL REALIZABILITY UNDER DEDUCTION

131

& VylWl(x, y)) 5 ) lVy-.IWo(x,y) V -.IVy-.IWl(x,y). By Lemma 10.7 (ii): (2) (y)Wo(x,y) & (y)Wl(x,y) (60 ,realizes +ylWo(x, y) & --f

whence (C) (y)TI(%,y). By (B), (2) and (1) : (D)f{g{he}[x]}[dO] ,realizes-x 1Vy1Wo(x, y) v lVYlWl(X, y). Put 4%) = ( ‘ { ~ { ~ ~ ) ~ ~ I ) Now ~~~l(~))O. (E) w(x) = 0, as otherwise (f{g{he}[x]}[~O])1,1would ,realize-% -_ lVylWl(x, y), so by Lemma 10.7 (i) (y)Wl(x,y), contradicting (C). Summarizing (from (A)): (3) (Ey)Wo(x,y) -+v(x) = 0. Similarly: (4) (Ey)Wl(x,y) +v(x) # 0. Thus D2 = R[w(x) = 01 and 0 3 = ~ [ v ( x )+ 01 = D2 are disjoint general recursive (a fortiori, recursively enumerable) classes, containing C O = 1(Ey)Wo(x,y) and C1 = i(Ey)Wl(x, y), respectively, whose union is all natural numbers. This contradicts I M pp. 311-312. REMARK 1 1.8. The reasoning under (a) fails with ,realizability, because not in general completely defined; e.g. it is not when A(%,y) 5 TI((+, (%)I, y). So ‘Ax eAca in general a function of order f. The reasoning under (b) fails for realizability, because v(x) = ({{e}[x]}[AxO](O))o is not a general recursive function, but only partial recursive. REMARK 11.9C. Theorem 11.7 (a) for a primitive recursive A(%,y) is immediate by a classical application of Lemma 8.4b (ii), if the symbolism and postulates of Group D are extended (within the framework laid down in 5.1) to give a prime formula A(x, y) expressing A (x,y). Therefore Lemma 8.4b cannot hold for ,realizability independently of how far we extend our Postulate Group D ; in particular, it fails if we extend that to reach the representing function of Tl(z, x, y ) , as the application just described would then contradict Theorem 1 1.7 (b). COROLLARY1 1.1OC. The following classically provable formulas are formally undecidable in the intuitionistic formal system of analysis : (a) (b)

(c)

VzVx[lVylTl(z, x, y) 5 ) 3yTl(z, x, y)] (Markov’s principle MI). v x [ l ( v y l w o ( x , y) VYlWl(X, y)) 3 l~YlWO(X, y) v lVYlWl(X, y)1 (an instance of De Morgan’s classical law). Vx[3yTl((x)o, (X>L y) V 4yTl((x)0,@)I,y)] (Law of excluded middle with one quantifier).

132

RE ALIZABILITY

CH. I1

(We show intuitionistically that each of the three formulas is unprovable, classically that its negation is unprovable.) PROOF.(a) Using "25.3, MI is equivalent to Vx[lVy-rT(x, y) 3 3yT(x, y)], which by (b) of the theorem with Theorem 11.3 (a) is unprovable. By (a) of the theorem with Corollary 9.4 and Theorem 9.3 (a), lVx[lVylT(x, y) 2 3yT(x, y)] is unprovable. (c) Similarly (using only $3 8, 9) from the results of 8.6 TI 6, adapted inessentially to this example. Note that (c) I- (a). Also it is not hard to see that (c) 1 (b). It seems to be an open question whether the double negations of these formulas are provable. By the results with 1945-realizability (8.6 7 6, I M p. 511), -11(c) is unprovable in intuitionistic number theory, even including the postulates of Group D (however extended) with only number variables.

CHAPTERI11

T H E I N T U I T I O N I S T I C CONTINUUM by RICHARD E. VESLEY7

Q 12. Introduction. We shall develop the intuitionistic theory of the continuum in the formal system of Chapter I. Our aims are, first, to investigate the adequacy of the system for this development, and, second, to provide an exposition of the theory. As Beth has observed (1959 p. 422), “the central place in intuitionistic mathematics is occupied by the theory of the continuum”. Any formal system for intuitionistic analysis should provide the means for a development of this theory at least through the well-known uniform continuity theorem (9 15 below). Such a development may clarify for some readers the intuitionistic sources, which (except for Heyting 1930, 1g3oa) are deliberately non-formal. Of the many sources (Brouwer 1918-9, 1924, 1927, 1928a, etc., Heyting 1952-3, 1956), we rely on two primarily, but not entirely. In $9 14, 15 we follow rather closely the exposition of Heyting 1956. In 3 16 we prove formally some less well-known theorems from Brouwer 1928a (not appearing in Heyting 1952-3 or 1956). Although the main objectives here are foundational analysis and clarification of the existing intuitionistic theory of the continuum, some additions are made to that theory. As one detail which seems to be developed here for the first time, we prove (*R14.11 below) the equivalence of the notion of sharp difference ( ZSin our symbolism), which Brouwer introduced in 1928a, to his notion of apartness #, introduced in 191&9 I1 and frequently used in intuitionistic writings (e.g. in his 1954, and Heyting’s 1956), of real number generators. Our proof of the equivalence uses Brouwer’s principle (9 7 above). This equivalence of fs to # we also use in simplifying the hy7 This chapter is a revised version of a Ph. D. thesis, written under the direction of Professor S. C. Kleene, and accepted by the University of Wisconsin May 28, 1962.

134

T H E INTUITIONISTIC CONTINUUM

CH. I11

pothesis in Brouwer’s formulation 1928a of the free-connectedness property (our *R14.13-*R14.14), and (in Remark 16.1) we indicate a reason why Brouwer could not have established the property quite as he formulated it.

Q 13. Real number generators and real numbers. Brouwer studies various formulations of the continuum in different papers. In general, he takes the following two steps. First, he describes a particular species of “points” (1927 p. 60) or “real number generators” (Heyting’s term in 1956 p. 16). (Other terms are also used.) These are infinite sequences and may be, for example, infinite sequences of rational numbers or dual fractions with some convergence condition (Brouwer 1928a p. 5, Heyting 1956 p. 16), sequences of nested dual intervals with a convergence condition (Brouwer 1927 p. 60, 1949 p. 122), or choice sequences of the rational numbers producing objects analogous to Dedekind cuts (Brouwer 1924-7 I1 p. 467). Second, he defines an “equality” (or “coincidence”) predicate over these infinite sequences and a “point core” (Brouwer 1927 p. 60) or “real number” (Heyting 1956 p. 37) as an equivalence class with respect to this equality predicate. In this form, as a “species of second order” (cf. Heyting 1956 p. 38), the continuum appears not to be an object for study in the formal system. But the elements of the underlying species of points or real number generators can be studied and the theory developed on this basis. Properties of real numbers will then appear as just those properties of real number generators which depend only on the equivalence classes with respect to the equality (or coincidence) predicate to which the generators belong. We use from now on the abbreviation “r.n.g.” for real number generator(s). Of the possible species of r.n.g. mentioned above we choose to consider the species of convergent sequences of dual fractions. As a preliminary we could without difficulty develop the theory of dual fractions (or of rational numbers in general) in the formal system. But we find it simpler to proceed directly to the convergent sequences. We consider only sequences of the form a(O),a(1)/2, a(2)/22, a(3)/23,. . ., which we study in the formal system through the sequences a(O),a ( l ) , a(2), 43), . . . of the numerators of successive fractions. We may write the convergence condition for these sequences as ( k ) ( E x ) ( f ~ ) \a(x)/236- a(x+p)/2“f27 < 1/2k, which formalized is the right side

9

13

REAL NUMBER GENERATORS

135

’ an abbreviation for this formula, *RO.l of “RO.1. Taking “ ~ E Ras holds by I M *19. We abbreviate agR & PER as “a, PER”, etc., and as “a$”). similarly with other uses of ‘‘E” (also k or~R Vk3xVp2k12Pa(x)-a(x+p) I <2X+P. *RO. 1.

-

This formulation of the convergence criterion corresponds to Heyting’s definition 1956 p. 16 of a “Cauchy sequence”, except that we are using sequences of dual fractions instead of sequences of rational numbers in general, and are confining our attention to non-negative dual fractions, for reasons of convenience. Cauchy convergence is usually written a little differently, namely, in our context, (k)(Ex)(p)(q)((ll(x+p)/2.+p-,(x+4)/2”fql < 1/2k.This formalizes as the right side of *R0.2, which we abbreviate “a~R1”. “R0.2.

t aeR1- Vk3xVpVq2kl2qa(x+p) -2pa(x+q) 1 <2X+p+q.

-

But these two formulations of Cauchy convergence are equivalent. I- ~ E R a ~ R 1 . “R0.3.

PROOF.I. Assume ~ E R and , after V-elim. and prior to 3-elim., (i) Vp2k+ll2Pa(x)-a(x+p)( <2X+P. Thence by V-elims., (ii) 2k+l12pa(x)-a(x+p)I<2x+P and (iii) 2k+112qa(x)-a(x+q)I <2x+q. Now 2k+1)2qa(x+p)-2Pa(x+q)l < 2k+l12qa(xfp)-2p+qa(x)I 2k+lI2P+qa(x)-2pa(x+q)I [*11.5; *145b with “3.91 = 2Q2k+112pa(x)a(x+p)I+2p2k+l12qa(x)-a(~+q)I [*11.9 with *3.3, *11.4] < 2q2X+P+ 2P2X+q [(ii), (iii), *145a, *144a, *134a] = 2-2X+P+q,whence by *145a, (iv) 2k12qa(x+p)-2pa(x+q)l<2x+p+q, and by V-, 3- and V-introd., aER1. 11. Use p=O in a ~ R 1 and , change the bound variable q to p. We follow Heyting 1956 p. 41 in considering also the “canonical” r.n.g. (abbreviated “c.r.n.g.”). These are sequences of the alreadydescribed kind with a prescribed rate of convergence given by (x)la(x)/F-a(x’)/F‘l< 1/2”’. This leads to the formula on the right side of *R0.4, which formula we abbreviate “ ~ E R ” ’ . 1 aER’ Vxl2a(x)-a(x’) I< 1. *R0.4.

+

-

Equivalent formulations are given by: *R0.5a-c. k 12a(x)-a(x’) 1 I 1 N [2a(x)
--

+

+

N

-

136

CH. I11

THE I N TU I TI O N I S TI C CONTINUUM

it will suffice to prove (a) A B, (b) B 2 C, ( c ) C 2 D, (d) D 3 B. (a) By “11.15a. (b) B gives four cases: (=, <), (<, <), (<, =), (=, =). But (=, =) is impossible, while (=, <), (<,<), (<, =) give the three cases of C, respectively (using * 138a). (c) Using *6.15, 2 u ( x ) ~2 l 2a(x) < 2a(x)+l. Using this, D follows in each of the three cases of C . (d) Assume D. Then 2a(x) 2 ( 2 a ( x ) ~ l ) + l[X8.1, “8.41 a(x’)+l [D, *144b]. Thus B. The reason for our considering two species of r.n.g. simultaneously is that the approach to the theory through R has some advantage for real number arithmetic (cf. Remark 14.1), while the approach through R’ permits us to bring in the spread concept in 14.1. These two approaches give rise to species of real numbers which are identical (in the sense of Brouwer 1924-7 I pp. 245-6), as will be established formally by showing that to each element of one species there is an equal element of the other. A mapping from R‘ to R is provided by *R0.7; one from R to R’ will be established in *R1.11 of 14.3 after introduction of the equality predicate for r.n.g. *R0.6. k ~ E R2’ VpVxl2pa(x)-a(x+p)I<2p. *R0.7. I- XER’ZI ~ E R . N

PROOFS.“R0.6. Assume aER’. We shall deduce Vx12pa(x)a(x+p)l<2P by ind. on p. I N D . STEP. I2P’a(x)-a(x+p‘)I I2P’a (x)-2pa (x’)I I2pa(x’) -a(x+ p’) I = 2p 12a(x)-a (x‘)I I2pa(x’)a(x’+p)l < 2p+2p [aER’, hyp. ind.] = 2p’. *R0.7. Assume CCER’. Then 2kl2Pa(k)-a(k+p)l<2k+p [*R0.6]. By V-, 3- and V-introd., Vk3xVp2kl2Pa(x)-a(x+p) I <2X+P, i.e. ~ E R .

+

+

Q 14. The spread representation; basic properties of the continuum. 14.1. In “R0.8 we show that the r.n.g. of the second kind constitute a spread. For the meanings of “Spr(o)” and ‘‘a~ts’’see 6.9. *R0.8. I- 3a[Spr(o) & o ( l ) = O & V a ( a ~ o ~ E R ’ ) ] . PROOF.We introduce o by Lemma 5.5 (c) (cf. the proof of *26.4a), using “23.5 [*23.2 with “6.31 to express a [a(b) for b
-

(A)

Vao(a)=

9

14

BASIC PROPERTIES O F THE CONTINUUM

137

By "22.3, *B4 (with *6.7), "22.1, *B6 and *143b (with "3.10, *18.5), Seq(a) & lh(a)> 1 3 IIi 1 . Also 12((t)lh(t)L2L

')I

l)-((t)lh(t)LIL

= 12((a)lh,a)L,L1)-2((a),h(a)i,-

=

0 I 1, and b(&O 3 Vso(a*2s+1)>0], assume Seq(a) & o(a)>O. By (A), a f l , whence lh(a*2s++1)>1. So, since = .(a) > 0, (A) gives o(a*2s+1)= 1 > 0. 4&Ii
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14.2. In *R0.9, p, XI, x are distinct variables, and A(p) is a formula in which x1 and x are free for p. "R0.9. *R0.10. *R0.11.

I- VpA(~l+p)8~X i I X 3 VpA(x+p). I- V p V q 2 k l 2 q ~ ( ~ l + p ) - 2 P ~ ( ~ l + q ) I < 2 ~ ' 8~ + ~X+l ~l X =I VpVq2k(2qa(x+p)-2pa(x+q) I <2x+p+q. I- Vp2k+l12Pa(~1)-~t(~l+p)I<2~'+~& X i l x =I Vp2kl2pa(x)-a(~+p)(<2~+p.

PROOFS. *R0.9. Assume VpA(xl+p) & x l l x . Assume x=xl+c (cf. 5.5 7 4). By V-elim., A(xl+c+p), whence A(x+p). By V-introd., V P W + P). "RO.10. Assume the antecedent and x=xl+c. Using V-elim. with c+p and c+q for p and q, 2k~2c+~a(x1+c+p)-2c+Pa(x1+c+q)I < 2x1+0+P+c+q. Dividing by 2c (i.e. using * 145a with * 1 1.9 and *3.9), and replacing xl+c by x, 2k12qa(x+p)-2Pa(~+q)I < 2x+P+q. *RO.11. Assume the antecedent. The steps from (i) to (iv) in the proof of "R0.3 and V-introd. give VpVq2kl2qa(xl+p)-2Pa(xl+q) I < zx1+P+q. SO by *RO. 10, V ~ V q 2 ~ 1 2 % ( ~ + ~ ) - 2 p ~ ( x + < q ) I2x+P+q, whence putting p=O, Vp2k12Pa(x)-a(x+p)I < 2x+P.

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THE INTUITIONISTIC CONTINUUM

CH. I11

14.3. The coincidence predicate for r.n.g. (Heyting 1956 p. 16) is expressed by the formula on the right in "R1.1, which formula we abbreviate as " a A p " . In "R1.3 a=p is Vx(a(x)=p(x)) (cf. 4.5). "R1.1. "R1.2. "R1.3. "R1.5. "R1.7.

I- a&@ Vk3~Vp2~la(x+p)-p(xfp)I<2~+p. I- Vpa(x+p)=p(x+p) 3 I- vt,,,a(t)=p(t) 3 a+p. I- a=@ 3 a 6 p . "R1.4. I-a+a. I- a&@ 3 PA... "R1.6. k a2.B & p+y 3 a'y. I- ~ E & R 3 PER. N

PROOFS. "R1.2. Assume Vt,,,a(t)=p(t). Then a(x+p)=P(x+p), so 2kla(x+p)-p(x+p)\ = 0 [*11.2] < 2x+P, whence by V-, 3- and V-introd., a&@. "R1.4. From a=a (4.5 T[ 5) by "R1.3. "R1.5. Use "11.4. . V-elims., and omitting 3x1, 3x2 pre"R1.6. Assume a+fi, p ~ y By ceding 3-elims. : Vp2k+1(a(xl+p)-p(x~+p)~<2x'+p, Vp2k+11P(~~+p) -y(x2+p)I <2x"p. Letting x=max(xl, x2) and using "R0.9, *8.4: Vp2k+lja(x+p)-p(x+p)1<2Xfp, Vp2k+11P(~+p)-y(~+p)I<2x+p. Now 2k+11a(x+p)-y(x+p)/ I 2k+11.(~+p)--P(~+~)I+2k+11P(~+p) -y(x+p)l [*11.5, etc.] < 2X+P+2x+p = 2.2X+P. So 2kla(x+p)y(x+p)I<2x+P. By V-, 3- and V-introd., a+y. "R1.7. Assume ~ E R a*p. , By V-elims., and preceding 3-elims.: Vp2k+2Ia(xz+p)-P(x2+p) 1 <2xa+p. Vp2k+312Pa(x1)-a(x1+p) I Letting x=max(xl, xa), and using *RO.ll and "R0.9, with *8.4: (i) vp2"+212Pa(x)-a(x+p)l<2x+p, (ii) Vp2k+2la(x+p)-p(x+p)I<2X+p, whence (V-elim. with 0 for p, *145a, etc.) (iii) 2k+212Pp(x)-2Pa(x)I < 2x+p. NOW 2'+212Pp(~)-p(~+~)I5 2'+212pp(~)-2%(~)If 2k+2\2Pa(x)-a(x+ p) I 2k+21a(x p) -p (x+ p) I < 2x+p 2x+p+2x+p [(iii), (i), (ii)] < 2x+p+2. So 2kI2PP(x)-p(x+p)I < 2X+P. By V-, 3and V-introd., PER.

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§ 14

BASIC PROPERTIES O F THE CONTINUUM

139

PROOFS. *R1.8. CASE 1 : y<x. Assume y+c=x, a=2Cq+r & r<2C

(using * 146a). SUBCASE1.1 : 2Y+1)2Ya-2xql<2Y+X. Use 3-introd. SUBCASE1.2: 2y+x < 2~+112Ya-2xql = 2~+1\2~(2cq+r)-2y+CqJ = 2~+1(2yr)[*11.8]. So assume 2y+x+d' = 2y+1(2yr). Then 2Y+112Ya2x(q+l)l = 2Y+112y(2cq+r)-2y+C(q+l)I = 2Y+l12Yr-2Y+CI [*11.7] = 2y+1(2x~2yr)[r<2C, x11.1, *6.11] = 2Y+1+X~2Y+1(2Yr)[*6.14] = (2~+~+2~+~):(2Y+x+d') = 2Y+Xld' [*6.8] < 2y+x [*6.15]. Use 3introd. CASE 2: y>x. Then 2Y+112Ya-2x2y'xa/ = 0 [*6.7, *11.2] < 2y+x. "R1.9. Assume (i) 2~'+312Pa-dl<2x+p and (ii) 2y+112ya-2xbl < 2Y+x. Then 2Y+312y+Xd-22x+pb( < 2y+312y+xd-2y+x+pal+ 2~+31 ~ Y + x + P ~ 22x+pbl = 2y+x2y+312pa- dl +2x+P+22~+1 I2ya- 2xbl < ~ Y + x ~ x + P + ~ x + P + ~ [(i), ~ Y + (ii)] ~ = 22x+p+~5. Thus 2y+312yd-2X+pbI < 2X+P+Y5. *R1.10. Assume a, ~ E and R (i) Vp2z+4la(w+p)-y(w+p)I<2*+p. First we shall deduce (a) 3b3c{3xVp2y+312Ya(x+p)-2x+PbI <2x+p+y5 & 3 ~ V p 2 ~ + 3 ~ 2 ~ y ( ~ + p ) - 2 x + P ~ ~ < 2 x + p + ~ 5 & ( y ~CASE ~ ~ b =1 ~: yz. Using a, ~ E Rassume , (prior to 3-elim.) formulas from which by *RO. 1 1 : (xi) Vp2Y+312Pa(x)a(x+p) I <2x+P, (xii) Vp2Y+312Py(x)-y(x+p) I <2x+P. Using *R1.8, assume (prior to 3-elims.) : (xiii) 2y+l12Yx(x)-2xbl<2Y+x, (xiv) 2~+112yy(x)-2Xc((2y+x. By (xi), (xiii) (or (xii), (xiv)) and *R1.9: 2y+312ya(x+ p) -2X+Pbl< 2x+p+y5,2y+3)2yy(x+p) -2X+pcI<2x+p+y5. Also, using case hyp., y
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T H E I N TU I TI O N I S TI C CONTINUUM

CH. I11

From (xv) we deduce K’ER’,thus. After V-elims. (with y and y + l for y) and preceding 3-elims. we assume formulas from which by “R0.9 (and V-elim.) : (xviii) 2y+312ya(x+p)-2X+Pa’(y) I <2X+P+Y5 and (xix) 2~+412y+la(x+p) -2x+pa’(y+ 1) I <2x+p+y+l5. Now 22y+x+p+4(2a’(y) .’(y+ 1)i = 22Y+412x+P+la’(y)-2x+payy+ i)i I 22y+412x+p+la’(y)2y+la(x+p)I +22y+412y+la(x+p)-2x+pa’(y+ 1 ) I = 2y+22y+312ya(x+p) -2X+Pc~’(y)I + ~ Y ~ Y + ~ ( ~ Y + ’ ~ c ( x + P ) -2X+P~r’(y+ 1) I < 2Y+22X+P+Y5+ 2~2x+p+y+15 [(xviii), (xix)] = 22y+x+p+115.Thence 812a’(y)-a’(y+ 1)1 < 15, whence (by contradicting /2a’(y)-a’(y+ l ) l > I ) , 12a’(y)a’(y+ I ) \ I1. Thence (xx) E’ER‘. Similarly from (xvi) : (xxi) TIER’. Toward deducing a’+a from (xv) and (xx), assume from (xv), after V-elim. (with t=13*2k for y) and prior to 3-elim., Vp2t+3)2ta(xl+p)2x1+Pa’(t) I <2X1+P+t5,whence by “R0.9, letting x=max(xl, t), (xxii) 2t+312ta(x+p)-2X+Pa‘(t)I <2X+p+t5. Assume (prior to 3-elim.): (xxiii) x=t+c. NOW t2tla(x+p)-~’(x+p)I < 2t2t+3)a(x+p)-a’(x+p)l [*3.10; a i b 3 a c s b c ] = 2t+312ta(x+p)-2ta’(x+p)I 2t+312ta(x+p) -2x+pa’(t) I +2t+312x+pat(t)-2ta’(x+p) I < 2~+P+t5+2t+32t12c+pa’(t)a’(t+c+p)) [(xxii), (xxiii)] < 2X+p+t5+2t+32t2c+p [*R0.6, (xx)] = 2X+P2t13. Thus 2k/a(x+p)-a‘(x+p)l<2X+P. By V-, 3- and V-introd.: (xxiv) d ~ a Similarly, . from (xvi) and (xxi): (xxv) y’Ay. Combining (xx), (xxi), (xxiv), (xxv), (xvii) : 3aA,ER’3y&4a‘*a & y’sy & v x x & w =y’(x) 1. “R1.11. I. Assume ~ E R Use . “R1.10 with a, a for a, y. 11. Use “R0.7, “R1.7. The law of the excluded middle does not hold for equality A of r.n.g. 01 and jj; indeed, when it is generalized on one of the variables, say a, we can refute it, using Brouwer’s principle “27.6. (Cf. Brouwer 1928a p. 7 item 1, with 1929 p. 161; 1954 p. 5 lines 7-9.) This we shall do in “R9.23. However, the law of double negation does hold for 01 5 jj (cf. 1928a p. 8 lines 23-35, 1954 p. 5 lines 6-7, Heyting 1956 p. 17); this we shall include in “R2.8. 14.4. We represent the apartness relation between r.n.g. (Brouwer 1954 p. 4 lines 13-14, Heyting 1956 p. 19) by the formula on the right

in “R2.1, abbreviated “a#p”. “R2.3, “R2.4, “R2.6, “R2.7 are from Heyting 1956 p. 20. Kleene shows in 18.2 of Chapter IV below that the converse of “R2.5 is not provable and (by classical reasoning) not refutable.

9

*R2.1. "R2.2. *R2.4. *R2.6. "R2.7. *R2.8.

141

BASIC PROPERTIES O F THE CONTINUUM

14 I- u #P

~ ~ ~ X V P ~ ~ ~ U ( X + P ) - ~22x+p. (X+P)/ I- i a # U . "R2.3. 1 0: #P 2 (3 #u. k u*p & a#y 2 p#y. *R2.5. k a # p 3 1ctAp. I- u, (3, ~ E &Ru # p 3 u#y V p#y. 1U , PER & ~ u # P 3 u+P. 1a, PER 2 ( l l u G ( 3 l a # p u*p). N

- -

PROOFS.*R2.4. Assume u ~ p u#y. , Prior to 3-elims., assume Vp2klu(xl+p)-y(xl+p)I 22x1+p;and after V-elim. and prior to 3elim., assume Vp2k+llu(x2+p)-P(xz+p) I <2X2+P. Letting x=max(xl, x2) and using *R0.9, V-elim. and *145b: (i) 2k+11u(x+p)-y(x+p)12 2.2x+P and (ii) 2k+llu(x+p)-P(x+p)l<2X+P. Now 2k+lIp(x+p)y(x+p) I22x+p, since otherwise 2k+llu(x+p) -y(x+p) I 5 2k+1Jct(~+p) -p(x+p)I+2k+llp(x+p)-y(x+p)I < 2x+p+2x+p [using (ii)]= 2.2x+P, contradicting (i). By V- and 3-introds., p#y. "R2.5. By *R2.2 and *R2.4. "R2.6. Assume a, P, ~ E and R u#p. Prior to 3-elims. from u#P, and after &- and V-elims. and prior to 3-elims. from u, p, ~ E Rassume , formulas from which, using "R0.9 (and *145b) and *RO.11: (i) 2k+312Pa(x)-2Pp(x) I28-2x+P, (ii) '2k+312Pu(~)-u(x+p)I <2X+P, (a) 2k+312~p(x)-p(x+p)I <2x+P, (iv) 2k+312py(x)-y(x+p)I <2x+P. Now 2k+3[2Pu(x)-2Py(x)I24.2X+P V 2k+3(2pp(x)-2Py(x)\24-2x+p, since otherwise ~ ~ + ~ I ~ P u ( x ) - ~ P P ( xI ) I 2k+312Pu(~)-2Py(x)I+2k+312py(x)2Pp(x)l < 4-2x+P+4.zX+P = 8.2X+P, contradicting (i). CASE 1 : 2k+312pu(x)-2py(x) I 24-2x+P.Then (a) 2k+31u(~+p) -y(x+p) I 22.2x+P, since otherwise 2k+312Pct(x)-2Py(x)I I 2k+312pu(~) -a(x+p)l+ 2k+31a (x+ p) -y (x p) I 2k+31y(x p) -2py (x)I < 2x+p 2*2x+p 2x+p [using (ii), (iv)] = 4-2X+P, contradicting case hyp. From (a) by *145b, 2k+2lu(x+p)-y(x+p)l~2x+p, whence by V- and 3-introds., u #y, and by V-introd., cc#y V p #y. CASE2: 2k+312pp(~)-2Py(x)I> 4 ~ 2 ~ + P . Similarly, using (iii) and (iv), p #y, and by V-introd., u #y V p#y. "R2.7. Assume u, PER and l u # p . After V-elims., and prior to 3-elims., from UER and PER, we assume formulas from which by *RO. 11 : (i) Vp2k+3/2pu(x)-u(x+p)I <2X+P, (ii) Vp.F+312PP(x)p(x+p)l < ~ x + P . For reductio ad absurdum, assume 2 k + l l ~ ( ~ ) - P (2 ~)I <2kf3/2P~(~)-~(~+~)I+ 2X. NOW 4*2X+P 2k+312P~(~)-2PP(~)I 2k+3iU(X+p) -p(x+ P)I 2k+31P(X+ P) -~ P P ( X I) I 2x+p+ 2k+3I ~ ( x + P ) -p(x+p)1+2x+p [(i), (ii)] I 2-2x+P+2k+31~(x+p)-P(x+p)I. Hence 2~+~<2k+2]~(x+p)-P(x+p)l. By V- and 3-introds., u#P, contra-

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142

THE INTUITIONISTIC CONTINUUM

CH. I11

dicting l a # P . Thus (iii) ~~+~IQ(X)-(~P(X)I<~X. So 2k+31a(x+p)P(x+p)I 5 2k+3/a(x+p)-2Pa(x)I+2k+312pa(x)-2PP(x) 1+2k+312PP(x) -P(x+p)I < 2x+P+4*2x+P+2X+P [(i), (iii), (ii)] = 6-2X+P< 8*2x+P. So 2k/a(x+p)-P(x+p)/
--

14.5. Of the operations of arithmetic we introduce only addition a+/3 and subtraction a-/3,

1a-81. There would be no difficulty in treating multiplication and division if required (cf. Heyting 1956 p. 21). Our new axioms are of the first form provided in 5.1. In "R3.3, *R3.4, *R4.3-*R4.5, *R5.3-*R5.6 we have analogues to "1 17, *119, "6.3, "6.5, "6.8, X11.1, "11.4, "1 1.7, "1 1.8, respectively (thus including all equalities from I M Theorem 25 and above X6.1-*6.21, x11.1-* 1 1.15b not involving 0, ' or multiplication). Each of these follows by V-introd. from (a substitution instance of) the corresponding theorem of I M or 5.5 above, and by *R 1.3 each has a version with = replaced by A.

xR3.1. *R3.2. "R3.3. "R3.5. xR4.1. "R4.2. "R4.3. "R4.5. "R4.6. xR5.1. "R5.2. "R5.3. "R5.5. "R5.7.

PROOFS. "R3.2. Use "RO.11, "11.5, *11.7. *R3.5-*R3.6. Use * 1 1.7. REMARK 14.1. As counterexample to "R3.2 with R replaced by R', I- IxlER' & llxl +IxlER'. Similarly, counterexamples to "R4.2 and "R5.2 with R' replacing R are obtained by taking a to be lx2X' 2 1 and P to be I x 1.

§ 14

143

BASIC PROPERTIES OF THE CONTINUUM

14.6. We shall study two ordering predicates for the continuum which are important intuitionistically: the “natural ordering” or “measurable natural ordering” < o (Brouwer 1928a p. 8, 1951 ; called the “pseudo-ordering” and written “<” in Heyting 1956 pp. 107, 25), and the “virtual ordering” < (written “<” in Brouwer 1928a p. 9, 1951, and “<” in Heyting 1956 p. 107). Informally the natural ordering predicate a<$ is ( E x ) ( E k ) ( p ) B ( x + p ) / ~~ + pa ( ~ + f i ) / F 1+ /2k. ~> We express this formally by the right side of *R6.1, abbreviated “a a ” . We write “aa’’ for a<+ V a+p; “ao>p, y” for ao>P & ao>y, etc. Analogously to “a=p” for Vx(a(x)=P(x)), we adopt the abbreviations “a(@” and “P2a” for Vx(a(x)(p(x)); since we shall avoid employing the corresponding abbreviations “a<@” and “P >a” for Vx(a(x)a with the disjunction a<@V a=p. We furthermore abbreviate 1ac.P by “a{+” and In number-theory there was no gain from such an abbreviation since Y a c b is equivalent to a > b and t o b , 2,02, KO). Cf. I M e n d s 26. Thus constructing an ascending chain from a to p establishes in any case that a o > P (if 5 and o> are not used, that a<+; if is used, that a<+). Such a chain appears in the proof of “R9.19; others will be used in 15.3 ff. Various inequality formulas in arithmetic have analogues in the natural ordering of the continuum, though we cannot derive these analogues in quite as uniform a manner as we did the analogues of equalities in 14.5. The proofs of *R6.16-*R6.20 provide examples. Most of the (next) theorems are from Heyting 1956 pp. 25-26. “PO>a”.

( 0 ,


*R6.1. *R6.2. *R6.3. *R6.5. *R6.6. *R6.7. *R6.9.

I- a<+

-

3k3xVp2k(P(x+p) Aa(x+p)) 22xfp.

I- a, PER & a#p 3 a < o p V p
I- a < o P 2 Ya’P.

I- a, PER & a++ & P+a ZI a 5 P . I- a<+ & P
144

CH. I11

THE INTUITIONISTIC CONTINUUM

*R6. *R6. *R6. *R6.

PROOFS. "R6.2. Assume u,PER and u # p . Prior to 3-elims. and after V-elims. assume formulas from which, using "R0.9 and "R0.11: (i) Vp2kI u(x p) -p (x p) I 2 2x+P, (ii) Vp2k+2I2Pa (x) -tc (x p) I <2x+p, (iii) Vp2k+212pp(x)-p(x+p)I<2x+p. By V-elim. from (i) with 0 for p (and * 145b), I2k+2+pu(x)-2k+2+pp(x)I24.2X+P. CASE 1 : .(X) 2kf2tc(x+p) +2*2X+p,or using (ii') and (iii') we could contradict (i'). Hence 2k+lp(x+p) 22k+ltc(x+p) >2x+p. By V- and 3-introds., t c < o P , and by V-introd., a<$ V p < o a . "R6.4. By "R6.3, "R2.5. "R6.5. By "R6.2, "R2.7 (with "63, "12). *R6.7. Use *R0.9. "R6.9. Assume u, p , y ~ Rand a<+. By "R6.3, "R2.6 and "R6.2, (tc2X+P. Thence by V-elim. with 0 for p, and "6.1 1, 0>2X, contradicting "3.9. *R6.16. Use "6.8.

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§ 14

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BASIC PROPERTIES OF THE CONTINUUM

*R6.17. By V-introd. from * 1 1.5, and "R6.15. *R6.18. Assume p . 2 ~ . CASE 1 : Po>y. Assume V ~ 2 ~ ( P ( x + p ) y(x+p)) 22x+p. Thence, after V-elim., P(x+p) >y(x+p). So by *6.6 with xR3.1 and ~R4.1,((a+P)1y)(x+p)=(a+((32y))(x+p). Hence 2hl((a+p) ~y)(x+p)-(a+(p'y))(x+p)l = 0 d x + P . By V- and 3introds., (a+p):y*a+(p-y). CASE 2: PAY. Then by *R3.5 (with *R3.4) and "R4.6: (i) (a+P) LyA(a+y) 'y, (ii) a+(P-l-y)&a+(y2-y). By "R4.3, (iii) ( a + y ) ~ y = a ; and by V-introd. from (a substitution instance of) a+(cLc)=a [*6.3a], (iv) a + ( y ~ y ) = a . Now, using the chain method with =, A : (a+P) "y A (a+y) -y [(i)] = a [(iii)] = [(iv)l .+(PAY) [(ii)l. .+(y-y) "R6.19. First we refine "6.19 t o the following (proved by cases a-c>(b-c)+d). b s c , b>c): "6.19'. d>O 3 (a>b+d,c+d I. Assume a o > P , y. Via "R0.9, V ~ 2 ~ ( a ( x +2(3(x+p))22x+P, p) Vp2k(a(x+p) ~ y ( x + p ) >2x+P. ) Thence 2ka(x+p) >2kP(x+p) +2x+P, 2ky(x+p)+2X+P. Thence by *6.19' and XR4.1, 2 k ( a ~ y ) ( x + p ) 2 2k(P ' y)(x+ p) 2x+p, whence a -yo >P 2y. *R6.20. First prove by cases b>c, bO 2 (a+d(cLb)+d). In "R6.21 A(=, a", P) is a " L a & a"

p(x) a"(x)=p(x)). In *R6.23 B(a, a", P) is a " A a & a">P & Vx(a(x)>P(x) a"(x)=a(x)) & Vx(a(x)
-

-

+

-

-

N

*R6.21.

-

t a , y, PER' & a, yo>P & VX,<~(X)=Y(X) 3 3 t ~ " , . ~ ~ . 3 y " , ~ , ~ , (a", A (P) a ,&

*R6.22. *R6.23.

*R6.24.

A(y, y", P) VX,,,~"(X)=Y"(X)). I- a, PER' 3 [ a o > P 3 a N a f l E R , ( ~&"."
I- a, y, PER' & a, yP)].

-

PROOFS."R6.21. Assume a , y , PER', (i) a o > p P , (ii) yo>pP, (%) Vx,,,a(x)=y(x). Introduce a" and y", using Lemma 5.5 (a) :

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T H E INTUITIONISTIC CONTINUUM

CH. I11

Assuming xp(x’). Then 2u”(x) = 2u(x) [(iv)] 2 2p(x) [case hyp.] < p(x’)+l [PER’, *R0.5a] = u”(x’)+I [(iv), case hyp.]. Also u”(x’) = p(x‘) < u(x’) < 2u(x)+ 1 [uER’, *R0.5a] = 2u”(x)+ 1. So by * 1 1.15a, 12a”(x)-u”(x’)I < 1. Cases 3 and 4 are similar. By V-introd. : (vii) ~ ” E R ’ Easily . from (iv) : (viii) uNp(x) u “ ( x ) = ~ ( x ) )Toward . u”*u, assume u “ # u , whence: (a) 2klu”(x+p)-u(x+p)/ >2x+p. Then u”(x+p) # Q(X+P). By (4, .(x+p)>P(x+p), whence by (iv), .”(x+p)=P(x+p). So (a) becomes 2k(u(x+p) ~ P ( x + p )22XfP. ) By V- and 3-introds., u o > p , contradicting (i). Hence l u ” # u . By “R2.7: (xi) u N A u . Combining (vii), (xi), (viii), (ix) and (x) : (xii) ~ ” E R&’ A(u, u”,p). Similarly: (xiii) ~ ” E R&’ A(y, y“, p). Combining (xii), (xiii) and (vi) : ~ u ” ~ ” ~ R ’ ~ ~ ” u”, ~ ”p)~ & ~ ,A(y, { Ay”, ( u P) , & VX,<,U”(X) =Y”(x)}. “R6.22. Assume u, PER’. I. Apply “R6.21 with u, u, p for a,y, p. 11. Use “R6.15, “R6.14. The virtual ordering predicate is expressed by the formula uo>p & l c r ~ p which , we abbreviate “uu”. *R7.2-*R7.4, “R7.10 and “R7.11 are a rearrangement of the axioms for virtual order given in Brouwer 1924-7 I1 p. 453, Brouwer 1928a p. 8, Heyting 1956 pp. 106-107. The first equivalence in *R7.8, *R7.9 is simply an unabbreviation of “ U ~ O ~ ’ ’ . N

“R7.1. “R7.2. “R7.3. “R7.4. “R7.5. *R7.7. “R7.8. “R7.9. “R7.10. “R7.11.

PROOFS. “R7.4. Assume u, p, ~ E Ru

p and (ii) l u n p , and p ~ ory simply (iii) Po>y. Using (i) and (iii) (besides u, p, ~ E Rin ) “R6.12, uo>y. Toward l u + y , assume u+y. Using this

9 14

BASIC PROPERTIES OF THE CONTINUUM

and (iii) in *R6.13, a<+; and using the latter and (i) in “R6.5, contradicting (ii). *R7.9. By “R7.8, *25, *49b.

147 a%P,

, expresses The formula on the right in *R8.1, abbreviated “ a ~ [ S 1&I”, that a , 81, 8 2 are r.n.g. such that a belongs to the closed interval [ S l , 621 (cf. Brouwer 1924-7 I1 p. 454, 1928a p. 9 lines 17-13 from below, Heyting 1956 pp. 40-41).

“R8.1. *R8.2. “R8.3. *R8.4. “R8.5. *R8.6.

t c ~ E [ S621 ~,

-

a, 61, S ~ E R &

1(a<61& a a 2 ) & 1(a=iS1& a>S2). t a+p & ~ ~ [ 6621 1 ,3 P E [ B ~ Sz]. ,

t- 6 1 A S i & a E [ S i , S 2 ] 3 a ~ [ S iS ,2 ] . k & aE[61, 621 3 a ~ [ 6 16,i ] . i- a, PER 2 a, P E [ ~ , PI. ‘I- a,61, 6 2 ~ R& S1o>S2 3 { a E [ S i , 621 (a<& ao>S2)}.

-

PROOFS. *R8.5. (Cf. preceding *RO. 1 .) Use *R7.6. *R8.6. Assume a,61, S ~ E R and (i) 610>62. I. Assume (ii) l ( a < 6 1 & a<&) and (iii) l(a>61 & a>&). If a<&, then by (i) and “R6.11, u62. 11. Assume a<& & a0>62. By *R7.8, -vx<61& ’1a>S2, whence -r(a<61& a<S2) & l ( a > S 1 & a>&); so a€[&, 621. 14.7. It will be convenient to have available formally some of the

theory of the species of finite dual fractions, or, more precisely, the theory of certain r.n.g. which correspond to finite dual fractions. It can be verified easily that the functor a-2-m (usually abbreviated “a2-m” ) of Axiom xR9.1 gives under the interpretation a r.n.g. corresponding to the dual fraction a.2-. The notation is unambiguous since there is no other way in which negative exponents have been used. We shall adopt the abbreviation “a” for 22-0 in contexts making it clear that a is a functor (not a term). Thus in *R9.18 “0” abbreviates 0-2-0. xR9.1. *R9.2. *R9.3.

(a-2-m)(x)= [a/2mzx].2xLm. 1 a2-m~R’.

t- 2ka2-(m+k)=a2-m.

148

THE INTUITIONISTIC CONTINUUM

CH. I11

*R9.4. *R9.5. *R9.6. "R9.7. *R9.8. *R9.9. "R9.10. *R9.11. "R9.12. *R9.13. "R9.14.

PROOFS. "R9.2. By *R0.4, xR9.1 we need to prove (prior to Vintrod.) : (a) /2[a/2m'x]2xzm- [a/2m"']2x''m1 I 1. CASE 1 : x<m. (ii) m-x'<m'-x [*6.20], (iii) m-x= Then (i) x'm=x'Lm=O, (mAx')+l. Using (i), (a) reduces to: (b) 12[a/2m'x]-[a/2m'x']1<1. By *13.4: (iv) a = 2m'x[a/2m'x]+rm(a, 2"'") = 2mLx'[a/2"'x'] rm(a, 2m'x'). By * 12.3 with "3.9: (v) rm(a, 2"-") <2"'=, rm(a, 2mzx') < 2m'x' < 2m-x [using also (ii) and *3.12]. So 2mAx'~2[a/2mLx][a/2mLx']I = 12m'~[~/2m'x] -2m'x' [a/2mzx']I [using (iii)] = 1 (a-rm(a, 2m-x ))-(al-rm(a,2m'x'))l [(iv)]= Irm(a,2m'x)-rm(a,2mzx')~ [*I 1.1 1 with (iv)] < 2mzx [*11.12; *8.6 with (v)]. Hence using (iii), /2[a/2m'X]-[a/2m'X']l<2, whence (b). CASE2: x 2 m . Now (a) reduces to I2 [a/20]2x1m- [a/20]2(xAm)+ 1 1 1 . Use "13.6, *11.2. "R9.3. We need to prove (a) [2ka/2(mfk)1x]2x'(mfk)- [a/2m'x]2x'm. CASE 1 : x<m. Then x I m = x (m+ k)=O and k+ (m 'x) =(m+ k) -x; and (a) reduces to [2ka/2(mfk)zx] = By * 13.5, this will follow if we establish 2ka=2(m+k)Lx [a/2"lx] +2krm(a, 2m'x) & 2krm(a, 2"lx) <2(rn+k)'x . But by "13.4 and *12.3 (with "107 and *145a), 2ka=2kf("'X)[a/2mLX] + 2krm(a, 2mLX) & 2krm(a, 2"'") <2kf(mzx). CASE 2: m<x<m+k. Now (a) reduces to [2ka/2(m+k)LX ] = [a/1]2X'm. But k = (m+k)-m 2 (m+k)Lx [*6.18]. So [2ka/2(m+k)'xI = [2kL((m+ k ) ' ~ ) ~ 2 ( r n +k)-x +0/2(m+k)-x] = 2k' ((m+k)'x) a [*13.8, X 1 3 . 1 1 = 2"-m a [*6.9, etc.] = [a/l]2x'rn [*13.6]. CASE 3: m+k<x. Using * 13.6, (a) reduces t o 2ka2x"(mfk)=a2x'm. "R9.4. Call this A B C. It will suffice t o prove (a) A 3 B, (b) B 13 C, ( c ) C zi A. (a) Assume a2-mnb2-n. Using "R1.1 and xR9.1, assume Vp2m+n/[a/2m'(xfn)12(x+p)'m- [b/2n' (X+P) 12(X+ P)'n I< 2x+p. Thence by V-elim., 2m+nl [a/2m.L("+m+n)]2(xfmfn)'m_

+

N

-

9

14

BASIC PROPERTIES O F THE CONTINUUM

149

[bpnl-(x+rn+n) 12( x + m + n F nI <2x+m+n. This reduces to 2m+n+xla2n-b2ml <2m+n+x, whence la2n-b2m(< 1, and thus la2n-b2mJ=0. So by * 11.2, 2na=2mb. (b) Assume 2na=2mb. CASE 1 : m l n . Let n=m+k. Now 2m+ka=2mb, whence 2ka=b. So b2-n = 2ka2-n = 2ka2-(m+k) = a2-m [*R9.3]. CASE 2: m>n. Similarly. (c) By "R1.3. *R9.7. I. As for (a) under "R9.4, using now "R6.1 instead of "R1.1, 2k(b2m-1-a2n) >2m+n, whence b2m:a2n>O, whence by "6.12, b2m>a2n. 11. First we prove (i) a 2ka2-n = etc. *R9.9. By *R1.2, it will follow from (a2-m+b2-n)(m+n+p)= ((2na+2mb)2-(m+n))(m+n+p), which is easily deduced using xR3.1, XR9.1, etc. "R9.10. From "R9.9 by *R9.3. "R9.15. *R9.16. *R9.17. *R9.18. *R9.19. "R9.20. *R9.21.

t ~ E R3' Ia(m)2-m-alo> 1.2-m. t ~ E R3 ' a ~ ~ ( a ( :1)2-m. m)

+

I- MER'3 ao=p(a(m) 1)2-m. I- aX.0. I- a,@R & a<+ 3 3a3m(a
+

PROOFS. "R9.15. Assume aER' and Ia(m)2-m-alo> 1-2-m. Assume, prior to 3-elims., Vp2k(l[a(m)/2mz(x+P) I2(x+p)zrn- .(X+P) I [l/2mz(x+P) 12(x+P)'rn)>2~+p.Using V-elim. with m+p for p, this reduces to 2k(la(m)2x+P--a(m+x+p) I ~ 2 x + p>2m+x+P, ) whence by "3.9 and "6.12, 12x+pa(m)-a(m+x+p) I >2X+P, contradicting XER' by "R0.6. hR9.16. Assume ~ E R and ' a0, whence [a(m)L 1/2rn'(x+rn+p) 12( r n + x + p ) A r n ~ a ( x + m>0, + p ) which reduces to (a(m)2x+p~2x+p) -a(x+m+p) >O. Then 2X+Pa(m)'a(x+m+p) > ~ x + Pwhence , J2x+Pa(m)-a(m+x+p) I >2x+P, contradicting ~ E R by ' "R0.6. "R9.19. Assume PER and (i) a
150

THE INTUITIONISTIC CONTINUUM

CH. I11

-af(x+2+k) 2 2xf2 > 3, whence (v) Pf(x+2+k) 1.1 >a'(x+2+k) +2. So a a' [(ii)] o > ( a ' ( ~ + 2 + k ) + 1 ) 2 - ( " ~ ~[*R9.17] +~)
O>

Then ii(y)=P(y). We shall deduce (a) 12a(x)--cr(x')[ < I . CASE 1 : xy. Then 12a(x)--cr(x')I = 12(2x1yP(y)L(2x'y1 2l))-(2x'lYp(y) L(2X'LYL I ) ) \ = 1(2x"Yp(y) '(2"''"L2))-(2X"YP(y) (2x'Ly:l))l [*6.14, "6.6 with case hyp.] < 12-11 [*11.13] = 1. From (a) by V-introd., ~ E R .Finally, 2kla(y+k'+p)((@(y) 11)2-Y)(y+k'+p)I=2k((2kf+Pp(y) 1(2k'+P-l))-(P(y)4)2k'+pl [xR9.1, etc.] I 2kI(2k'+p_1.1)-2k'+PI [*6.14, *11.13] I 2k11-O[ [*11.13] = 2k < 2y+k'+p. By V-, 3- and V-introd., a"(P(y)'1)2--Y. "R9.21. Similarly, letting

Using *R9.20 and *R9.21 with Brouwer's principle (for numbers) we next refute the (generalized) law of the excluded middle for 5 . This proof of *R9.22 via "R9.20 and "R9.21 is essentially due to Kleene, who in March 1963 gave a simpler proof of *R9.22 than that in the author's thesis (where the result appeared right after the present "R1.11). We have adapted Kleene's proof to obtain successive proofs of "R9.20, "R9.21 and *R9.22. *R9.22. *R9.23. *R9.24. *R9.25.

I- PER' 2 ~ t / t c , , ~ ( c ~ +VPl a * P ) . I- PER 2 IVCC,,R(CI~PV i a A P ) . t- PER 2 IVK~~R(O( P ) . t PER 3 ~V-cr,,~(aP).

PROOFS. *R9.22. Assume (i) PER' and VaaER'(a'P V la'-P). Using "27.6 with *R0.8, and omitting 32 prior to 3-elim., assume (c) Va,,H.3y{Vx[~(~(x))>0 2 y=x] & {(ZAP & ~ ( ~ ( y ) ) =Vl )( l a * @ & ~(ii(y)) = 2))). Thence, using (i) and omitting 3y: (G) Vx[~(p(x))>O

3

15

151

T H E UNIFORM CONTINUITY THEOREM

3 Y=X] Lk {(PAP & ~(D(y))=l)V (1PA.B & ~(B(y))=2)}.By *R1.4, PAP. So by (iii): (iv) ~ ( F ( y ) ) = lUsing . *R9.20 and "R9.21 with (i),

assume (v) a1ER & &(y)=p(y) & al*(p(y) 2 1)2-Y and (vi) QER' & iiz(y)=P(y) & az*(P(y)+ 1)2-~.Now (P(y) 1 P - y P- a1 [(v)] A P [(v), (iv), (ii)]* az [(vi), (iv), (ii)] A (P(y)+1)2-Y [(vi)]. So by *R9.5, p(y)2 1 =P(y) 1, which by cases (P(y)< I , p(y) > 1) is absurd. *R9.23. Assume PER. Using *R1.11, assume prior to 3-elim.: P'ER & P'SP. By *R9.22, (i) ~ V a a E R t ( a ~V Pla*P'). ' By *R0.7 and (3'5P: [ ~ E R 2 a+@ V la*P] 3 [ ~ E R3 ' a+P' V -1a5P'l. Thence by *69 and *12: (c) ivUaEw(cX*P' v ia*P') 3 ivaa,R(a*p v ia'p). Use (i) with (ii). *R9.24. Assume PER and Vaa,R(a P ) . Using*R6.4, VaaeR(a*P V 7aA.P). But by 3-elim. from *R9.23, lVaa,R(a+P V

+

la+

p),

Q 15. The uniform continuity theorem. 15.1. We shall establish formally the theorem on the uniform continuity of a function defined for every real number represented by r.n.g. in the closed interval [Sl, 821 where 610342. (For the theorem without the condition 61 62, cf. Heyting 1956 p. 46; the original versions in Brouwer 1923a p. 5, 1924 p. 193, 1927 p. 67 are for [0,1].) A preliminary result *R10.1 states that for each pair of r.n.g. a and y sufficiently close together in the sense that la-yl is small, there are c.r.n.g. a' and y' (with a' + a and y' * y ) "close together" in the sense that initial segments coincide. O>

*R10.1.

t- a,yER & Ia-yl<01.2-(~+~)3 3a'a,Ew3y'ys,R'(a'*a & y'+ Y Vx,..a'(x)=y'(x)).

PROOF.Assume a,ycR, Ia-yI <01.2-("+~).Assume vp2k((i.2-(z+4))(x+p) ~(la-yl)(x+p)) 22X+P. Using V-elim. with Z+4+p for p, writing w=X+Z+4, and reducing, 2k(2(w+P)'(z+4)~ la(w+p) -y(w+P) 1) 22w+p. so 2(w+p)'(z+4)2Ia(w+P)-Y(w+P)I >O? and by Vwhence by *6.12, etc., 2z+41a(w+p)-y(w+p)l<2W+p, introd., Vp2z+4la(w+p)-y(w+p)I <2w+p. Now use *Rl.lO. Let 61 and 8 2 be c.r.n.g. with 61 5 62. Then "R10.2 asserts the existence of a certain fan. In the proof of the uniform continuity theorem "R10.3, we establish that the closed interval [Sl,621 coincides (Heyting 1956 p. 42) with this fan. (Cf. Brouwer 1924 p. 192,1928a p. 5.)

152

THE INTUITIONISTIC CONTINUUM

*R10.2.

-I 81,82~R'& 81182 =I 3o{Spr(o) & o(l)=O & Va[o(a)=O =I 3bVs(a(a*28+1)=0 2 s g b ) ] & VK[KGS CCER' & Sl<~r<Sz]).

-

PROOF.Assume S~,SZER', S I I S Z . Introduce (cf. the proof of "R0.8):

(A)

CH. I11

Q

via Lemma 5.5 (c)

Vao(a)=

& IIb. Assume o(a)=O. CASE 1 : a = l . Then Seq(a*281(0)+1) 1h(a*2s"o)+1)=1 & 81(0)=(a*2S'(0)+1)o~1 I S z ( 0 ) [ S ~ I S Z ]So . by (A), ~(a*2'~((')+~)=0, and by %introd., 3sa(a*28+1)=0. CASE 2: a# 1. Using (A), Seq(a) & lh(a)> 1 & Sl(lh(a) 1) <(a)lh(a)Ll1.1 <&(lh(a) 1 1 ) . SUBCASE 2.1 : S1(lh(a)> 1) = (a)lh(a)L1 2 1. Letting t be a*261(1h(a))+1, we deduce Seq(t) & Ih(t)> 1, 12((t)lh(t)L21l)-((t)lh(t)AIL1)l = 1261(lh(a) I)-Sl(lh(a))l I 1 [B~ER'],Q(ni<1h(t)Llpit)*) = .(a) = 0, 81(lh(t) 1) = 8l(lh(a)) = (t)1h(t)2lA1I Sz(lh(a)) [81182] = &(Ih(t)Al). So by (A), o(t)=O. By %introd., 3s~(a*2~+1)=0. SUBCASE 2.2: (a)lh(a)Ll-1 = 8z(lh(a)2 1). Similarly. SUBCASE2.3: 81(lh(a)2 1) < (a)lh(a)zl 2 1 < Sz(lh(a) 1). Let t be a*22((a)lh(a)'1'1)+1. Then Seq(t) & lh(t)> 1 & o(ni 1)+ 1 < (a)lh(a)Ll 2 1 < Sz(lh(a)2 1) 1. So 81(lh(t)2 1) = Bl(lh(a)) I 281(lh(a) 1 ) + 1 [ ~ I E R '*R0.5a] , < 2(Sl(lh(a)2 I ) + 1) I 2((a)lh(a)+11 1) [(i)] = (t)lll(t)Ll 2 1 I 2(6z(lh(a)11) 11) [(i)] = 282(lh(a)1 .1) 1 2 I 262(lh(a) 1 ) 1 [*6.18] I Sz(lh(a))[SZER',*R0.5a-c] = 82(lh(t)2 1). So by (A), o(t)=O. By %introd., 3so(a*2s+1)=0. IIIa. Assume KEG. As before, CCER'. Also by V-elim. with 1 for x, and (A), 61(0) I ~ ( 0I )S z ( 0 ) ; and by V-elim. with x+2 for x, and (A), Sl(x+l) I M ( x + ~I ) Sz(x+l). Hence by induction cases (IM p. 186) and V-introd., S1O), s<8z(lh(a)).

3

15

THE UNIFORM CONTINUITY THEOREM

153

15.2. Let F(a, B) be a predicate expressed formally by the formula F(a, p). Suppose 61 and 8 2 are r.n.g. A necessary condition that F(a, B) be the representing predicate of a function from real numbers, represented by r.n.g. a in [Sl, 621, to real numbers, represented by r.n.g. B, is given formally by the following formula (provided y,C are free for a, P in F(a, P) and do not occur free in F(a, p)).

%(F,6 ~ 6 2:) V~VYVPVC{~E[~~, 621 & a+y & F(a, p) & F(y, C) 2 P, CER & PAC). REMARK 15.1. However, %(F, 61, 62) does not entail either of the replacement properties (a) a*y & F(a, P) 2 F(y, p) and (b) & F(a, p) 2 F(a, C). For, letting F(a, p) be agR' & a+P, I- %(F,61, 62) (by "R0.7, "R1.7, "R1.6 and *R1.5), but I- i l x l ~ i l x 2& F(ilx1, 1x1) and I- 1F(ilx2, ilxl) (so the closure of (a) isrefutable). Similarly, letting F(a, P) be PER' & a+p, k %(F, 61, 62), but t Ixl+ilx2 & F(ilx1, Axl) and I- lF(ilx1, ilx2). The fourth member of the conjunction in the hypothesis of "R10.3 asserts that the function represented by F(a, P) is completely defined in [Sl, 621. The conclusion corresponds under the interpretation to a familiar form of the definition of uniform continuity. "R10.3.

t 61,82ER & 610>62 & %(F,61, 62) & VK{~E[&, 621 3 W ( a , P)} 3 Vn3mVaVyVPVC{la-yl <01-2-m 8~a, YE[&, 621 8~F(a, P) F(y, C) 2 lP-Cl<01.2-~).

PROOF. Assume 81,62~R, (i) 610>62, (ii) %(F,61, 64, (iii) V a { a ~ [ 8 1621 , 2 3PF(a, p)}. Using "R1.11, assume (prior to 3-elims.): (iv) 6iER' & SiA.61, (v) 6 b ~ R& ' 6&&. By (i) and *R6.13-*R6.14, Sio>69. Using *R6.22, assume: (vi) 6;gR' & 6qA;Si & 6:<6b. By

"R6.14, (vii) 6;0=pSb. Using (vi), (v) and *R10.2, assume (the conjunction of) : (viii) Spr(a) & (. 1) =O & Va[a(a)=O ZI 3bVs(a(a*28+1)=O I)sib)] and (ix) Va[aEa ~ E R&' S;89 [*R6.15], KER'& a ~ [ 6 ;691 , [*R8.6, (v)-(vii)] and MER'& a€[&, 621 [*R8.3, (vi), (iv); "R8.4, (v)]. Using (iii), assume F(a, P). From (ii) (and *R1.4), PER. So, writing b=2x3P(x), assume V~2n+212p(b)~-p((b)o+p)(<2(b)o+p.Writing "G(n, p, b)" for this, and using &-, 3-, 2-and V-introds. : (x) VnVa,,,3b3P(G(n, P, b) & F(K,P)). Consider "27.8 as of the form A & B & C ZI D, and write "A(n, a, b)" N

154

CH. I11

THE INTUITIONISTIC CONTINUUM

for its A(a, b) (so C, D become C(n), D(n)); by “69 and “89, A & B &VnC(n) s,VnD(n). This with A & B given by (viii), and VnC(n) by (x), gives VnD(n). So assume: (xi) Va,,,3bVy,,,{Vxx,zy(x) =a(x) 2 3p(G(n, p, b) & F(y, p))}. Toward the conclusion of “R10.3, assume (xii) la-y/<01-2-(~+~) & a,y~[61,621 & F(a, p) & F(y, C). Using “R10.1 (and *R8.1), assume: (xiii) a’,y’~R’& a ’ n a & y ’ ~ & y VxX5p’(x)=y’(x). Then (xiv) V X ~ < ~ ~=y’(x). ’ ( X ) Using *R8.2-*R8.4 and (iv)-(vi), a’,y’~[G;,$4, whence by (vii) and “R8.6: (xv) (xvii) y ’ 4 & (xviii) y’o>6i. By (xiii) and (v), a’<06;, (xvi) a’,y’,GieR’. So using “R6.21 with (xvi), (xviii) and (xiv): (xix) a”,y”~R’,(XX) a”*a‘ & a“6;(x) X) ~’’‘(x) =a”(x)) & Vx(a”(x)<6/;(x) a’’’(x)= y’”(x)=y”(x)) & s;(x)), (xxv) y”’+y” & y”’2BI; & Vx(y”(x)2 q X ) Vx(y”(x)S;(x). Then a’”(x)=a’’(x) [(xxiv)] < 6i(x) [(xx)]. So, using also (xxiv), G;(x) 2n+lfx+p~2x+n+p [(xxx), “6.20 withitshyp. aO] = 2 x f n + p ( 21~) 2 2x+n+p.So by V- and3-introds., (xxxi) lp’-C’l
-

-

-

-

+

-

3 15

155

THE UNIFORM CONTINUITY THEOREM

15.3. Using *R10.3, we shall establish that the continuum is “indivisible” (“unzerlegbar”; Brouwer 1927 p. 66, 1928a p. 1 1 lines 4-9, Heyting 1956 p. 46), at least by any predicate C(a) expressible by a formula C(a) of the system.

*R10.4.

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I- 81,82~R& 810>82 & VaVy{a€[81,82] & a A y 2 (C(a) C(y))} & Va{a€[81, 821 3 C(a) v -IC(a)) 3 Va(a~[8i, 821 2 C(a)) V Va(a~[81,821 3 lC(a)).

PROOF.Assume 81,82~Rand the other hyps. of the implication. Using *R1.11, assume (i) 8 i ~ R&’ 8;+81. Now using “R6.14 and *R8.3: (ii) 8io=p82, ( 3 ) VaVy{a@i, 821 & a + y 3 (C(a) C(y))}, (iv) Va{a~[8i,821 3 C(a) V lC(a)}. Let F(a, p) be [p+O & C(a)] V [PA 1 & lC(a)]. Toward (v), assume a€[&, 821 & a s y & F(a, p) & F(y, C). By (iii), C(a) C(y). Now from F(a, p) & F(y, C) by cases (C(a), YCta)), P A C and (using *R9.2, *R0.7, “R1.7) P&R. By &-, 2-and V-introds., (v) %(F,Si, 82). Assuming aE[S;, 821 and using (iv), C(a) V l C ( a ) , whence by cases (using *R1.4), 3pF(a, p). So (vi) Va(a~[Gi,821 3 3pF(a, p)}. Using 8 i ~ R(from (i)), &ER, (ii), (v) and (vi) in “R10.3, assume: (vii) VaVyVPV<{(a-yl<01-2-m & a,y~[6i,821 & F(a, p) & F(y, C) 2 Ip-Cl<01.2-1}. We shall deduce (a) VaVy{la-yll-2-1. Toward C(a) 3 C(y), assume C(a). By F(a, @), p A 0 . Now C+ 1 would lead to Ip-CI 10- 1 I A 1 [*R9.14] 1.2-1 [*R9.7]. So 7 C n 1, whence from F(y, ?J,C+O & C(y), whence C(y). Thus by 3-introd., C(a) 3 C(y). Similarly, C(y) r)C(a). By &-introd., C(a) C(y). By 2- and V-introds., (a). By *R8.5, (viii) 8 i ~ [ 8 i821. , Thence by (iv): (ix) C(8i) V lC(8;). CASE 1 : C(6;). We deduce (b) ~ E R&’ a(m+2)=8i(m+2)+a & EE[~;, 821 3 C(a) by induction on a with a double basis (IM p. 193), thus: FIRST BASIS.Assume a E R , (i’) a(m+2)=8i(m+2)+0 and (ii‘) a ~ [ 8 i821. , Now la-8il o > Ia-a(m+2)2-(m+2)/+ja(m+2)2-(m+2)-8i1 [*R6.17] = ~a-a(m+2)2-(~+2~~+/8~(m+2)2-(~+2~-6~~ [(i’), *R9.5, etc.] o> 1*2-(m+2)+1*2-(m+2) [*R9.15 twice, with ~ E R ’or 8 i ~ R ’ , “R6.16 (twice)] 1*2-(m+l) [*R9.10, *R9.3]
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CH. I11

THE INTUITIONISTIC CONTINUUM

+2)2-(m+2)I + Ia(m+2)2-(m+2) -8; (m+2)2-(m+2)I + 18; (m+2)2-(m+2)-8; I [* R6.1710> 1.2-(m+2)+ Ia(m+2)2-(m+2) -S; (m+2)2-(m+z)I + 1 -2-(m+z) [* R9.151 = 1-2-(m+2)+ I (Sf(m+ 2) + 1)2-(m+z)-Sf (m+2) ju -a(m

*

2-(m+2)1 + 1.2-(m+2) [(i”)]P 1-2-(m+2)+ 1.2-(m+2)+ 1.2-(m+2) [*R9.10, *R5.6] 3*2-(m+z)[*R9.10] < o 1-2-m [*R9.7]. So, using also (ii”)and (viii), la-Si1<01-2-m & a , S i ~ [ S i821. , By (a), C(u) C(Si), and by Case 1 hyp., C(a). IND. STEP.Assume MER’,(i”) a(m+2)=Si(m+2)+a”, (ii”) u ~ [ S i 821, , and the (second) hyp. ind. By *R9.2: (iii”) (S;(m+2)+a‘)2-(’+2)~R’. By XR9.1, etc.: (iv”) ((Si(m+2)+a’). 2-(m+2))(m 2) =8; (m+2) +a’. Also : (v”) (8; (m+ 2) +a’) 2-(m+z) 4-r (8;(m+2)+ 1)2-(m+z) [*R9.8] 8; [*R9.17]. And: (vi”) (8i(m+2)+ a’)2-(m+2) = ((6;(m+2)+af’) 1)2-(m+2) [*R9.5] = (a(m+2) 2 1). 2-(m+z) [(i“)] a [*R9.16] o> 82 [(ii”), *R8.6 with (ii), and a,8;,82~R]. Combining (v”) and (vi”) by *R8.6 (with (ii), (iii’”), etc.) : (vii”) (Si(m+2)+a’)2-(m+2k[&, 821. From (iii“’), (iv”) and (vii”) by the hyp. ind. : (viii”) C((S;(m+2)+a’)2-(m+2)). Further: (ix”) 1 ct -(6;(m+ 2) a’)2-(m+2)I I a- (6; (m 2) +a”) 2-(m+2) I I(Si (m+ 2) +aN)2-(m+2)-(6; (m+2) +a’)2-(m+2)I 5 101- (Si(m+2) a”)2-(m+2)I 1.2-(m+2) = la-~(m+2)2-(m+2)1+ 1.2-(m+z) [(i”)] .=p 1-2-(m+z)+1. 2-(m+z) A 1-2-(m+l) 1-2-m. By (a) with (ix”), (ii”) and (vii”), C(a) ,- C((Bi(m+2)+a’)2-(a+2)), whence by (viii’”), C(a). Now from (b) we deduce (c) Va(aE[S;, 821 C(u)) thus. Assume a E [ S i , 821. Using “R1.11, assume ~ ’ E R&’ C C ’ + M . By *R8.2, u ’ ~ [ S i 821, , whence by (ii), etc. and “R8.6, a’<4f. Using “R6.24 assume ~ ” E R&’ uN+u’ & a ” 2 S i . By “R8.2, a ” ~ [ S f821. , From u“>Si, assume a”(m+2)= Si(m+2)+a. By (b), C(a”). Also a”*a. By (iii), C(a). By 2-and Vintrod., (c). From (c), using (i) and *R8.3, Va(a~[81,821 2 C(a)). By V-introd., Va(a+31, 821 XIC(a)) V Vu(ct+31, 821 3 y C ( u ) ) . CASE2: Similarly, Va(a~[G1,SZ] 2 -,C(a)). N

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Q 16. The structure of the continuum. 16.1. In “Die Struktur des Kontinuums” (Brouwer 1928a), Brouwer discusses seven properties of the continuum (pp. 6-7), giving in most cases both intuitionistic counterexamples to classical theorems and (in general, without proof) intuitionistically true analogues of these classical results. For the property of discreteness, our “R9.23 corresponds to Brouwer’s counterexample (bottom p. 7 item 1) ; no intuitionistic analogue is given. For the next property, that of ordering (item 2 pp. 7-9), see 14.6 above where Brouwer’s axioms for virtual order are derived for <. For the

9

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157

remaining properties we concentrate on obtaining the intuitionistic theorems rather than the counterexamples. The results appear as 'R12.2 (density in itself), "R12.4 (compactness), "R13.8 (everywhere density), "R14.12 with "R14.11 (separability in itself), "R14.13 (free connectedness). As remarked in the introduction, in the case of the latter two we simplify and (in the case of the last) amend Brouwer's formulation. (These are respectively Brouwer's items 3, 7, 6, 4, 5.)

16.2. The relations of inclusion and proper inclusion for closed intervals are expressed formally by the formulas on the right (abbreviated on the left) in *R11.1 and "R11.2, respectively. "R11.3 is proved in Brouwer 1924-7 I1 p. 454 Footnote 1.

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Pi, 821 5 [qi, qzl &h,ql,qz~R8~Va(aE[&, 821 2 a ~ [ q iulzl). , "R11.2. b P i , 621 c [qi, 1 2 1 [Sl, 621 5 [Tl, 3 2 1 471,q z l c [Sl, 821. "R11.3. 1[Si, 821 C [TI, q z ] Sl,&E[qi, 7121"R11.4. b 81,Sg~R2 [Sl, Sz] c [Sl, 621. "R11.5. k i[81, 621 c [Sl, Sz]. "R11.6. k Pi, 821 C [qi, 3 2 1 & [ql, q2] E [el, (321 3 [sl, s21c [el, e2i. " ~ 1 1 . 7 . 1[al, azl = [ql, 3 4 & [ql, q21= [el, eZi "R11.1.

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[sl, sZ1= [el, e2i.

PROOFS. "R11.3. I. Assume [Sl, SZ] c [ql, qz], whence 81,Sz~Rand 81,&&31,&1 ~ l , S 2 ~ [ q l , q 2 1whence, , using "R8.5, 81,82~[q1,qz]. 11. Assume Sl,Sz~[ql,321, whence (i) S ~ , S Z , ~ I , ~and ~ E(ii) R 1(81q1 8~Sl>q2) and (iii) 1(82q1 & Sz>qz). Prior to 3- and V-introds., assume (iv) a~[S1,821, whence (v) -1(a<:S1&a<&) & l ( a > 8 1 & or>S2). Assume, for reductio ad absurdum, (a) u < q l & u& V u+S1, we deduce by cases using 'R7.4 (with (i), (iv)) and "R7.2, 8181 V a*&), whence by "63, l a 5 3 1 & l a + S ~ . By "R7.11 (and (i), (iv)), a<S1. Similarly a<&, and by &-introd. a < & & a<Sz, contradicting (v). So, rejecting (a), l ( a < q l & aq1& a>q2). By &-introd., l ( a < q l & aq2), whence with (i) and (iv), o r ~ [ q ~ 121. , "R11.6. Assume (i) [61,8~]E [ql, q2] and (ii) [TI, Y Z ] c [el, 821.

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T H E INTUITIONISTIC CONTINUUM

By (i) and "R11.3, 8 1 , 8 ~ ~ [ qq2]. i , So by (ii), 61,8~~[01, 021, and by "R11.3, [Si, 621 E [el, e~]. "R11.7. Assume (i) [ h , 621 c [ql, 3 2 1 and (ii) [ql, 921 c [el, 021. By "R11.6, [Sl, SZ] E [el, eel. Assuming [el, 021 c [Si, &I, [el, 021 c [ql, 321 follows by (i) and "R11.6, contradicting (ii). Hence 1[01, 021 E [Sl, 821. 16.3. We shall represent sequences of real numbers by sequences ilxy(203%), ilxp(213"), Axy(223"), . . ., where (n)lxy(2"3")~R. Y [ ~ I ( X= ) ?(2aX).

XR 12.1.

When "rp[n~" is used without argument, it shall abbreviate ilxcp[n](x) (i.e. ilxft(n, x, rp) for the function symbol ft introduced with the axiom xR12.1 ; cf. 5.1). 16.4. We now establish the property of the continuum of being "dense in itself" ("in sich dicht"; Brouwer 1928a p. 9 line 6 from below to p. 10 line 2, with p. 7 lines 11-12). That is, we show that, for each r.n.g. a there is a sequence of properly nested closed intervals each containing a, such that each r.n.g. contained in each of the intervals SL a.

1

"R12.2.

3 3$+{Vn[~[n+ll, +[n+lll C [ ~ [ n l ,+[rill & Vna~[cp[nl> +[nlI & vP(VnP€[~nl> hn11 2 PA.)).

PROOF.Assume aeR. Using "R1.11, assume: (i) ~ E R&' a ' n a . Using Lemma 5.3 (a), introduce cp and +: (ii) Vacp(a)=((a'((a)o)-l). = 2-("9 ((a)1), (iii) Va+(a)=((a'((a)o)+2)2-ca)0)((a)l). Now ~p[~](x) ((a'(n)2 1)2-")(x), whence by (XO. 1 and) V-introds., (iv) Vncp[ni= (a'(n) 2 1)2-n. Similarly, (v) V11+[~]=(a'(n)+2)2-n. By *R9.2 (and *R0.7), (vi) Vncp[n],+[n~€R. Using *R9.8, (vii) Vncp[nl+[nl. I. Using "R0.5a-c and (i) : (ix) 2a'(n) 22
(0

3

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THE STRUCTURE OF THE CONTINUUM

159

2-n [*R9.17] o> (a’(n)+2)2-n [*R9.8, *R6.7] = +[n]. SO a’+[nl, whence by “R8.6 (with (i), (vi), (viii)), a’~[cp[n], +[n]], and by *R8.2, a~[cp[nl,+[nl]. By V-introd. : (XV) Vna~[cp[nl,+[nl]. 111. Assume VnP~[cp[~l, +[n]]. Using *R1.11 : (xvi) ~ ’ E R&’ p’+p. By *R8.2, (xvii) Vnp’~[cp[~l, +[n1]. Assume for reductio ad absurdum, (a) a‘<+’. Assume Vp2k(P’(x+p) La’(x+p)) 22X+P. By V-elim., 2k(p’(x+2+k) :a’(x+2+k)) >2x+2+k, whence (p’(x+2+k) -a’(x+ 2+k)) 2 2x+2 > 3. So P’(x+2+k)>a’(x+2+k)+3, and P’(x+ 2+k)~ 1 >a’(x+2+k)+2. NOW ~ ’ K (p’(~+2+k) o 1)2-(x+2+k)o > (a’(x+2+k)+2)2-(x+2+k) = +[x+2+k]. So by *R8.6 (with (viii), (vi)), p’$[cp[x+2+k~, +~~+2+k]], contradicting (xvii). Rejecting (a) : (xviii) a’Kop’. Similarly (using “R9.17, (iv)): (xix) a’o>p’. By (xviii), (xix) (with (i) and (xvi)) and *R6.5, p‘Aa‘, whence p A a . By 2- and Vintrod. : (XX) Vp(Vnp~[cp[nl,+[nl] 2 PA.). 16.5. We establish a form of compactness defined in Brouwer 1928a p. 12 lines 20-32. The assertion is that in the continuum there is no “hollow interval nest” (“hohle Intervallschachtelung”), that is, no sequence of nested closed intervals I , such that, for each r.n.g. a, there is an integer n, such that 01 is not in I,,. This is expressed formally by *R12.4. *R12.3 is a useful lemma.

*R12.3. *R12.4.

~ 2 Vn[q[n+l],+[n+111 s [ ‘ ~ [ n l+[n11 V n V ~ s n “ p ~ nh119 1 s [ ‘ P I ~hm11. I, I- -dq+{Vn[cp[n+ii, $[n+i]l5 [~[nl,#[rill & VaaeR3ba$[cp[bl,+lbll)-

PROOFS. *R12.3. Assume the hyp. (i). Using *R11.1, q[m],+[ml~R. +[m]] by i d . on p, thus. BASIS. We deduce [y[m+p],+[m+p]] G [(~[m], *], c [cp~~+~], Use *R11.4. IND. STEP. Using *R11.6, [ ~ p [ ~ + ~+[m+p.]] Jl[m+pll [(ill 5 “ P c ~hI m , l l PYP. ind.1. “R12.4. Assume (prior to 3-elims.) : (i) Vn[cp[n+l],+[n+l]] 5 [(P[n],+[n]], and (ii) Vaae~~ba$[V[bl, +[bl]* By (i): (iii) Vncp[nl, +[nlERUsing * R l . l l , assume: (iv) cp’~R’& y’”p[o], (v) +’ER’ & +“+[o]. Using Lemma 5.3 (a), introduce: (vi) 81=0 and (vii) 82=9’(0)+ +’(O)+l. By *R9.2: (viii) 81,82~R’.By *R9.18: (ix) 610>82. Using xR9.1, &(x+ 1) =282(x) ; so by V-introd., (x) Vx82(x+ 1) =262(x). Further 61(x) = 0 62(x); so by V-introd., (xi) 81182. We deduce (a) [ ~ [ n l +[nl] , c [81, 821 by induction on n, thus. BASIS.By *R9.18, (cp’(O)+ 1)2-0 [*R9.17, (iv)] (cp’(O)++’(O)+ 1)2-0 cp‘<&. Also, 9’ O>

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[*R9.8, "R9.5; "R6.7, "R6.81 = 82. So by "R8.6 (and (ix), etc.), 821, and by "R8.2 and (iv), 'p[01~[81, 821. Similarly, +[0]~[81, 821. Using "R11.3, [cp[ol, +roll E [SI,821. IND.STEP.Using "R11.6, [qqn+ll, [81, 821 [hyp. ind.1. Using (viii) and (xi) in +[n+l]l c ['P[n],+[Ill1 [(ill *R10.2: (xii) Spr(o) & o(l)=O & Va[a(a)=O 3 3bVs(a(a*2s+l)=O 3 sib)], (xiii) Vcr[cr~o XER'& 81
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(xvii)

Vxu(x) =

J(a)x-l if x
Px

A

Thus (xviii) V X ~ < ~ U ( X ) = ( ~ ) ,1.I -We shalldeduce (b) 12u(x)-u(x')I l . So x ' < (z-l)+l = z. So )2u(x)u(x')I = ~ 2 ( ( a ) x ~ 1 ) - ( ( a ) x< ~ ~1 l )[(xvi), ~ case hyp.]. CASE 2: x=z'l. SUBCASE 2.1: z=O. Then by (xvi), a = l . So 12u(x)-u(x')I = 10-01 < 1. SUBCASE 2.2: z > l . Then xz-l. Then x' ( a L 1) x 2 z . s o 12u(x)-u(x')I = 12.2xL("Ll) ((4GlL1)-2 ( ( a L 1'1)l = 0 < 1. From (b) by V-introd., (xix) UER'. We next deduce ( c ) u(x)<&(x) by ind. on x, thus. BASIS:By cases (z=O, z>O), u(0) = (a)o-1 [(xvii)] < Sz(0) [(xvi)]. IND.STEP.CASE 1 : x'z, x'=z] < 2 6 4 ~ ) [hyp. ind.] = 82(x') [(x)]. From (c) by V-introd., ~ 1 6 2Similarly . to (xi), Sl +[~(a)lIlIV[ l B ( a ,Z, 82)&P(a)=oI}* Let t=maxis~,c,)P(i).We deduce a contradiction thus. By (a) and "R11.3, cp[t1~[81, 821. Using "R1.11 : (xxii) &R' & cpL~qqt-j. Now &81, 621, whence by "R8.6 (with (ix), etc.), &0>62. So using "R6.22,

§ 16

THE STRUCTURE OF THE CONTINUUM

161

(xxiii) (P&R& rp:~cp[t] & &‘<62. Similarly to (xi), 611rg;. So using (xiii): (xxiv) &EC. Using (xxiii) with X23.1, *B21, *3.12, etc., r&’(z)&(z). So by *H7, letting s=P(x(z)): (xxv) s l t . Employing *23.5 and *23.2, we readily deduce B(Z(z),z, 62) from (xxiii), using cases (211, z > l ) for the last part. So by (xxi) and *23.2: VY,,,~~XX<,Y(X) =CPZ(x) 3 Y4“p[sl* +rsll}. Thence with (xxiv), (PW t ‘P[slJ 4qsfJ, whence by (xxiii) and *R8.2, (P[~I$[(P[~I, +rsl]. But by *R12.3, (i) and (xxv), [(~[t], 9rt1ls [qqs],+[s~!, whence by *R11.3, (PctlE“‘Plsl9 (C[Sll.

16.6. The formula on the right of “R13.1, abbreviated “a~(S1,62)”, expresses the assertion that a is an r.n.g. in the open interval (81, 84, or that a is “between” 61 and 82 (Brouwer 1928a p. 9 lines 13-10 from below; for *R13.5, cf. lines 8-6 from below). *R13.1. *R13.2. *R13.3. *R13.4. *R13.5.

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I- aE(61, 62) aE[61, 621 & l a 4 3 1 & laA62. I- a A p & a€(&, 62) 3 @~(61,62). I- 6196i & aE(S1,62) 3 aE(6i, 62). 16296i & aE(61, 62) 3 aE(61, 6i). I- a,61,62~R& 3 {aE(61, 62) 61
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PROOFS.“R13.5. Assume a,61,62~Rand (i) 6 1 ~ 6 2 .I. Assume (ii) a~[61,621, (iii) -rccGS1& l a & & . Using (i) and (ii) (and *R7.4), la<S1 & 1a>62. Thence with (iii) and *R7.11, S1
I- Vx12a(x)-a(xf)I 1 2 3 VpVxl2Pa(x)-a(x+p)I <2p+l. I- Vxl2a(x)-a(x’)I 1 2 3 ~ E R . I- a$ER & 1 a A P 3 3yYERy~(a, p).

PROOFS. *R13.6. Assuming (i) Vx12a(x)-a(xf)I 2 2 , we deduce Vxl2pa(x)-a(x+p)l<2p+l by ind. on p, thus. I N D . STEP. 12P’a(x)a(x+p’)I I 2PI2a(~)-a(x’)I+12~a(x’)-a(x’+p)I < 2~+1+2p+l [(i), hyp. ind.] = 2p’+1. *R13.7. Assume Vx12a(x)-a(xf)l 1 2 . Using “R13.6, 2k12pa(k+ 1)a(k+ 1 +p) 1 <2k+l+p, whence aER. *R13.8. Assume a,PER, (i) la+p. Using *R1.11: (ii) a ‘ E R & a’&a,

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CH. I11

(iii) P’ER’& p’6.B. Introduce y by Lemma 5.3 (a):

(A)

VXY(X) = min(a’(x), P’(X))+

[max(a’(x), p’ (x))2 min(a’(x), (3‘ (x))/2].

Thus we can write y(x) = b + [ a - ~ b / 2 ] [where a 2 b by *8.8] 2 b + [ a l b / l ] [*13.11] = b+(aAb) [*13.6] = a [*6.7]. So (iv) Vx min(a’(x), p’(x))a‘(x)+l. NOW2y(x) = ~ K ‘ ( x ) + ( ( ~ ’ ( x ) L ~ ’ ( x ) ) 1) = 2a’(x)+(p’(x)~ ( d ( x ) 1)) + [*6.5] = (2a’(x)+p‘(x)) (a’(x)+l) [*6.6] = (a’(x)+p’(x))Ll [*6.5, “6.31. For (vi): CASE A‘: a’(x‘)<2a’(x) & p’(x’)<2p’(x). Then 2a’(x)2 1 & 2p’(x) 2 1 and ~’(x‘) = 2a’(x) 2 1 & p’(x’)=2p’(x) 2 1 [*R0.5a-b, (ii), (iii)]. Soy(x’) = (2a‘(x)2 1) + [(2p’(x)L 1) 1(2a’(x)A 1)/2] = (2a’(x)2 1)+ [2p’(x)12a’(x)/2] [*6.9, *6.7] = (2a’(x)~l)+@’(x)-~a’(x)) [*6.14, *13.8, x13.11 = (2a’(x)+ 1 [*6.6 twice, *6.5] = (a’(x)+p‘(x)) 11 [*6.5, *6.3]. B’(x))~ ( +a’(x)) CASE B’: a’(x’)<2a’(x) & p’(x’)=2p‘(x). Similarly y(x’) = (2a’(x) 2 1) [2p’(x) -2 (2a’(x)2- 1)/2] = (2a’(x)2 1) [(2P’(X) 2) 2 (2a‘(x) 1)/2] = (2a’(x)2 1) [((2p’(x)+2) 22a’(x))2 1/2] = (2a’(x)2 1) (((p’(x) 1) La’(x)) 1) [*13.9, “13.71 = (a’(x)+p‘(x))’l. CASE C’: a’(x’)<2a’(x) & a’(.’) >2p’(x). Similarly y(x’) = (2a’(x)2 1)+ [(2p’(x)+ 1) -(2a‘(x) 1)/2] = (2a’(x)2 1)+ [(2p‘(x)+2) ~2a’(x)/2],etc. Cases D’-I‘ are treated similarly. CASE 2: a’(x)Zp’(x) & ~ ’ ( x ‘>P’(x‘). ) Symmetric to Case 1. CASE 3: a’(x)p’(x’). SUBCASE3.1 : a’(x)P’(x), so this subcase comes under Case 2. CASE 4: a’(x)>p’(x) & ~ ’ ( x ’ y(x+p) and VpP’(x+p) >y(x+p). Thence a‘@)>y(x) and P’(x)>y(x),

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THE STRUCTURE O F THE CONTINUUM

163

so by *7.6, min(a’(x), p’(x))>y(x), contradicting (iv). So, rejecting @), yQ=op‘. By 3-introd., discharging (a) : (viii) y < a ’ 3 ygop‘, and by contraposition ly{=op‘ 3 lya’ & y>p‘). By &-introd. with (ii), (iii) and (vii), yE[a’, p’], and thence by (ii), (iii) and *R8.3-*R8.4: (ix) y ~ [ ap]. , Toward l y e a , assume (c) ysa’. We shall first contradict (d) y<+’. Assume from (c), (d) and *R0.9: (e) Vp2k+lly(x+p)-cr’(x+p)l< 2x+p, and V~2~l(p‘(x+p) -y(x+p)) 22x+P. Then Vpp’(x+p) >y(x+p). Using also (iv) (since min(a, b) =a V min(a, b) =b) : (f) Vpa’(x+p) 2 y(x+p)) 1 [*6.6, (91. So (h) (1 +y(t)) l a ‘ @ )>P’(t) l y ( t ) . mlence 2kIP‘(t)--Y(t)l = 2k(P‘(t)LY(t))[(f)l I 2k((l+Y(t))-”) [(h)l = 2k(l+(y(t)’a‘(t))) [*6.6, (f)] < 2t [(g)]. By V-, 3- and V-introd., p‘ey, contradicting (d) by *R6.4. Hence yQ=+’. Similarly, y.>p‘. By *R6.5, ynp’. By (c), a’+@’, and with (ii), (iii), a&@,contradicting (i). Hence, rejecting (c), -yea’. By (ii), YyAa. Symmetrically, l y n p . By &-introd. with (ix): (x) YE(., p).

+

+

16.7. To express the “sharp difference” of two r.n.g. (Brouwer 1928a p. 10 lines 27-33), we use the formula on the right of *R14.1, abbreviated “ct#sp”. (For *R14.2, cf. 1928a p. 10 Footnote 7.)

*R14.1. *R14.2.

I- a#& - l a + @ & VyYER{y$(a, p) 3 ( l y > a & l y > p ) V (1y-0 & lycp)}. I- a,f%R & u # $ 3 a<@ V p
PROOFS. *R14.2. Assume PER and a#&. Then (i) la*Zp and (ii) .$(a, p) 3 ( l a > a & lcr>p) V (lcr P V -.la
I- a , p E R 3 Vx(p(x)-a(x)=2 3 P(x’)“a(x’)22). k a , p ~ R3 ‘ Vx(p(x)-a(x)>2 Z)P(x’) -a(x’)>2).

164

CH. I11

T H E INTUITIONISTIC CONTINUUM

*R14.5.

t-

PER' 3 V X ( ~ ( X~) a ( x ) = & 2 P(X‘):~(X’)=~ a(x‘)=2a(x)+ 1 & p(x’)=2p(x) 2 1).

3

PROOFS.*R14.3. Assume a,p~R‘ and p(x)‘a(x)=2. Now 2p(x)2 1 =2a(x)+3. So p(x’)Aa(x‘) 2 (2p(x)2 1) ~ a ( x ’ ) [PER’, *R0.5a-c, *6.17] 2 (2p(x)- 1) -(2a(x)+l) [aeR, *R0.5a-c, *6.18] = (2a(x)+3)-(2a(x)fl) = 2. > *R14.4. Assume Q,PER’ and p(x)-a(x)>2. Now 2 p ( x ) ~ 1 2a(x)+3 > 2a(x)+1. So p(x’)~a(x’)2 ( 2 ( 3 ( ~ ) ~ 1 ) 2 ( 2 a ( x ) > +l) (2a(x)+3) 1(2a(x)+ 1) [*6.19] = 2. *R14.5. Assume E,PER‘, p(x) 2a(x)=2 and p(x’)‘a(x’)=2. If a(x’)#2a(x) + 1, then by *R0.5a, a(x’)<2a(x), whence 2 = p(x’)2 a(x’) 2 P(x’) -2a(x) 2 (2p(x) 1) 22a(x) = 2(p(x) -a(x)) 2 1 = 3. Hence a(x’)=2a(x)+ 1. Similarly, p(x’)=2p(x) 2 1. _I_

*R14.6.

I- a , p ~ R& ’ a
3

3a‘a’ER3p‘B’ER{a‘~~ & p”p

Vx@‘(x)La‘(X)=2

& u‘g3‘ &

3 P‘(X‘) -a’(x‘)

>2)}.

PROOF.Assume (i) ~ E R(ii) , PER’, (iii) a s p . Introduce a’ and p‘, using Lemma 5.5 (b) and (a), and letting A(a, p, x) be p(x)~ a ( x ) = 2 & P(X’)-a(x‘)=2.

(Al) (A2)

(B)

u’(0)=Ia(O)+l if A(% p, 01, \a(O) otherwise.

+

+

a@’) 1 if A(a, p, x’) & a(x’)#2a’(x) 1, VX~’(X’) = a(x’) otherwise. a(x)+l if A(a, p, x) & a(x)=2a’(x-1)+1, x) otherwise.

{

Then easily: (iv) Vx(a(x)i a ‘ ( x )
+

9

16

165

THE STRUCTURE O F THE CONTINUUM

a(x)=2a’(x-l)+1. We deduce (a) a’(x)=a(x) thus. CASE A: x=O. By (iv), a’(O)>a(O). Hence a(O)#2a’(O)+l, whence by Case 2 hyp., -A(%, P, 0). By (Al), u’(O)=a(O). CASE B: x>O. Use Case 2 hyp. SUBCASE 2.1 : A(a, P, x’) & a(x’) #2a’(x) 1. Then by (v) : (i’) a’(x’)= a(x’)+l. By (a), a(x’)#Za(x)+I. By (i) and*R0.5a-b, a(x’)+l=2a(x) V a(x’)=2a(x). We deduce (b) ar(x‘)=2a(x) V a‘(x’)=2a(x)+ 1. CASE A: a(x’)+l=2a(x). Then a’(x’) = a(x’)+l [(i’)] = 2a(x). CASE B: a(x’)=2a(x). Then a‘(x’) = a(x’)+l = 2a(x)+l. Now by cases from (b), using (a): 12a’(x)-a’(x’)I
+

“I.

11. We shall next deduce 12P’(x)-P’(x’) 1 I 1, whence (by V-introd.) P’ER’.CASE I : A(a, P, x) & a(x)=2a’(x’ 1)+ 1. SUBCASE 1.1 : A(a, p, x’) & a(x’)=2a’(x)+l. By*R14.5, a(x’)=2a(x)+l. Now 12p’(x)-p’(xf)I = I2(a(x)+l)-(a(xr)+1)1 [(B)] = 1(2a(x)+2)-(2a(x)+2)1 = 0 I 1. SUBCASE 1.2: lA(a, P, x’) V a(x’)#2a’(x)+ 1 . Then a(x’)=2a(x) 1 (using *R14.5) and P(x‘)=a(x‘)+2 (using case hyp.). So l2p’(x)-pf(x‘)I = 12(a(x)+l)-~(x’)I [(B)] = I(a(x’)+I)-(a(x’)+2)1 = 1. CASE 2: -rA(a, P, x) V a(x)#2a’(x- 1) 1. SUBCASE 2.1 : A(a, p, x’) & a(x’)=2a’(x) 1. If a’(x)+a(x), then by (iv), a’(x)=a(x)+l, a(x‘) = 2a’(x)+l [subcase hyp.] = 2(a(x)+l)+l = 2a(x)+3, contradicting (i) by *R0.5a. So (Y) a‘(x)=a(x). So by subcase hyp.: (ii”) a(x’)=2a(x)+I, and also using *R14.4 (with A(a, @, x’)) : (iii”) P(x)-a(x) 5 2 . Assume for reductio ad absurdum (a’) P(x)1-a(x)=2. Then, using A(a, P, x‘) (from subcase hyp.), A(a, (3, x). So by case hyp., a ( x ) # 2 a ’ ( x ~ l ) + l . Now by (v), a’(x)=a(x)+ 1, contradicting (i“). Thus, rejecting (a’), (iv”) P(x)-a(x)#2. If P(x)la(x)=O, then P(x’) = a(x’)+2 [subcase hyp.] = 2a(x)+3 [(ii”)] 2 2P(x)+3 [*6.11], contradicting (ii) by *R0.5a. Thus P(x)-a(x) #O. Then by (iii”) and (iv”), p(x)-a(x) = 1. so I2Pf(X)-P’(X‘)l = I2P(x)-(a(xf)+1)l [(B)1= 12(a(x)+l)-(a(x’)+l)I = 1(2a(x)+2)-(2a(x)+2)1 [(ii”)] = 0 1. SUBCASE 2.2: lA(a, p, x’) V a(x’)#2a’(x)+l. Then 12p’(x)-P’(x’)I = 12p(x)-p(x’)I I 1 [(ii)]. 111. From (iv) by *11.15a, Ja‘(k’+p)-a(k’+p)I
+

+

+

+

+

+ +

166

THE INTUITIONISTIC CONTINUUM

CH. I11

V. We shall deduce a’(x)2.By 3- and V-introd., (x) Vx(P’(x)‘a’(x)=2 3 P’(x’) -a’(x’)>2). In “R14.8 A(y, a, P, y) is Vx[(xy 3 y(x)=

P(4)I. t PER' & P(x)-a(x)>2 3 IP-a14=01.2-X. I- %,PER’& a<@ & VX(P(X) ’Cc(X)=2 3 P(X’)-a(x’)>2) & IP-aI <01*2-(9+2) 3 3YY€FL4Y> P, Y) v 3Yy,RA(Y>P>a>Y).

“R14.7. *R14.8.

a 9

PROOFS. *R14.7. Assume (i) K,PER’, (ii) P(x)Aa(x)>2. Then 3-2-X 5 0IP(x)-a(x)12-X [(ii), *R9.8, *R9.5] 5 I~(x)2-X-a(x)2-Xl [*R9.14] IP(x)~-”-PI+I~-~I+I~-.(x)~”~ [*R6.17] 1.2-Xf Thence, using Ip-aI+1.2-x [*R9.15, (i); “R6.161 2.2-X+Ip-al. *R6.16, IP -a1 4 1-2-X. “R14.8. Assume (i) ~ E R(ii), PER’, (iii) a<@, (iv) V x ( p ( x ) ~ a ( x ) = 2 3 p(x’) -a(xf) >2), (v) IP-al < 0 1 - 2 - ( ~ + ~ Toward ). (vi), assume (a) x2.Now IP-al K O 1-2-=’ [*R14.7] 1-2-(yf2)[(a), with “R9.5, *R9.7, *3.12], contradicting (v). By 3- and V-introd. : (vi) Vx(x
O>

0

0

2

§ 16

THE STRUCTURE OF THE CONTINUUM

167

Then A(y, a, p, y). We shall deduce 12y(x)-y(x’)[y. Use (ii). By V-introd., ~ E RBy . &-, 3- and V-introd., (c). CASE 2: p(y)+x(y)=l. Then (i’) 2p(y)2 1 =2a(y)+ 1. Using (vi), p(y’)l a ( y ‘ ) I1. SUBCASE 2.1 : p(y‘)’a(y’)=O. Introduce y by (A) above, whence A(y, a , p, y). We shall deduce 12y(x)-y(x’)Iy. Use (ii). By V-introd., ~ E R ’By . &-, 3- and V-introd., (c). SUBCASE 2.2: p ( y ’ ) ~ a ( y ’ ) = l Then . (ii’) P(y’)>a(y’). If 2a(y)>a(y’), then by (i’), 2p(y)2 1 >a(y‘), and thence 1 = p(y‘) Aa(y’) [subcase hyp.] 2 (2p(y)2 1) l a ( y ’ ) [(ii), *ROSa-c, *6.17] > (2p(y)2 1) -2a(y) [*6.20] = (2a(y)+ 1) -2a(y) [(i‘)] = 1. Hence 2a(y)2p(y) A 1, then 1 = p(y’)la(y‘) [subcase hyp.] > ( 2 p ( y ) ~ l ) ~ a ( y[(ii’), ‘ ) *6.19] = (2a(y)+ l ) l 2 a ( y ) [(i’), sub2case hyp.] = 1. Hence p(y’)<2p(y) 1, whence by *R0.5a-c and (i’): (iii’) p(y’)=2a(y)+l. Introduce y by (A) above, whence A(y, a, p, y). We shall deduce 12y(x)-y(x’) I < 1. CASE B: x=y. Then 12y(x)-y(x’)l = 12a(y)-p(y’)I = 12a(y)-((2a(y) +1)1 [(iii’)] = 1. By V-introd., ~ E R ’ By . &-, 3- and V-introd., (c). SUB~CASE 2.2.2: 24y)
*O if a= 1 V {Seq(a) & lh(a)= 1 & [ ( a ) o l l=a(O) V ( a ) o l l=p(O)]} V {Seq(a)& lh(a)> 1

(A)

Vac(a)=.

12((a)lh(a)“211)-((a)lh(a)ill l)lgl & c(&
168

THE INTUITIONISTIC CONTINUUM

CH. I11

IIb. Assume o(a)=O, toward deducing 3sa(a*28+1)=0. CASE I : a = l . Then by (A), ~r(a*2~(~)+')=0. CASE 2: a f l . Then by (A), Seq(a) & lh(a)2 1 8~ [(a)lh(a)Ll 2 1 =a(lh(a) 2 1) V (a)lh(a)L1 2 1 =P(lh(a) 1l)]. SUBCASE2.1 : (a)lh(a)Ll 2 1 =a(lh(a) 2 1). Using (A) and (i), ~ ( a * 2 ~ ( ' ~ ( ~ ) ) + SUBCASE ')=0. 2.2: (a),h(a)Ll2 1 =p(lh(a) 2 1). Using (A) and (ii), ~ ( a * 2 ~ ( ' ~ ( & =O.) ) + ~ ) IIIa. Assume (a) y ~ o .By V-elim., a(y(x"))=O, whence using (A), 12y(x)-y(x')I i 1. By V-introd., yeR'. By V-elim. from (a), o(y(x'))=O, whence from (A) by cases (x=O, x>O): y(x)=a(x) V y(x)=p(x). By V-introd., Vx(y(x)=a(x) V y(x)=p(x)). IIIb. Assume y€R' & Vx(y(x)=a(x) V y(x)=p(x)). By ind. on x with a double basis, o(y(x))=O. By V-introd., y ~ o . *R14.10. Assume (i) Vx(y(x)=a(x) V y(x)=p(x)), and for reductio ad absurdum (a) y ~ ( a , p ) , whence ~ E R and ~ E R If . yo>a, then assuming Vp2k(y(x+p) ~ a ( x + p )22X+P, ) we deduce successively Y(X+P) #a(x+p), y(x+p)=P(x+p) [(i)l?and Y G p [V-introd., *R1.21, contradicting (ii). Thus yo>.. Similarly, y
I- a$ER 3 (afsP

-

a#P).

PROOF. Assume a,PeR. I . Assume (i) a#*@. Using *R1.11, assume (ii) E'ER' & a'+a, (iii) P'ER' & p ' ~ p .By (i), *R14.2 and *R7.2-*R7.3, ct'2). By (ii)-(v) : (vi) a b + a " ~ a ' + a & p"+p'+p. Applying *R14.9 (with (v)), assume: (vii) Spr(o) & o(l)=O & V y [ y ~ o yeR' & Vx(y(x)=a"(x) V y(x)=p"(x))]. Toward (viii), assume y ~ a . Then yeR' & Vx(y(x)=a"'(x) V y(x)=p"(x)). By *R14.10, y$(a'", p"), whence by (vi) with *R13.3-*R13.4, y$(a, p). So by (i) with "R14.1 and yeR', ( l y > a & l y > p ) V ( - t y < a & l y < p ) . By 2and V-introd., (viii) Vy,,,{(ly>a & l y > p ) V ( l y e x 81lyO 3 y=xl&{(ly>.&-ry=ip&7(y(y))=1) v(ly 20 y1=x] & {(Ta"'>a & la"'>>p & .(."(y1))=1) v (+
-

§ 16

@(yl))

169

THE STRUCTURE O F THE CONTINUUM = 1. Similarly to

(x) and (xi) : (xii) VX[T(~(X)) >O

2 y ~ = x ]&

{ ( l p “ > a & l p ” > p &7(pN(yz))=l) v ( l p ” < a & l p ” < p &‘C(p7y2))=2)}, (xiii) 7(v(yz))=2. Let y=max(yl, yz). Assume for reductio ad absurdum: (xiv) ~p”-a”~<~l.2-(y+2). Thence by “R14.8 with (v), 3yY,R,A(y,a’”, P”, Y) V 3yy,wA(y, P”, a’”,Y).CASEA : 3y,,,,A(y, a”‘, p”, y). Assume (xv) ~ E R&’ A(y, a’”, p”, y). Then Vx(xO 1 z=x] & {(-,y>cc & l y > p & T(?(z))= 1) V ( l y < a & l y < p & 7(?(2))=2)}. Using (xvi) in (xvii), z=yl. So by (xvi), T(?(Z))= 1 , whence by (xvii): (xviii) ly>a & -~y>p. But using (xv), 2kly(y’+p)-P”(y’+p)l = 0 < 2y’+p, whence y 5 p” A p’ [(vi)] > a’ [Case 1 hyp.] * cc [(vi)], contradicting (xviii). CASE B: 3yYewA(y,p“, a”’, y). Similarly. Thus rejecting (xiv), Ip“-a 111 I K O1~2-(~+’) o> l~2-(’+~) [*R9.7]. So assume

=I

Vp2k((lP”-ar”l)(x+p) 1(1-2-(~+~))(x+p)) 22X+P. Thence by “6.15, etc., 2k(lP”-a”l)(x+p) 2 2 X + P , whence by XR5.1 and V- and 3-introds., p”#a”’. Thence by (vi) with *R2.4 and *R2.3, a # p . CASE 2: p‘ a & ly>P) V ( l y < a & lyp. By *R6.9 (and a, PER, (iii)), ap. Thence using *R8.6, (v) and (vi): (c) yE[a, p]. By (b) and “R6.4, 1 y A p . Thence using (vii) and (c),yE(a, P), contradicting (iv). So rejecting (b), y<+. With (vi) and *R7.8, l y < a & -ry a & l y > p ) V ( l y < a & -.ly
t- a$ER & u#P

3 3a3m(a2-%(a,

p)).

PROOF.Assume u,PER and u # p . By “R6.2, a<+ V pea. CASE 1 : a<+.

Using *R9.19, assume (preceding 3-elims.) a
170

THE INTUITIONISTIC CONTINUUM

CH. I11

By *R7.7, a
16.8. We define the unit continuum [0, 11 to be “freely connected” (“freizusammenhangend”) if, for each pair A ( y ) and B ( y ) of species of r.n.g. such that (1) A ( y ) & B ( [ )--f ya - + A ( y ) , or, for each y in [0, 11, y

q --f B(y). (Brouwer 1928a p. 1 1 lines 14-3 from below.) Brouwer required the above only of A ( y ) and B(y) out of which [0, 11 is “composed” (“zusammengesetzt”, p. 10 bottom), i.e. such that there does not exist a y in [0, 11 for which neither A(?) nor B ( y ) . We do not need to make this restriction in establishing the free connectedness of [0, 13 in *R14.14. In “R14.13 we consider a generalization of Brouwer’s result to the whole continuum [0, co). We add the hypothesis that B ( y ) has an element. (For [0, 00) the theorem would not hold otherwise.) Instead of a#& & a

a we use yo>a (equivalent by *R7.8). Further, in the conclusion instead of +: we put < o , for a reason to be given in Remark 16.1.

-

*R14.13* t- VyyeRV&ER{A(y) & B(C) 3 yT

I)B(y)l}*

PROOF.Assume the hyps. (i)-(iii). Using (ii), *R9.2 and *R9.8, Vyy,R[yo=Pa2-m3 A(y)] V Vyy,,[y:y4=0(a+ 1)2-” I) B(y)]. By cases and V-introd., VaVm3y{[{VyY,,[yo>a2-m 3 A(y)] & y=O) V{Vy,,R[yQ=~ (a+ 1)2-m 3 B(y)] & y= l}] & y l l}. Applying *25.7, assume: (iv) VaVm@“y,R[y.>a2-m I)A(y)l &x() =O}V{Vy,,R[y4=~(a+1)2-m 2 B(y)] & x()= l}] & x(b2-o I) A(y)]. Using “R9.2 and *R9.8, A(b2-0). So by (i) and (v), b2-0
171

(A)

-2q(x)2 1 if x(<2q(x) 1, x’>)= 1, 2q(x) if x(<2q(x)L 1, x’>)=O Vxq(x’)=. & x(<2q(x),x’>)=l, 2q(x)+l if K(<2q(X)-1, x’>)=O & x(<2q(x),x’>)=O.

Using *R0.5b-a, (vii) ~ E R ’whence , (viii) qeR. We shall deduce (ix) 1)2-x 3 A(y)l vYy&t[yo>(q(x)f 1)2-x B(y)l by ind. on x. BASIS. By (A), q(O)=M. Toward (a) assume yeR and yO. So M - l < M . Hence by (vi) and *E5, x(<M 1, 0)) # 1. So by (iv) (and using *R6.7), A(y). By 3- and V-introd., (a) VyYeR[y(M+1)2-0. By (vi) and *E5, Toward (b), assume ~ E R x(<M,O>)= 1. So by (iv), B(y). By 3- and V-introd.: (b) Vy,,,[yo> (q(O)+ 1)2-0 3 B(y)]. IND.STEP. We use cases (the same as in (A)) 2 1)2-x’ 2 A(y)] and (d) VY,,,[YO> to deduce (c) VY,,~[Y (q(x‘)+ 1)2-=’ = ((2q(x)2 I ) + 1)2-=‘. By (iv) and case hyp., B(y). By 3- and V-introd., (d). CASE2:x(<2q(x)21, x’>)=O &x(<2q(x),x’>)=l. Similarly. CASE^: x(<2q(x)L 1, x’>)=O & x(<2q(x),x’>)=O. Similarly. By V-introd. (and *87) from (ix) : (x) VXVY~,,[Y (q(x)+1)2-X Z)B(y)]. Toward (xi), assume ~ E R and y < q . Using *R1.11, assume ~ ‘ E R&’ y“y. Then y’2x+k+2, whence q(x+k+2) ~ y ’ ( ~ + k + 2 ) > 4NOW . YAY‘ (y’(~+k+2)+ 1)2-(x+k+2) [*R9.17] (q(x+k+2) 21)2-(x+k+2) [*R9.8]. Using (x), A(y). By 3-and V-introd. : (xi) VyYpR[yq 3 B(y)]. Combining (viii), (xi) and (xii): 3~,,R{VyycR[y<~~ 3 A(y)] & VY,cR[YO>Y1 3 B(Y)ll. Using *R14.13, we prove the free connectedness of [0, 11. We let a d abbreviate aER & a o + 1 ; the latter is equivalent to aE[O, 11 by *R8.6, *R9.18, etc. vYy~RIY
‘



*R14.14.

1Vy{A(y) V B(y) 3 ~ € 1&)VyVC{A(y) & B(C) Z)yY 2 B(y)l).

172

THE INTUITIONISTIC CONTINUUM

CH. I11

PROOF.Assume the hyp. (i)-(iii). Let B'(y) be l l ( B ( y ) V y > I ) . Toward (iv), assume y&R and A(y). By (i),y d . Assuming B(<)V <>1, we deduce y<< by cases (using (ii) in the first case). So by *12 and *R7.9: B'(<) 3 y<<. Hence (iv) VyYeR&,R{A(y) & B'(C) 3 y<<). Toward (v), assume PER and a<.@. By *R6.9, a

a =I A h ) ] V VyYeR[y4:oP3 B'(y)l. CASE 1 : a < ~ l . Using *R9.19, assume a<~a12-m1<~l and a < 0 a 2 2 - ~ ~ <+. By *R9.4, *R9.7, etc. : a12-m1a22-ma. Thence by cases we deduce a formula from which we can assume: (i') aa 3 A(y)] V Vy,,I[y-K~a2-m 3 B(y)]. SUBCASE 1.1 : Vy,,I[yo>a 3 A(y)]. Then VyysR[y~>a3 A(y)], whence (a). SUBCASE R y4Coa2-m. If y d , then 1.2: Vyy,,[y l . If y+I, then -1y0>1, whence by *R7.9, y >1, whence B(y) V y > 1. So y d V ye1 2 B(y) V y > 1. Thence by *12 and "63, +(y) V y > l ) 3 ye1 & 7 ~ 4 1 .Thus l l ( B ( y ) V y > l ) , 3 B'(y)]. Using (ii'), VyYeR[y4:=.p 3 i.e. B'(y). So VyYGR[y4:=a2-m B'(y)], whence (a). CASE 2: 1<+. We deduce VyyeH.[yQC~p 3B'(y)]. Thus (v) vaaeHvP~€R{CCa A(y)l vYy~R[y ~=I B'(y)]. For reductio ad absurdum, assume (b)yp>l. Using "R9.19, assume 1 ) in the conclusion is replaced by < (>), and Brouwer's hypothesis l3yy,R[lA(y) & lB(y)] is added. For, we show that if the instance (A) of this for ycop, yo>p as the A(y), p p . Then ya. Then yp]. CASE 2: p<+. Similarly, assuming ~ E and R y<+, and deducing yo>p. Ic. Using (i) and *R1.1I , (

0

N

3

16

THE STRUCTURE OF THE CONTINUUM

173

assume ~ ' E R&' p ' ~ p . Then p + p'0>(p'(O)+1)2-0 [*R9.17] (p'(O)+2)2-0, whence 3b(b2-00>p). Id. Prior to 3-elim. and 7-introd., assume yeR & -Iy p . Using *63, yQ=op& 1yAp. Using *R6.5, y*p. 11. Now using (A), after 2-elim. and prior to 3-elim.: (i +R ) Vy,,R[yp]. Assume for reductio ad absurdum, (a) p < q . Using *R13.8, assume y e R & ye(p, 3). Then by *R8.6 and (a), (b) y{op & y o > ? & ly*p & -1y5-q. So yq. By *R7.7, p>q. By (ii), po>p, contradicting *R6.8. Hence p>q. Now by *R6.5, p+. So by (ii), VyyER(y>p2 yo>p). This with *R7.7 gives VyYER(pp2 y">p). Assume (ii) a$eR and (3) u>p. Putting y = l - ~ ( l - l _ ( a - ~ p(cf. ) ) X7.1), we can establish (iv) y d , (v) y>O [using (iii)] and (vi) yo>O 2 ao>p. Using OEI, (iv) and (v)in (i), and the result in (vi), we have after 2-introds.: a,fkR 2 (a>p 2 ao>p).
-


CHAPTER I V

O N O R D E R I N T H E CONTINUUM by S. C. KLEENE

Q 17. Introduction and preliminaries. 17.1. Brouwer in his paper “On order in the continuum, and the relation of truth to non-contradictoricity” 1951 considers five pairs of properties of real numbers, and makes various claims about the relations between the properties in the pairs. (There he cites his Dutch articles 1948b and 1949a for the proofs. But three of the pairs are also discussed in his English paper 1948 p. 1248.) We shall list these claims, and determine (independently of 1948b and 1949a, or of 1948) exactly which ones can be established on the basis of the formal system of Chapters I and I11 above. (Our results here, except those (in 18.2) based on 33 10, 1 1 of Chapter 11, were obtained in preliminary form in May 1955, before Vesley’s Chapter I11 was written.) We chose Brouwer’s paper 1951 for this investigation (but could have chosen 1948), because it contains accessible examples of results which Brouwer reached only by a new method, which he introduced in 1948, and used in 1948a, 1948b, 1949, 1949a and later papers. This method rests on “defining” a choice sequence a(O),a(l),a(2), . . . in such a way that a(x+l) depends on whether or not the “creating subject” between the choices of a(.) and .(.+I) will have solved a certain problem, the solution to which we do not now know how to obtain. The method is analyzed and discussed by van Dantzig 1949 and Heyting 1956 8.1.1, in connection with an application of it in Brouwer’s Dutch paper 1948a, which is claimed t o show the unprovability of a ”- 0 + a # 0 for real numbers a. To give this example, essentially as in Heyting 1956, say P is a proposition-such that no method is known which will lead to a proof of p or of p.Then the “creating subject” can choose successively the integral part and digits in the dual expansion ag.ap2a3.. . of a real

9

17

175

INTRODUCTION AND PRELIMINARIES

number a according to the following instruction: always choose 0, unless between the choices of ax and az+l the truth of P or of first becomes evident to you, in which case choose that az+l = 1. Now a * 0 would mean that P will never become known to the (any!) creating subject, so F, and also that will never become known, so ?, contradicting P ; thus a 0. But we are in no position to infer 01 # 0 (as would follow if a * 0 + LY # 0 were proved), since a # 0 would mean that we can find an interval separating a from 0, which we would know to be the case only if we already knew how to find the solution to the problem whether P or p . Thus a # 0 is unproved, although a * 0 has been proved. (But a # 0 is not absurd, i.e. not a # 0, or then we would have a A 0; cf. *R2.7 in Chapter 111. Likewise, not 0 -+a # 0, or we would have a # 0; cf. I M *60d.) An immediate objection to Brouwer’s argument is that a particular real number a has not been defined mathematically; the alleged “definition” makes “a” depend on the unpredeterminate activity of a “creating subject”. van Dantzig 1949 attempts to deal with this objection by giving an “objectivistic” or “formal” version, as contrasted to Brouwer’s “subjectivistic” version. In this, van Dantzig introduces as parameter a sequence w = (wo, 01, 0 2 , . . .) of finite sets of “deductions”, where the deductions in w z are performed between the choices of ax and ax+l, with ax+l = 1 if a deduction of P or of P occurs in ox but no such deduction occurs in wy for any y < x (= 0 otherwise; and a0 = 0). Thereby a becomes a m (notation ours). van Dantzig, after alluding p. 955 to “the existence of propositions which are undecidable within a definite formal system” (Godel), says that the “deductions” should be “according to the rules of a given semantical metasystem”. To the present writer, there is a fundamental vagueness of concept here, despite some detailed assumptions which van Dantzig lists. But if we attempt to proceed notwithstanding this vagueness, van Dantzig’s reformulation of Brouwer’s argument seems to go as follows. One can quantify over all such sequences o of “deductions”. If in every such sequence there is no “deduction” of P (of F), then therefore, (i) ( 0 ) a m A O . Also, if a m # 0, then there is a (then “deduction” of P or of in o ;thus, so long as we do not know how to solve the problem whether P or F, (ii) (Eo)am#O is unprovable.

a

A

3

4;

176

ON ORDER I N THE CONTINUUM

CH. IV

But, as we see it, the unprovability of a 9 0 + a # 0 does not follow now. We can quantify this to obtain (Z) (Eo)amAO + (Ew)am#O. But to utilize (iii) with (i) to contradict (ii), we would first need to transform (i) into ( E o ) a m ~ Owhich , we are only able to do classically, Heyting affirms that it is not very important whether we express the result in Brouwer’s words or in those of van Dantzig, or whether we call it a mathematical result or not, provided we understand what it means. Then he says that it shows it would be foolish to seek a proof of the equivalence of the relations a j3 and a # j3 between real numbers a and j3. (The implication 01 # j3 --f a A j3 is known ; cf. “R2.5.) In informal intuitionism, care is necessary to identify the grounds for statements of the form “ A does not hold (or is unprovable)”, as contrasted to “ A is absurd”, which is simply 2.In a case like “the law of the excluded middle A v A does not hold”, a ground is simply that A v 2, although not absurd itself (cf. *51a), has absurd consequences, e.g. ones of the form (a)(A(a)v A(a)) (cf. *27.17). If we admit quantification of proposition variables, its closure ( A ) ( A v A) is absurd. In the case of a * + a # p, no absurd consequence by intuitionistic deductions of the older kind has been exhibited. In formal intuitionism, “unprovable” takes on its metamathematical meaning, so there is no problem here unless one is proposing to vary the formal system under consideration. The semantical methods we have available will enable us to show that VpBERVa,,R(ycc*p 3 cc#p) is unprovable, but (classically) has no absurd consequences, in the formal system of intuitionistic analysis of Chapters I and 111. (A related observation is made by Kreisel in 1962c, but his =, # are not the 5 ,# for real number generators; cf. Remark 18.6.)

17.2. We list in Table 1 opposite the five pairs of properties (A$, Bi) considered by Brouwer 1951, in the symbolism of Vesley’s Chapter I11 (extending I M and Chapter I) used for the moment informally. I t is to be understood that cc, p range over R (cf. “R0.1). The first four properties are from Brouwer’s table p. 358, and the fifth (As, B5) is from his Footnote 3. Brouwer’s continuum includes all the real numbers, while Vesley in Chapter I11 includes only the non-negative real numbers. Thereby Vesley simplifies the formalization, without evading any of the

9

17

177

INTRODUCTION AND PRELIMINARIES

fundamental problems; the inclusion of the negative reals would have added only a little uninteresting detail. Consequently in Vesley's formalization 0 (i.e. 0-2-0) is a boundary point with a somewhat special status. For this reason, in rendering one of Brouwer's claims involving BI, it would not do to use 0 as Brouwer did. We could simply substitute 1 for 0 throughout the properties; but we shall generalize to any PER (with p>O when required). In place of l a < p (as direct translation of Brouwer's into our symbols), we usually write aQ=op, which is equivalent to it by "R7.8. For the first four pairs, Brouwer claims that -,-,Bi is equivalent to At, but that T - I B is ~ not equivalent to Bt. The claimed equivalence we write

--

at : iiBt At, which (with f3,u~R3 prefixed) we shall show in 17.3 to be provable. By *51b, *49a and *25, l l ( l l B t Bt). So the claimed non-equivalence is not to be rendered by -I(-,-,B~ Bt) or V a a E R l ( l l B t Bt),

-

178

ON ORDER IN THE CONTINUUM

-

CH. IV

which leaves us lVaaeR(llB t Bi) as the plausible rendering. Using *49a with *45, and at, the latter simplifies to 1VaaeR(Ac3 Bc). Brouwer’s 1948 p. 1248 Footnote 1 says that by “non-equivalence” he means the absurdity of equivalence (not just the unprovability of equivalence). For the second and fourth pairs, we shall find that his claims of non-equivalence (as rendered above) are not tenable in this sense. For the fifth pair, if we understand correctly Brouwer’s language (which speaks of the “converse” of a proposition not explicitly written as an implication), he claims non-equivalence of B5 to A5 (untenable), equivalence of 1B5 to 1A5, and non-equivalence of 11B5 to A5. Using

B5 3 A5

b:

(proved below), the equivalence simplifies to

1B5 3 1A5.

C:

Using b, c, * 13 and *45, the claimed non-equivalences rendered in the plausible way simplify to 1V/aac~(A5 3 B5) and - 1 v a , , ~ ( l 4 3 5 3 As). (The renderings i(B 5 A5) and l ( l l B 5 As) are refuted thus: By *51b, etc., 11(71A5 -As). Thence by the said equivalence, l l ( l l B 5 As). From this and l l ( l l B 5 B5), by *25 and *24, i i ( B 5 A5).) Thus, to substantiate Brouwer’s claims concerning the five pairs of properties in 1951, we should prove six positive formulas a1, a2, as, a4, b, c and six “non-implications” (more precisely, negated universalized implications). The six positive formulas we shall prove in short order in 17.3. Of the six non-implications, we shall prove only the three shown in the right column of the table with their negation signs retained (“1V”). The other three we have written omitting the 1 before V and in closed form; the resulting formulas we shall show to be formally undecidable. The work will be arranged as follows. Four other formulas have been inserted into the right column of Table 1, including one expressing the implication 01 5 -+01 # (cf. 17.1). The deducibilities d-j shown by arrows in the table we shall quickly establish in 17.3. The first and last (underlined) formulas we shall prove in 18.1. The second underlined formula we shall show (classically) to be realizable, and the third and fourth underlined formulas to be un,realizable, in 18.2. N

--

-

§ 18

OF CERTAIN CLASSICAL ORDER PROPERTIES

179

17.3. al: P,uER 2 ( l l B 1 - A1). Assume p , u ~ RI. . By *R1.4, u A a , i.e. A1, whence by *11 l l B 1 3 A1. 11. Assume ao>p & a<+ Thence by *R6.6 a < a , contradicting hR6.8. So l ( a o > P & u P V a++), i.e. -1lB1. a2. Assume P,aeR. Now l + < p V a>p) l ( l a < P & l u > P ) [*63] N -.~u*P [*R7.10]. a3. Assume P,uER.Now - 1 1 ( a s PV u o > P ) ~ ( l u ' p & m>p) [*63] i a < P [*R7.1] ~ + o p [*R7.8]. a4. *R7.9. b. "R7.7 (without assuming p , a ~ R ) . c. Assume P,uER. Using *R7.8, lu*fi & uo>P 2 1uA.P & l u > P . Now use "63. e. I. *R6.2. 11. "R6.3, "R2.3. i. We need that P,aER, 11B5 3 A5 t l u < P Z, As. But by *R7.8 and a3, p,aER k ~ a < p3 l l B 5 .

-

-

-

-

Q 18. Refutation or proof of independence, of certain classical order properties. 18.1. After we have *R15.2 and *R15.3, three similar results will follow by the arrows d, i, j in Table 1 (and three others *R9.23*R9.25 were established in Chapter 111). The following proofs of *R15.1-*R15.3 are by Vesley. They employ some of the later results in Chapter 111, and are shorter than the author's original proofs (which used nothing after *R9.2). *R15.1. *R15.2.

t- PER' & P>O

3 iVauER'(ao>p V a++). k PER & p>O 3 lVa,,,(uo>p V a++).

PROOFS. *R15.1. Assume (a) PER'&P>O and VaaER.(aO>PVa++3). Applying *27.6 with "R0.8, and omitting 37 prior to 3-elim., assume (b) VauER.3y{Vx[7(E(x))>02 y=x] & {(ao>p & +(y))= 1) V (u<+ & 7(E(y))=2)}}. Using this with (a) and omitting 3y prior to 3-elim.: (c) @>p & ~ ( p ( y ) ) = 1) V (p++ & 7(@(y))=2).We begin with the slightly more complicated CASE 2: ~ ( p ( y ) ) = 2 .Using "R9.20 with (a), assume (d) ~ E R ' (e) , E(y)=@(y),(f) a'(P(y)~1)2-y. By *R9.16, (a) and (f): (g) P + o a . Using (d), (e) and case hyp. in (b): (h) a<+ Now (i) p * a [(g), (h), *R6.5] (p(y)'1)2-Y [(f)]. Thence by (a) and "R7.1: (j) P(y)>l. Let (k) y=(P(y)'1)2--Y [Lemma 5.3 (a)]. By *R9.2: (t) ~ E R ' . So assume from (b): VX[T(~(X))>O2 z=x] & {(yo>p & T(~(z ) ) =1) V (y<=.p & 7(7(2))=2)}. SUBCASE 2.2: 7(7(z))=2. Let (m) w=max(z, y). Using (€) and 3-elim. from *R9.20 with y, w

180

ON ORDER I N THE CONTINUUM

CH. IV

for p, y, assume (n) ~ E R ' (, 0 ) 8(w)=p(w), (p) 6+(y(w) 11)2-W. Using (m), (0) and "23.4 a(z)=p(z), whence by subcase hyp. ~(8(2))=2. Using this and (n) in (b), 6<$. But 8 5 ((p(y)'1)2-Y(w)i1)2-W [(p), (k)] = ((p(y)11)2wzy21)2-w [xR9.1, (m), etc.] < o (p(y)1 1)2wzy2-w [*R9.8, (I), "6.161 = (p(y)2 1)2w1y2-(y+(w'y)) [(m), "6.71 = (p(y)11)2-Y [*R9.3] + p [(i)]. SUBCASE2.1 : ~ ( p ( z ) ) =1. Similarly, using "R9.21 instead. CASE 1: ~ ( F ( y ) ) = lSimilarly . to Case 2, using *R9.21 first (and not deducing (1)). "R15.2. From "R15.1 (using "R1.11, "R7.3, "R1.5, "R0.7, "R6.13, "R6.14) as *R9.23 from "R9.22. "R15.3.

t PER 13 i V a , , R ( i a < p

13 a+p V

~c>p).

PROOF.Assume PER and (a) Va,,R(lai:p 13 V a>p). Now (b) p , p + l ~ R[*R3.2, "R9.2, "R0.71. Also p = p+O [XR9.1, X13.11 < o p + 1 [*R9.8, "R6.16, "R3.41, so (c) p o > p + l [*R6.7] and (d) ++p+l [*R6.4]. Using "R1.5 and "R1.6, (e) VaVy[a~[P,p+l] & a+y 13 ( a A p --*@)I. Toward (f), assume a ~ [ pp+1]. , By "R8.1, ~ E RSo . by "R8.6, (b) and (c) C C K O whence ~, by "R7.8 l a < p , whence by (a) a + p V u > p , whence by "R7.1 u + p V l a + p . By 13- and Vintrod., (f) V a ( a ~ [ pp+ , 11 13 a'p V -.la+@). By "R10.4 (for a'p as the C(a)) with (b), (c), (e) and (f), V a ( a ~ [ pp+1] , 13 N A P ) V Va(a~[p,p+ 11 13 CASE 1 : V a ( a ~ [ pp+ , 11 3 a s p ) . Then using "R8.5 with (b), p+I+p, contradicting (d). CASE 2: V a ( a ~ [ pp+1] , ZI l a + (3). Similarly, ++p, contradicting "R1.4. 18.2. When we have established Theorems 18.1, 18.2 and 18.4, the formal undecidability of the five formulas without -IV in the table of 17.2 will follow (the unprovability intuitionistically, the irrefutability classically) by Theorem 1 1.3 (a), Theorem 9.3 (a) and Corollary 9.4, and the arrows e, f , g, hinTable 1 (with proofsin 17.3). "R15.4. "R15.5. "R15.6.

t a,pER' 13 (ao>p t a@R' 2 (a#p t a,@&' II( a e p

--

-

3xa(x)~ P ( x ) > 2 ) . 3xla(x)-p(x)1>2). Vxla(x)-p(x)/<2).

PROOFS. "R15.4. Assume a , p ~ R ' .I. Prior to 3-elims. from uo>p, assume Vp2k(a(x+p) 'p(x+p)) 22x+p. Thence 2k(a(x+k+2) 2 p(x+k+2)) 2 2x+k+2 > - 2k+2, whence a(x+k+2) 1-P(x+k+2) 2 22 > 2, whence 3xa(x)1 p ( x )>2. 11. Prior to 3-elim., assume a(.) ~ p ( x>2. ) Thence a(x)~ p ( x23, ) so (a) 3.2P I 2p(a(x)'-p(x)) =

9 18

O F CERTAIN CLASSICAL ORDER PROPERTIES

181

2pa(x)-2pP(x). Now a(x+p)+2p > 2Pa(x) [*R0.6, "1 1.151 2 3-2P+2Pp(x)[(a)] = 2*2P+2PP(x)+2P > 2-2P+P(x+p) [*R0.6, "1 1.151, whence a(x+p) ~ p ( x + p22P. ) So 2X(a(x+p) ~ P ( x + p )22x+P, ) whence by V-, 3- and 3-introd., a o > p . "R15.5. Assume CC,PER'. I. Assume u#P. By *R6.2 (with *R0.7), fb>a V a o > p . CASE 1 : Po>. By "R15.4, 3xP(x)'a(x)>2, whence 3xla(x)-P(x)1>2. 11. Prior to 3-elim., assume Ia(x)-P(x)1>2. By *11.14a, a(x)>P(x)+2 V P(x)>a(x)+2. CASE 1 : a(x)>P(x)+2. Then a(.) ~ P ( x ) > 2 ,whence by 3-introd. and "R15.4 m>P, whence by *R6.3 (with *R2.3) a#P. *R15.6. Assume U,PER'.Then a+@ l a # P [*R2.8] -dxla(x)P(x)1>2 [*R15.5] Vx+(x)-P(x)1>2 "861 Vxla(x)-P(x)112 [* 139-* 1411. THEOREM 18.1C. Classically,

-

-

(l)

VPBeRVaaeR(la*P

-

.#P)

is realizable. PROOF.Using "R1.11 with *R1.6, *R1.5, *R2.4 and *R2.3, (1) is deducible from (2) VPpER,VaaeRt(la~P 3 a#@). Using *R15.6 and *R15.5, (2) is deducible from VPVa[-rVxla(x)-P(X)I 1 2 3 3xla(x)-P(x)I>21. Using #15 and #D in 5.5, we can take the scopes of Vx and 3x in (3) to be prime. Classically, (3) is true. Hence by Lemma 8.4b (ii), (3) is realizable. By Theorem 9.3 (a), so is (1). THEOREM 18.2. The formula (3)

(4) VPBERVaaeR(a>P 2 ao>P) is ~n~realizable. PROOF.From (4) by "R0.7 and "R7.1, we can deduce P,ueR' & a<=.P & la*@ ZI w>P, (5) and thence by *R6.15, *R15.6 and "R15.4, (6)

P,a€R' 8~a 2 P & lVyla(y)-P(y)112 3 ~ Y ~ ( Y ) ' P ( Y ) > ~ -

Thence we shall further deduce

(7)

VX[lVYlT(X, y)

3YT(X, Y)1

for the T(x, y) of Theorem 11.7 (b).

182

ON ORDER IN THE CONTINUUM

CH. IV

Accordingly, we assume (a)

lVylT(x, y),

and set out to deduce 3yT(x, y) from (a) and (6) with x held constant. By Lemma 8.8 Tl(z, x, y) V -rTl(z, x, y), whence by substitution T(x, Y) v l T ( % Y). (b) So, for any natural number b, we can assume, preparatory to 3-elims. from Lemmas 5.3 (b) and 5.5 (b), (c)

(4

P(0) =b, VYP(Y‘)=2P(Y) a(0)=b 1, 24y)

+

f

1 if l T ( x , y), if T(x, y).

Using respectively *R0.5a-b, ind. on y, and (f): (e)

P,a€R’.

(f)

dY)2P(Y)+1.

(g)

a2P.

Next we establish (h)

4 Y ) >P(Y)

+ 1 3 4Y’) >P(Y’) +2. +

Assume a(y) >P(y) 1. Then a(y’) 2 2a(y)5 1 2 2(p(y)f 2 ) 2P(Y)+3 = P(y’)+3 > P(y’)+2. - Next, (i)

2

1=

VYla(Y)--P(Y)l(2 3 VY.(Y))=P(Y)+l.

Assume (A) Vyla(y)-P(y)1<2. Now a(y)=@(y)+l, or else by (f) .(Y)>P(Y)+ whereupon by (h) .(Y’)>P(Y’)+2> so I.(y’)-P(y’)l>2, contradicting (A). By V-introd., Vya(y)=P(y) + 1. - Next we establish, by ind. on y, 1 9

(j)

MY)=NY) + 1 M Y ) >P(y) + 1

--

VzZ<,-rT(x,41 & 3ZZ
Using *140 [*58b, “861, the left [right] members of the two equivalences are mutually exclusive. So if we deduce both members of one equivalence, that one will follow by * 1 1, * 16, the other by * IOa, * 16. IND. STEP. By (f), we have two cases. CASE 1 : a(y)=P(y)+ 1. By hyp. 1.1: ind., vz,<,-~T(x, z). By (b), we have two subcases. SUBCASE T(x, y). Then 3z,<,,T(x, z). Also a(y’) = 2a(y) [(d)] = 2(P(y)+l) [Case 1 hyp.] = 2P(y)+2 = P(y‘)+2 > p(y’)+l. CASE 2: a(y)>

3 18

+

O F CERTAIN CLASSICAL ORDER PROPERTIES

183

p(y) 1. By hyp. ind., 3z,<,T(x, z), whence 3z,<,nT(x, z). Using (h), a(y') >B(y') 1. - Using (j), Vya(y)=P(y) 1 2 VylT(x, y). Combining this by *2 with (i), and using contraposition *12 and (a),

(k)

+

+

-rVYla(Y)-P(Y)l12.

If 3ya(y)AP(y)>2, then yya(y)>P(y)+l, so by (j) 3yT(x,y). So

(1'

3Ya(Y) 'p(Y)>2

(4

3YTk Y).

yYT(xJ Y). Using (e), (g) and (k) in (6), and the result in ( I ) ,

Using 3-elim. from Lemmas 5.3 @) and 5.5 (b) to discharge the assumptions (c) and (d), 2-introd. to discharge (a), and V-introd., we have the deducibility of (7) from (6), and thence from (4). By Theorem 11.7 (b) (7) is un,realizable. By Theorem 11.3 (a), so is (4). REMARK 18.3. By (c), p = b [= b2-0, XR9.11, so the proof shows that VaaGR(a>b2 aO>b) is un,realizable. Modifying the proof inessentially, 3xVp2p(x+p) p 2 , ao>p) is un,realizable. THEOREM 18.4. The formzlla

(8)

a>p)

v p ~ ~ R v a a ~ R2( aKp ~ a ~ ~

is zcn,realizable. PROOF.From (8) by *R0.7 and "R7.1, we can deduce (9)

P,aER' & 1aA.P 3 a o + p V a<+,

and thence by *R15.6 and *R15.4 (10)

PmR' & lvYla(Y)-p(Y)l12 2 13Y4Y)>MY) +2

v -3YP(Y) > d Y )

+2.

Thence we shall further deduce (1 1)

v x [ l ~ y l w o ( xy) , i 3 VYlWl(X, y)) 3 l v y l w o ( x , y) v l v y l w l ( x , y)]

for the Wo(x, y) and Wl(x, y) of Theorem 11.7 (d). Accordingly, we assume (a)

l(Vylwo(x, y) & VylWl(x, y)).

184

CH. IV

ON ORDER I N THE CONTINUUM

By Lemma 8.8 and substitution, and middle Remark 4.1, (bo)

WO(X,y) V iWo(x, y),

(bl)

Wl(X, y) v lWl(X, y).

The proof of I M p. 308 (51) is readily formalized to give (c)

y) & 3yWl(x, y)),

+yWo(x,

whence

(4

l(WO(X,Y) 8z WdX, Y)).

So, for any natural number b > 0, we can assume, preparatory t o 3-elims. from Lemmas 5.3 (b) and 5.5 (b),

(f )

P(O)=b, P(Y’)=2P(Y), a(0)=b, 2a(y)1.1 if WO(X, y) & lWl(X, y), Vya(y’)= 2a(y) if 1Wo(x, y) & l W i ( x , y), 2a(y)+ 1 if lWo(x, y) C% Wi(x, y).

1

Using respectively “R0.5a-b and ind. on y: (g)

P,aeR’.

(h)

P(y) = 2Yb

> 0.

Next we establish, by ind. on y,

0)

MY>P(Y)

---

~ Z ~ < ~ W41 O& (X, vzz
Using “140, *141 [(c) etc.] the left [right] members are mutually exclusive; etc. (cf. (j) for Theorem 18.2). IND. STEP. CASE 1 : a(y)O from (h)] = P(y‘). SUBCASE 2.2: lWo(x, y) & lWl(x, y). By hyp. ind., Vzz
P(Y) >.(Y) = NY”)2 4 ~ ” ) + 4 , ~ ( Y ) > P ( Y )2 4 ~ >P(Y”)+~”)

For, assume P(y)>a(y). Then by (i) and (c), -dyWl(x, y). Hence .(y’)+2 I 2a(y)+2 = 2(a(y)+l) I 2P(y) = P(y’), and .(y”)+4 <

9

18

OF CERTAIN CLASSICAL ORDER PROPERTIES

2a(y’)+4

(k)

= 2(dY’)+2)

185

I W Y ’ ) = MY”). -

VYl~(Y)--P(Y)lI22 VYa(Y)=P(y).

For, assuming a(y)4. y) 8~-~Wl(x,y)), By (k) with (9, Vyla(y)--P(y)li2 2 vy(-.~Wo(x, whence by *87, contraposition and [a), 1 V YM Y ) -B (Y) I 1 2 . (1) Next we establish

(mo) 13ya(y)>B(y)+2 2 l~YlWO(X> Y)> (m1) 1 W q y ) > a ( y ) + 2 2 lVYlWl(X>Y). Assume dya(y)>P(y)+2, whence Vya(y)IP(y)+2, and by (j1): (A) Vya(y)Y) i?l lWl(x, y)). This with “87 contradicts (a). - Using (g) and (1) in (lo), and (mo) and (ml) with the result, (n)

l v y l W o ( x , y) V l V y l W l ( x , y).

REMARK 18.5. By the proof, or for m > 0 an easy modification,

VaasR(1a*b2-m I)a ~ b 2 - mV a>b2-m) is un,realizable for any b > 0 and any m. REMARK 18.6. Combining several deductions (namely, e+g in 17.3, the deduction of (7) from (4)in the proof of Theorem 18.2, beginning proof of Corollary 1 1.10 (a)), from (I)

vPBsRvRaeR(la*P

a#P)

(in Theorem 18.1) we can deduce Markov’s principle

Mi: VzVx[iVYiTi(z, X, Y)

2 WTi(Z, X, y)1

(beginning 10.1). - Modifying inessentially the proof of Theorem 18.2, we can instead deduce from (4),and so by e+g from ( l ) , (3’)

~ p ~ ~ l I l ~ y l l ~ ~ 2Y3Yla(Y)--P(Y)I>21. ~ - B ~ Y ~ 1 ~ ~

Conversely, from (3’) we can deduce (3),and thence (by the proof of Theorem 18.1) (1). Thus: Each two of ( I ) , (4), (3’) are interdeducible. Formula (3‘) is similar in form to MI, but involves function variables. A direct extension of Markov’s principle from algorithms for one-place number-theoretic functions to algorithms for type-2 functionals is

186

ON ORDER IN THE CONTINUUM

CH. IV

where Ti(w, z, y) is chosen by (the method of proof of) Lemma 8.5 to numeralwise express Ti(w, z, y) I M p. 292. This formula M i is deducible from (1) essentially as M1 and (3'). Conversely, by informal reasoning (which is presumably formalizable) in the theory of recursive functionals, (3') follows from Mi. - Thus, since the converses of (1) and (4) are provable (*R2.5, *R7.7) : Markov's principle extended to functionals as (3'), and presumably as Mi, i s interdeducible with (i) the equivalence of inequality -.luAP to apartness a # @ , and with (ii) the equivalence of the virtual ordering cc


(3")

VPVa[-.lVy4y)=P(y) 2 3YdY) +P(Y)l

(cf. van Rootselaar 1960, Kreisel 1962c), we can deduce (3")

Va[-IVy1a(y)=o

3)3ya(y)=O],

using V-elims. with P1=LyO, and al=lyg(a(y))+2 or ai=lyg(a(y)) resp. ; and conversely, using ul=lyt(a, P, y) for a standard formula t(a, P, y)=O equivalent to lu(y)-P(y)(>2 or to a(y)#P(y), resp. (cf. 5.5 preceding "14.1). Thus: Each two of (3'), (3"),(3") are interdeducible.

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BIRKHOFF, GARRETT 1940. Lattice theory. American Mathematical Society Colloquium publications, vol. 25, New York, v + 155 pp. 2nd ed., 1948, xiv+283 pp. BOREL,~ I L E 1898. Leqons sur la th6orie des fonctions. Paris (Gauthier-Villars),8+ 136 pp. 4th ed. (LeGons sur la thkorie des fonctions; principes de la thhorie des ensembles en vue des applications B la thhorie des fonctions). Pans (Gauthier-Villars), 1950, xii+ 295 pp.

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logic, vol. 25 (1960), no. 4 (publ. 1962), pp. 389-390. 1962. O n weak completeness of intuitionistic predicate logic. Ibid., vol. 27,

pp. 139-158. 1962a. Foundations of intuitionistic logic. Logic, methodology and philosophy

of science, Stanford Calif. (Stanford Univ. Press), pp. 198-210. 1962b. Review of Beth 1959a. Zentralbl. Math. Grentzgeb., vol. 91, pp. 9-11. 1962c. Review of van Rootselaar 1960. Ibid., vol. 95, pp. 242-243. 1962d. Proof theoretic results on intuitionistic first order arithmetic ( H A )

(abstract). Jour. symbolic logic, vol. 27, pp. 379-380. 1962e. Proof theoretic results on intuitionistic higher order arithmetic (abstract).

Ibid., p. 380. 1962f. Consequences of Brouwer’s bar theorem (abstract). Ibid., pp. 380-38 1.

See Dyson and Kreisel, Kreisel and Putnam. KREISEL.G. and PUTNAM, H. 1957. Eine Unableitbarkeitsbeweismethode fur den intuitionistischen Aussagenkalkiil. Archiv f iir mathematische Logik und Grundlagenforschung, vol. 3, nos. 3-4, pp. 74-78. KURODA, SIGEKATU 1951. Intuitionistische Untersuchungen der formalistischen Logik. Nagoya mathematical journal, vol. 2. pp. 35-47. LEBLANC, HUGUES and BELNAP. NUELD., JR. 1962. Intuitionism reconsidered. Notre Dame journal of formal logic, vol. 3, pp. 79-82. B. DE LOOR, 1925. Die hoofstelling van die algebra van intuisionistiese standpunt (The fundamental theorem of algebra from the intuitionistic standpoint). Thesis Amsterdam, 63 pp. EUKASIEWICZ, JAN 1952. On the intuitionistic theory of deduction. Kon. Ned. Akad. Wet. (Amsterdam), Proc.. ser. A, vol. 55 (or Indag. math., vol. 14). pp. 202-212.

N. LUSIN,NICHOLAS 1927. S u r les ensembles analytiques. Fund. math., vol. 10, pp. 1-95. 1930. Lesons sur les ensembles analytiques et leurs applications. Paris (Gauthier-Villars),xv 328 pp.

+

MAEHARA, SH6JI 1954. Eine Darstellung der intuitionistischen Logik in der klassischen. Nagoya mathematical journal, vol. 7, pp. 45-64. Cf. Mathematical reviews, vol. 16, p. 325.

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MARKOV.A. A. 1951c. Teorija algorifmov (Theory of algorithms). Trudy MatematiEeskogo Instituta imeni V. A. Steklova, vol. 38, pp. 176-179. Eng. tr. by Edwin Hewitt, American Mathematical Society translations, ser. 2, vol. 15, 1960, pp. 1-14. 1954. Teorija algorifmov (Theory of algorithms). Trudy Matem. Inst. im. V. A. Steklova, vol. 42; Izdatel’stvo Akademii Nauk SSSR, MoscowLeningrad, 375 pp. Eng. tr. by Jacques J. Schorr-Kon and PST staff, pub. for the U. S. National Science Foundation and Department of Commerce by the Israel Program for Scientific Translations, Jerusalem, 1961, vi +444 pp. 1954a. 0 nepreryvnosti konstruktivnyh funkcii (On the continuity of constructive functions). Uspehi matematieeskih nauk, vol. 9, no. 3 (61), pp. 226-230. MCCALL,STORRS 1962. A simple decision procedure for one-variable implicationlnegation formulae in intuitionist logic. Notre Dame jour. formal logic, vol. 3, pp. 102-122. MCKINSEY, J. C. C. and TARSKI,ALFRED 1946. On closed elements in closure algebras. Ann. Math., vol. 47, pp. 122-162. MEDVEDEV, Ju. T. 1962. Finitnye zadati (Finite problems). Doklady Akademii Nauk SSSR, vol. 142, pp. 1015-1018. Eng. tr. by Elliott Mendelson, Soviet mathematics (Amer. Math. SOC.),vol. 3, pp. 227-230. MOSTOWSKI, ANDRZEJ 1961. (an organizer) Infmitistic methods. Proceedings of the Symposium on Foundations of Mathematics, Warsaw, 2-9 September 1959. Oxford, London, New York, Paris (Pergamon Press), Warszawa (Padstwowe Wydawnictwo Naukowe), 1961, 362 pp. Lacking the names of the Editors, we use that of one of the Organizing Committee of the Symposium. TARSKI,ALFRED NAGEL,ERNEST;SUPPES,PATRICK; 1962. (editors) Logic, methodology and philosophy of science. Proceedings of the 1960 International Congress [Stanford University, August 24-Sept. 21, Stanford Calif. (Stanford Univ. Press), 1962, ix+661 pp. IWAO NISHIMURA, 1960. On formulas of one variable in intuitionistic propositional calculus. Jour. symbolic logic, vol. 25, pp. 327-331.

OHNISHI. MASAO 1953. On intuitionistic functional calculus. Osaka mathematical journal, vol. 5, pp. 203-209. Cf. de Iongh 1948. PI~TER, R6sz~ 1951. Rekursive Funktionen. Akadkmiai Kiad6 (Akademischer Verlag) Budapest, 206 pp. 2nd ed., Budapest 1957, 278 pp.

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PIL'EAK, B. Ju. 1950. 0 probleme raareSimosti dlja isEi'slenija zadaE (On the decision problem for the calculus of problems). Doklady Akad. Nauk SSSR, vol. 75, pp. 773-776. 1952. Ob istislenii zadaE (On the calculus of problems). UkrainskiI matematiEeskiI iurnal, vol. 4, pp. 174-194. Cf. Kreisel-Putnam 1957 p, 78. PORTE,JEAN 1958. Une propribti du calcul propositionnel intuitionniste. Kon. Ned. Akad. Wet. (Amsterdam), Proc., ser. A, vol. 61 (or Indag. math., vol. 20), pp. 362-365. RASIOWA, HELENA 1951. Algebraic treatment of the functional calculi of Heyting and Lewis. Fund. math., vol. 38, pp. 101-126. 1954. Constructive theories. Bulletin de 1'AcadCmie Polonaise des Sciences, Classe 111, vol. 2, pp. 121-124. 1954a. Algebraic models of axiomatic theories. Fund. math., vol. 41, pp. 291-310.

See Rasiowa and Sikorski. RASIOWA, HELENA and SIKORSKI, ROMAN 1953. Algebraic treatment of the notion of satisfiabilily. Fund. math., vol. 40, pp. 62-95. 1954. O n existential theorems in non-classical functional calculi. Ibid., vol. 41, pp. 21-28. 1955. A n application of lattices to logic. Ibid., vol. 42, pp. 83-100. 1959. Formalisierte intuitionistische elementare Theorien. Constructivity in mathematics, Amsterdam (North-Holland Pub. Co.), pp. 24 1-249. RIDDER, J. 1950-1. Formalistische Betrachtztngen iiber intuitionistische und verwandte logische Systeme, I - V I I . Kon. Ned. Akad. Wet. (Amsterdam), Proc., ser. A, vol. 53 (1950). pp. 327-336, 446-455, 787-799, 1375-1389 (= Indag. math., vol. 12, pp. 75-84, 98-107, 231-243, 445-459) and vol. 54 (1951) (or Indag. math., vol. 13), pp. 94- 105, 169-177, 226-236. RIEGER, LADISLAV 1949. On the lattice theory of Brouwerian propositional logic. Acta Facultatis Rcrum Naturalium Universitatis Carolinae, no. 189, Prague (F. RivnAE), 40 pp. (with Czech summary). ROBINSON, T. THACHER 1963. Interpretations of Kleene's metamathematical predicate I'lA in intuitionistic arithmetic N. Dissertation Princeton (mimeographed), iii+ 75 pp. ROGERS, HARTLEY, JR. 1964. Recursive functions and effective computability. McGraw-Hill, New York. ROOTSELAAR, B. VAN 1952. Un probl2me de M. Dijkman. Kon. Ned. Akad. Wet. (Amsterdam), Proc., ser. A, vol. 55 (or Indag. math., vol. 14), pp. 405-407.

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1960. O n intuitionistic difference relations. Ibid., vol. 63 (or Indag. math.,

vol. 22), pp. 316-322. Corrections to the paper “On intuitionistic difference relations”, ibid., vol. 66 (or Indag. math., vol. 25) 1963, pp. 132-133. Cf. Kreisel 1962c. ROSE,GENEF. 1953. Propositional calculus and realizability. Trans. Amer. Math. SOC.,vol. 75, pp. 1-19. The publ. version of 1952. Cf. JaSkowski 1936, Pil‘Eak 1952, G&l-Rosser-Scott1958, Medvedev 1962. ROSSER,J. BARKLEY 1957. (anonymous editor) Summaries of talks presented at the Summer Institute of Symbolic Logic in 1957 at Cornell University (mimeographed). 3 vols., 432 pp. 2nd ed., Princeton N. J. (Communications Research Division, Institute for Defense Analyses) 1960, xvi+427 pp. See Church and Rosser; GAl, Rosser and Scott. E. SACKS, GERALD 1963. Degrees of unsolvability. Annals of Mathematics studies, no. 55, Princeton Univ. Press, Princeton N. J., xi+ 174 pp.

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H. ARNOLD SCHMIDT, 1958. U n procbdb maniable de dkcision pour la logique propositionnelle intuitionniste. Le raisonnement en mathhmatiques et en sciences expbrimentales, pp. 57-66. Colloques Internationaux du Centre National de la Recherche Scientifique, LXX. Editions du Centre National de la Recherche Scientifique, Paris. SCHROTER, KARL 1956. Ober den Zusammenhang der in den Implikationsaxiomen vollstandigen

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+98 pp.

B. A. (Trakhtenbrot, B. A.) TRAHTENBROT, 1960. Algoritmy i mdinnoe regenie zalaE (Algorithms and machine solution of problems). 2nd ed., ed. by S. V. Jablonskii, Gosudarstv. Izdat. Fiz.-

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199

Mat. Lit., Moscow, 119 pp. Algorithms and automatic computing machines, Boston (D. C. Heath) 1963, viiif101 pp. is an Eng. tr. and aduptation (in which notes and some additional references were supplied, and the text was cut, rearranged and paraphrased in places) by Jerome Kristian, James D. McCawley and Samuel A. Schmitt (ed. by Alfred L. Putnam and Izaak Wirzup, under a grant from the U. S. National Science Foundation) of the 2nd ed. UMEZAWA, TOSHIO 1955. Obey die Zwischensysteme der Aussagenlogik. Nagoya math. journal, V O ~ . 9, pp. 181-189. 1959. On intermediate propositional logics. Jour. symbolic logic, vol. 24, pp. 20-36. 1959a. On logics intermediate between intuitionistic and classical predicate logic. Ibid., pp. 141-153. V. A. USPENSKI?, 1960. Lekcii o vyEislimyh funkcijah (Lectures on computable functions). MatematiEeskaya logika i osnoviniya matematiki, Gosudarstv. Izdat. Fiz.Mat. Lit., Moscow, 492 pp.

DALEN.DIRK See Dalen, Dirk van.

VAN

DANTZIG, DAVID See Dantzig, David van.

VAN

ROOTSELAAR, B. See Rootselaar, B. van.

VAN

VESLEY,RICHARD E. 1963. On strengthening intuitionistic logic. Notre Dame jour. formal logic, vol. 4, p. 80. ZEGALKIN, I. I. 1936. 0 probleme razrefimosti v Brouwer’ovskoi logike predlobnii (On the decision problem in Brouwer’s propositional logic). Trudy 2-go Vsesojuznogo Matematireskogo S’ezda, Leningrad 24-30 ijunja 1934, vol. 2, Moscow-Leningrad 1936, p. 437. An abstract, stating the existence of a decision procedure for Brouwer’s propositional calculus. (Gentzen 1934-5 was received 21 July 1933.)

SYMBOLS AND NOTATIONS [The notations “used formally” include ones used also informally (possibly changing] Roman to italic type). A bold-faced page reference indicates that the notation is also indexed in IM p. 538. Notations of IM taken over tacitly are indexed only in 1M p. 538. USED N

: v

11,30 8,30 8,30 8,30

8, 30 ;,I 8,30 3! 89 a, b, . . . . Y,Z a,9 a,b ,..., x 8,9 2, s, . . . 9 a, 9 $j:Eix), ... 11

10 r(x), r(t) 11 9, 10 fi, kt, 12 pj, sj, f=j, 5 11 0 8, 147 1,2,.. 11,120,147 157 c, E (4 (t)

L -

FORMALLY

1, 8, 15, 27

,

8,114

11

+ 14, 120, 142 14

163 #S 140, 141 # 23,27 <, > < > 23, 143 -2 143 143 G,~2 146 -c> 3 143 KS o> ad 171 aER 135 aER1 135 aER 135 aEd 57 147 a4?h, 621 161 cc@&, 82) a, PE.., a$.. 135

a*b 37, 122 ab 14,24 aexpb 14,24 a-2-m, a2-m 147 a! 23,24 24, 142 i-b, 26 142 I-PI 28 0 26 WI lh 36 max, min 25,31 24 pd 34 Pi Pr 31 l-m 26

138

#

23,37 25 sg, sg 56 SPr 57 SPd T1, T, . . . 119,130 seq

wo, W l

130

8, 10 30 28,29 23,38 23,38

ilX

PY fl, x a iti, a G4ip (ah,1 (cC)i, . . . 40, E!i 40,92 <x0, . . . . G> 40,92 [a0, . . . . ~140,45 158 ‘Plnl

‘=

JPI ifQ1 41 [PZ if QZ

USED ONLY INFORMALLY

91 93 91 AfTi v[@, aI Au y[O, a] 92 92 A VII@1 flu l...azcp[...] 92 A% v(b, “a) 122 $l”cxq~[b,%] 123 :~xd3,4 124 124 2AplPl

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92 122 124 124 b(ec)[Cy,dd] 1 2 4

1 a, b, . . . %, “/Y, . . . (a0, al) a-l Q =a CEO 0 q-,f--,C-

200

92 121 121 120 121 121 122 112 14

64

EE

-[q QE

c.r.n.g. conv E I

IM r.n.g. Tl>l 1

101 102 126 135 ia 52 52 1 134 93

INDEX

Names mentioned only in Footnotes 2 and 3 are not indexed. abbreviations (principles governing) 9, 10, 11, 12. absurdity 176 ; cf. negation. Ackermann, W. 3. addition (natural numbers) 14, 23, (real numbers) 142. algorithm 2, 44, 70, 71, 79, 90, 119, 185. analysis, set theory: classical 1, 7, 8, cf. classical; intuitionistic (Brouwer’s) 4, 5, 7, 43, 76, 82, 83, 118, 133. analytic predicate 118. apartness 133, 140, 141, 143, 163, 168. 174, 175, 176, 177, 178, 180, 181, 185,

186.

Aristotle 1. arithmetic (number theory) 4, 7, 14, 22 (5.5), 86 (*27.21), 99, 118, 119, 120,

without the - theorem 79 (Kleene 1957), 113, 116, 117, 120.

barred sequence number 46, 52,63, 69. basic (formal) system 8, 13, 52, 54, 110. Belinfante, M. J. 2. Belnap, N. D., Jr. 6. Bernays, P. 3, 8, 22, 48. Beth, E. W. 2, 4, 6, 81, 82. 133. binary fan 49, 60. Birkhoff, G. 6. Bore& 8. 1. bounded quantifiers 13, 15, 30. bound variables 11, 12, 17. Brouwer, L. E. J. 1, 2, 4, 5, 7, 43, 44, 45, 46, 47, 64, 65, 66, 76, 82, 86, 140, 143, 157, 158, 172, 174,

93, 94, 97, 98,

131, 132; fundamental theorem of - 23, 32, 35. arithmetical: function 47, 114, 116; predicate 47, 114. atomic inference 64. axiom: of choice 14 (x2.1), 17, 41, 72, 88; particular -s 14, 19, 20, 24, 25,

48, 50, 51, 52, 56, 57, 59, 67, 69, 70, 71, 72, 73, 75, 87, 90, 120, 133, 134, 136, 146, 147, 151, 155, 156, 159, 161, 163, 169, 170, 175, 176, 177, 178; -‘s

principle (and -‘s principle for numbers) 69, 70, 71, 72, 73, 74, 75,

76, 77, 78, 79, 80, 81, 84, 85, 88, 89, 90, 91, 121, 133, 140, 150, 153 (*27.8), 163; --)s principle for

26, 28, 31, 34, 35, 36, 37, 38, 142,

147; schemata 13, 14, 51, 54.55, 63,

decisions 74, 80; ---Is principle for functions 72, 73, 74, 80, 88, 89, 90, 91, 121; cf. analysis, bar theorem, choice, continuum, creating subject, fan, intuitionism, law, species, spread, uniform continuity theorem.

64, 67, 70, 73, 79, 80, 88.

bar : recursion 128 ; theorem (Brouwer) 51, 52, 54, 55, 56, 57, 59, 63, 64, 65, 69, 77, 78, 79, 87; intuitionism

201

202

INDEX

canonical real number generator 135, 136. Cantor, G. -'s diagonal method 70. cardinal number 45, 70. cases, definition by 31, 41, 42. Cauchy, A. L. - convergence, sequence 135. chains (equalities, inequalities) 143. choice: axiom of - 14 (x2.1), 17, 41, 72, 88; law 44, 56, 57, 62, 76; sequence (Brouwer) 7, 43, 44, 45, 46, 47, 48, 49, 55, 56, 57, 59, 65, 66, 69, 70, 80, 174. Church, A. 2, 3, 4, 12, 18, 19, 83; -'s &operator 8, 10, 11, 12, 16, 18, 19; --Is thesis 2, 47 (lines 14, 32), 71, 94. classical: (formal) system 8, 13, 52; -results (informal) 47, 71, 98, 1 14, 115, 116, 117, 118, 119, 129, 130, 131, 132, 176, 178, 180, 181; -VS. intuitionistic (logic etc.) 1, 2, 5, 7, 8, 13, 14, 46, 47, 48, 52, 59, 70, 77, 81, 82, 83, 86, 118. closed: formula etc. 12; interval 147. coincidence (r.n.g.) 134, 138. compactness 157, 159. completeness 1 19; cf. in-. computable functions 2. congruence 17, 86, 87. conjunction cf. propositional connectives. consistency 90, 1 10, 118, 129. constructiveness 1, 2, 4, 5, 99; degrees of non---, relativized - 80,. 95, 97, 98. continuum (intuitionistic, Brouwer's) 120, 133, 134, 136, 143, 155, 156, 158, 159, 170, 174, 176. contrary-to-fact conditional 13, 97. convergent sequence 134, 135. conversion (lambda) 18. correlation: law 44, 45, 56, 59, 66, 76; functions t o functions 72, 73, 91; functions t o numbers 14, 17, 41, 88; numbers t o functions 69, 70, 71, 72, 75, 79; numbers t o fan elements 75, 76; numbers t o numbers 17; cf.

Brouwer's principle. countable functional 7 1, 9 1, 121. course-of-values: functions 38, 45, 93; recursion 39, 42. creating subject (Brouwer) 174, 175. Curry, H. B. 6, 19, 81. Dalen, G. van 2. Dantzig, D. van 2, 174, 175, 176. Davis, M. 3. decision procedure 82, 83; cf. Church's thesis. Dedekind, R. - cut 134. degree (Kleene-Post) 118. Dekker, J. C. E. 3. De Morgan, A. -'s law 131. density: everywhere 157, 161 ; in itself 157, 158. difference cf. inequality. subtraction; sharp - 133, 163, 168, 169. Dijkman, J. G. 2. discreteness 156. disjunction cf. propositional connectives. divisibility 28, 32. division 26. double negation, law of 13, 84, 118, 140, 141, 175, 177, 178, 192 (line 6). dual fractions 134; finite - 147, 148, 149. Dyson, V. H. 6, 81, 187. element 44. empty: choice sequence 56, 66; spread 56, 57. equality: (functions) 15, 16; (functions of order a) 122, 123, 124, 125; (natural numbers) 1, 8, 16, 20, 27; (real numbers) 134, 138, 180; axioms cf. replacement property. 177, 178. equivalence 1 1, 16; nonEuclid --Is first theorem 23, 32. everywhere density 157, 161. excluded middle, law of 7, 15, 43, 47, 54, 63, 76, 77, 83, 84, 104, 118, 131, 132, 140, 150, 176, 192 (line 6). existence cf. quantifiers.

INDEX

explicit : barredness, securability 50, 51, 52, 53, 69; definition 19, 20,

39, 41. exponentiation 14, 23, 24, 147, 148,

149. extensions of formal systems 5, 8, 19,

203

Hardy, G. H. 32. Harrop, R. 3, 6,7, 81. Henkin, L. 6. Herbrand, J. 2. Herrnes, H. 3. Heyting, A. 1, 2, 3, 4, 5, 6. 7, 44, 46,

82, 98, 99, 118. 119, 131, 176.

56, 65, 79, 82, 133, 134, 135, 138, 140, 142, 143, 146, 147, 151, 155, factorial function 23,24. 174,176. fan 59, 60, 62, 77, 151; - theorem Hilbert, D. 3, 8,48. (Brouwer) 47,59,60,62,70, 75, 76, hollow interval nest 159. 95, 112, 113, 115, 116, 117, 120 hyperarithmetical function 47. (Kleene 1957), 153 (*27.8). Feys, R. 19 immediately secured sequence number finitary spread cf. fan. 46,50. finite: sequence 40, 45,92;set cf. fan. implication 97, 120; cf. propositional formal: symbols 8,9, 10, 1 1 ;system 5, connectives. 8. inclusion (intervals) 157. formalization 5,8 (3.l),82,90,99,110. incompleteness 5, 83 (end 7.9); cf. formation rules 8. Godel’s - theorems. formula 10, 12. independence results (bar theorem) 51, free: connectedness 134, 157, 170, 87. 1 10, 1 16, 1 17, (Brouwer’s prin171, 172, 173;substitution 1 1 ; variciple) 70, 81, 1 10, (law of excluded ables 11. middle) 1 18, (Markov’s principle Freudenthal, H. 2. etc.) 120, 131, 132, 186, (order function: -s (formal treatment) 14, properties) 140, 176, 178, 179, 180, 22,23,cf. correlation; symbols 8,9, 185, 186;cf. Godel’s incompleteness 10, 19, 23, 24, 38, 39, 41, 42; theorems. variables 7, 8, 9, 10, 11. 38, 39, 41, indivisibility 155. 42, 121, 122. induction 14,51,52,59,60,62,65,67, functional 71, 72, 79, 121, 185; 69, 123;cf. bar theorem. recursion 106; - variables 72, 79. inductive : barredness, securability 50, functor 10, 12,95. 51, 52, 53, 69; definition 50, 52, fundamental theorem of arithmetic 23, 123. 32,35. inequality (natural numbers) 1 1, (real Gbl, I. L. 6, 197. numbers) 140, 141, 163, 169, 174, generality, cf. quantifiers. 175, 176, 177, 178, 181, 183, 185, general recursive function 2,3.4,9, 10, 186; cf. apartness, order relation. 12, 47, 71, 91;cf. relative. infinitely proceeding sequence 46; cf. Gentzen, G. 6, 7, 81, 82, 83, 199. choice sequence. Glivenko, V. 6. infinity 44,45. Godel, K. 2,3, 5, 6, 7,23, 81,82, 119, interval: 147, 161, 171. 120; -’s incompleteness theorems intuitionism 1, 2,4,5,6,7,44,45,47, 5. 82, 175; - numbers 23. 93, 94, 70, 72, 80, 133, 176; cf. Brouwer, 95, 122. intuitionistic. Griss, G. F. C. 2. intuitionistic: formal system 8. 52, 73; mathematics cf. intuitionism; cf.

204

INDEX

analysis, arithmetic, continuum, predicate calculus, propositional calculus. Iongh, J. J. de 6, 195. JaSkowski, S. 3, 6, 81, 197. jump operation 1 14, 1 16. Kabakov, F. A. 6. Klaua, D. 3. Kleene, S. C. 1, 2, 4, 6, 7, 8, 9, 10, 19, 23, 40, 43, 44, 47, 52, 53, 71, 76, 79. 81, 82, 83, 88, 91, 92, 93, 94, 95. 96. 97, 99, 100, 101, 106, 110, 111, 112, 114, 115, 116, 118, 119, 120, 121, 132, 133, 140, 150, 172, 187. Kolmogorov, A. N. 3, 6. Konig, D. -'s lemma 59, 115. Kreisel, G.3,4, 6, 7, 8, 79, 81, 119, 120, 176, 186, 187, 196, 197. Kronecker, L. 1. Kuroda, S. 6. lambda: conversion 18, 19; definable functions 2; normal 18; normal form 18, 19; operator, prefix 11, 12, 16, 18, 19. law (Brouwer) 4, 44, 47, 56, 62, 66, 70, 90. Leblanc, H. 6. least-number: operator 30, 1 14; principle 86. logical symbol 8. Loor, B. de 2. Lukasiewicz, J. 6. Lusin, N. N. 1. Maehara, S. 3. 6. Mannoury, G. 6. Markov, A. A. 2, 3, 99, 1 19; principle 119, 120, 131, 185, maximum 25, 31. McCall, S. 6. McKinsey, J. C. C. 6, 81. measurable order 143, 144, 145, 150, 177, 178, 179, 181, Medvedev, Ju. T. 3, 6, 197.

-'s

186.

146, 186.

metamathematics (formal systems) 5, 6, 8, 66, 90, 110. minimum 25, 30, 31, 114 (line 20). model theory 6, 7, 90, 1 10 ( 6 ) . Moschovakis, J. R. cf. Rand, J. Mostowski, A. 3, 6, 81. multiplication 14, 23. Myhill, J. 3. Nagel, E. 3. natural: number 8, 45; order 143, cf measurable order. negation 1, 6, 8, 13. 30, 82, 97, 98, 176, 178, cf. double negation, excluded middle; -less intuitionistic mathematics 2. Nelson, D. 3, 4, 6, 7, 22, 81, 98, 99, 110. nested intervals 158. Nishimura, I. 6. normal (lambda) 18; - form 18, 19. normal form theorem (recursive functions) 91, 122. notation (formal) 9, 10, 11, 12. number: theory c arithmetic; variables 8, 9, 10, I; cf. natural -, real -. numeralwise: exp ssibility 22, 102, 104, 11 1 ; reprt. atability 21, 102, 103. 104. Ohnishi, M. 6. open: formula etc. 12; interval 161. operator 121; cf. lambda -, propositional connectives, quantifiers. order of a formula 120, 124, 125. order of a function 120, 121. order relation between natural numbers 23, 27. order relation between real numbers (measurable, natural) 143, 144, 145, 146, 150, 177, 178, 179, 180, 181, 183, 186, (virtual, pseudo-) 143, 146, 150, 177, 178, 180, 181, 183, 185, 186. pair order 121. parentheses etc. 8, 10, 92. partial recursive function 10, 12, 91. past secured sequence number 46.

205

INDEX

path 49, 60. Peano, G. -'s axioms 7, 14. Phter, R. 3, 4. Pil'M, B. Ju. 6, 81, 197. Poincarh, H. 1. point 134. p-order 121. Porte, J. 3, 6. Post, E. L. 2, 112, 114, 118. postulates cf. axiom, rules of inference. predecessor function 24. predicate: 2; calculus (logic) 6, 7, 9, 13, 14, 15, 81, 82, 83, 84. 119; Symbol 11, 12; variable 52, 67. prime: formula 12, 15, 27, 42, cf. standard formula; number 31, 32, 34.

primitive: recursion 20, 38, 39. 41, 42; recursive function 9, 10, 12, 19, 20, 23, 91, 93; recursive predicate 11, 22, 23, 27; symbols 8, 11. product 14, 23; finite - 28, 29. proof 8 (3.1), 14 (4.4), 64, 65, 66. proper inclusion (intervals) 157. properly defined 91, 92, 121. propositional:calculus 9, 13, 15, 8 1, 82, 83, cf. predicate calculus; connectives 8, 13, 15, 30; variables 176. pseudo-order 143, cf. measurable order. Putnam, H.3, 6, 81, 196. quantifiers 11, 46, 47, 48, 79, 90, 93. 94, 95; alterations of - 15. 17, 40, 41, 72, 120, 131; bounded - 15, 30; cf. correlation. quotient 26. Rand, J. (Moschovakis, J. R.) 68, 88. Rasiowa, H. 6, 81. realizability 96, 97, 98, 99, 100, 111, 120; 1945 definition of - 81, 93, 94, 97, 98, 99,

110, 119,

120; 1957

definition of - 95, 97, 100. 119; formalization of - 99, 110; special -, g 120, 125, 126; E truth 101, 102, 119, 125, 126, 131; - under

deduction 105, 127; cf. realizable, realizes-Y-. realizable 96, 99; (1945-)- 94; -/T 96, 99; -@, -@IT 100; C/-, 125; C/-/T, C/--@/T 111; 8C/,--O/T 126; cf. realizability. 125. realization function 99; realizes-Y 96; (1945-)realizes94; C/111; 8125; C/#- 126. realizing-Y function 99, 101, 102; *126 (10.6, 10.7). real number 134, 136, 151, 153, 176; -generator 134, 135, 136. recursion 20, 38, 39, 41, 42, 106, 128. recursive functions: general 2, 3. 4, 9, 10, 12, 47, 71, 91, 93; partial 10, 12, 91,93; primitive 9, 10, 12, 19,23, 91, 93; relativized 93, 96, 97; special 121, 122; special classes of 3. reduction (lambda) 19.

relative: constructiveness, recursiveness 10 (line 7), 80, 93, 95, 96, 97, 98, 111, 114, 116, 121.

remainder 26. replacement property: = 16, 20; t138, 141. 142, 144, 146, 147, 153, 161. representing function 11. Ridder, J. 6.

Rieger, L. 6. Robinson, R. M. 3, 4. Robinson, T. T. 7. 193. Rogers, H., Jr. 3, 4. Rootselaar, B. van 2, 186. Rose, G. F. 3, 6, 81, 119. Rosser, J. B. 3, 6, 19, 197. rules of inference 14, 64. Sacks, G. E. 3. Sanin, N. A. 3, 4, 7. 99, 119. Schmidt, H. A. 6, 81. Schroter, K. 6. Schiitte, K.3, 6. Scott, D. 3, 6, 81, 197, 198. securable:sequence number 46,47,48, 50, 51, 52, 53, 60, 64. 65, 66; -E, -1 52.

206

INDEX

secured sequence number 46. semantics 6, 7, 90, 110 (6). sentence 12. separability in itself 157, 169. sequence: finite 40. 45, 92; finite - of function values 38, 45; infinite -of function values cf. choice sequence; - number 37, 45. set (Brouwer) 4, 43, 65, cf. spread. sharp: arrow 47; diiference 133, 163, 168, 169.

Sikorski, R. 6. Skolem, Th. 3, 6, 22, 81. Smullyan, R. 3, 23. s-order 120, 121. special: realizability 120, 125, recursive function 120, 121, species (Brouwcr) 65, 80, 86, 134, 147; - of higher order 5, 7, Spector, C. 3. 8. spread (Brouwer) 4, 43, 44, 45,

126; 122. 136, 134.

48, 55, 56, 57, 64, 65, 66. 74, 80, 90, 136, 163, 167; cf. fan. ,realizability 120, 125, 126. srealizes-!P 125. standard formula 27, 28, 30. 31, 37. sterilized choice sequence 43. Stone, M. H. 6, 81. subfan 59. substitution 11, 12. subtraction (natural numbers) 24, 25, 26, (real numbers) 142. successor: function 8, 10, 14; order 120, 121. sum (natural numbers) 14, 23, (real numbers) 142; finite - 28, 29. Suppes, P. 3.

Tarski, A. 3. 6, 81. term 10, 12. terminated choice sequence 44, 56, 66.

topological interpretation 8 1. Trahtenbrot, B. A. 3. transformation rules cf. postulates. transitive laws 16, 138, 143, 144, 146, 157.

tree (of sequence numbers) 48, 49, 50, 51, 60, 65, 66, 115, (proof) 65, 66. true-Y (formula) 101. Turing, A. M. 2, 83. Umezawa, T. 6. undecidability cf. decision procedure, Godel's incompleteness theorems, independence results. uniform continuity theorem (Brouwer) 72, 133, 151, 153.

universal: quantifier cf. quantifiers; spread 45, 51, 55, 57, 64, 66, 70, 74, 80.

unprovability

172,

cf. independence results. 173;

Uspenskii, V. A. 3. variables (formal use of) 8, 9, 10, 11, 15, (functional) 72, 79, (informal) 91, 11 1, 121, 122, (prcdicate) 52, 67, (propositional) 176. vertex 48, 49, 60. Vesley, R. E. 5. 6, 65, 72, 73, 76, 88, 174, 176, 177, 179.

virtual order 143, 146, 150, 156, 177, 178, 180. 181, 183, 185, 186.

Wajsberg, M. 6, 81. well-ordering 48, 51, 65, 66, 69. Wright, E. M. 32. Zegalkin, I. I. 6. zero (degree) 112, 113, (natural number) 8, 10, 14, (real number) 147, 170, 171, 177.