Intuitionistic Theory

-*Q‘ 5\‘\“‘3 .///X ‘\ \._ ‘ >‘\ \ I/I/ W \ .»-x\‘‘ as\‘\‘~_\ ‘I’ /I f‘./,'/L,‘ I , ‘ I / /7 ‘5I'?.?I:'.I',. I I I ' T _.‘.‘..~_\_\‘(.,\~.$\‘|5“ 4/5'» ' // ..''.‘_\-' / ’//2’’!","", . /I K //////7’:/"‘,-"7 1’ . ’/ I‘ I ‘,.,. ‘ I" ' I , I////I’ "'1 ‘ I/'’'I'''I, ,2’ r ' /I 5//’r'«’I'; IZ /I/// . /4L’ I, /({..vI, INVESTIGATIONS IN INTUITIONISTIC HIERARCHY THEORY WI m Veldmon INVESTIGATIONS IN INTUITIONIQTIC HlERARCHy THEoRy Promoter; Prof‘. II. de Ionqh. INVE§TIGATlO.NS IN INTUITIONISTIC HlERARCH\} THEORy PROEFSCHRIFT TEE VERKRUGIMG VAN DE GRAAD VAN DOCTOR IN DE WlSKuNDE EN NATUURWETENSCHHPPEN AAN DE KATHOLIEKE UNIVERSITEIT TE NUMEGEN OP GEZAG VAN DE RECTOR MAGNIFICUS PROF. DR. P.G.A.B. WIJDEVELD VOLGENS BESLUIT VAN HET COLLEGE VAN DECRNEN IN HET OPENBAAR TE VERDEDIGEN OP WOENSDRG 20 ME! I98: DES NAMIDDAGS TE 2.00 uua PREClE§ Dook WILLEM HENRI MARIA VELDMAN GEBOREN TE MHRSTRIC HT :99: KRIPS REPRO MEPPEL in een ernstcq bestaan Lou hi; (:2 gronde qaan hield Net het zwevende plezier hem levende. hang lodeczen. Hwalz nimmen Sczze. ken yh Fry‘ bineamen, Ik hce an moeol en soe Cl:(-‘oar jL'mm’rcme? Unnoonzélbarn dat sect: Cinefndickheit, As it de stap hat op de Pjcrde trimel, Obe Postma De omslaq cs zen ontwerp van mgn kunstbroeder kees §treeH:erk, TABLE OF CONTENTS 0 Introduction 1 1 A Short apology For intuitionicticz analysis 5 2 Al-_the bottom of the hierarchy. A discussion" of Brouwer-kriplce-'S axiom 12. 3 The second level 0F the arithmetical hierarchy 15 LiSome activities of disjunction and Conjunction 21 5 An aside on implication 3” 6 Arithmetical sets introduced ‘*5 7 lhe arithmetical hierarchy established. '5“ 8 Hyperarithmeticol sets introduced 61 9 We hyperarithmetical hierarchy established 53 Io Analytical and co-analytical set»; 3"’ ll Some members of’ the analytical Pamily 96 I2 Ah outburst of‘ disjunctive, conjunctive and implfcative productivity “-9 13 Brou.wer’s thesis, and some oF its consequences 155 H lhe collapse of the projective hierarchy 157 15 A contrapositioh of’ countable choice W9 16 The truth about oleterminacx; 139 47 Appendix: Strange lights in a dark alley 2” Notes I references 115/ 113 Synopsis 223 Subject index] index ol‘ symbols 237-l?-3” Curriculum vitae 7.35 0. INTRODIACTION Thisthese; is concerned with constructive reasoning in descriptive set theory. The venerable Subject of‘ descriptive set theory was developed in the early decades of this century, mainly by French and Russian mathematicians. It started from the following observation; once the class of‘ continuous real Emotions has been established, one naturally comes to think of‘ the class of‘ real Functions which are limits of? everywhere convergent Sequences OF Continuous Functions. This wide: class can be extended in its turn, by the same operation of-’ forming limits of everywhere convergent sequences. This goes on and on, even into the transpinite. Thus a splendid structure arises, called =' §aire’s hierarchy. The same story may be told in terms of‘ sets. Looking at the subsets oF Baire space ‘°w which are Forced into existence when we allow For the clopen (= cJosed- and-open) neighbourhoods and then apply the operations o(-‘countable union and intersection again and again, we may wonder once More, because there is no end ol-‘it. One aFl:er another, the classes oF Borel; hierarchy present themselves, each Containing subsets of °"w not heard of‘ before, No Borel class exhausts the possible subsets o[3 “’w l7\is can be proved in a Few lines; one shows that each class contains at universal elemgn_t and diagonalizes. (cf. chapter b, esp. blu) However, the very ease of the proof-‘ arouses suspicion. People like Borel, Baire, Lebesgue, who were the First to raise and answer many questions in this Subject, spent much thought on the plausibility oF their arguments. Diagonalizing was Felt as cheap Pea-Sohing, especially by Baire. Avoiding the diagonal argument, only relying on methods ,,From practice’: one succeeded in showing up members of’the first three 0'” Four classes oF Baire. DLaqona[[2£nq’ OF Coup-ge’ wag nolj the WOV’StOF €VtlS. Iﬂ l-USCHISCatalogue, to be Round.on page 55 of I930, it comes immediatelyafter ,, normal constructive argument‘: before such horrible things as-. the use of R1 as a well-deﬁned, Completed mathematical set, or, even worse, the essentially incomprehensible argument by, which Zermelo established o. well- ordering of any set, from the axiom OF choice. Now, For heanI?Jn’s sake, what might be wrong with the diagonal argument? From a classical point of‘ view, one cannot bring up much against it. In fact, as soon as we agree upon the meaning 0? negation (P and ‘'l’ cannot hold together, whatever be the proposition P) we have to accept it. But in intuilzionisrn we may Find. an explanation for our uneasiness. 7. Let us remark that, classically, we may build up the Borel sets in ‘”w From the Closed-and-open neighbourhoods, using only countable union and intersection. Complementationcan be missed 0.9 an operation For making new sets out of already existing ones: as the complement of any closed—and-open set is closed- and- open, the complement of any Set bull-t From the closed- oind- open gets by countable union and intersection, is such a set again. This certainty is given by such wonderful guardians of-‘classical symmetry as are de Morgan's laws. De Morgan's laws are not acceptable, intuitionistically, apart From Some V91)! Qimplesituations, From which they were derived by 0. crude generalization. Wecannot explain away complernentation, or, more qenerally, the analogue of logical implication, as methods of‘ constructing sets. But we might try to do without them. We will do so in this treatise. When negation and implication are put aside, the possibility of diagonalizing is taken from our hands, and the hierarchy problem is open again. A solution is given in chapters 6-9. There is good reason to consider negation and implication with some caution. Many unsettled questions in Lntuitionistic loqic are connected with them. (Compare the discussion in the appendix, chapter l7- We are not able to decide how Far the divergence between classical and intuitionistic logic goes. Also, at curious role is played. by negatibn in the recent discussion of the intuitionistic completeness of‘intuitionistic predicate logic, cf. de Swart l9'2‘6, Veldman I976) The intuitionistic hierarchy has a very delicate structure. The class of the closed Subsets of Baire space,For instance, is no longer closed under the operation of Finite union. One has to distinguish between closed sets, binary unions of closed. sets, ternary unions of closed Sets, and So on. This phenomenon is discussed in chapter 4. The productive force of disjunction and conjunction is explored f-‘urther in chapter 11.2.0-7.6 and chapter 120-}. zlmplication, although absent From chapters 6-9, L9 "05 Completely lc°"9°tb€", and, We will see, in chapter 5 and chapter l2.8‘9 that it Shares in some of the properties established For disjunction and conjunotlbn. Distrust of diaqonalization is one OF many points on which early descriptive Set theorists and intuitionists have similar views. Their common basic concern might be described as-. exploring the Constructive CON-'-Uvuum. _ Brouwer’s rejection of classical logic is, of‘ course, a major point 0F dif-‘Ference. 3 But one is tempted to ask not the main theorem of this essay, which . establishes the intuitionistic hierarchy (chapter 9, theorems 9.? and 9.9) Wqbt have delivered Baire From his $¢WP'€9 Since Addison 1955 it has become customary among logicians to Consider descriptive set theory For its Connection with recursion theory. We will bypass this development. From an intuitionistic point of view, recursion theory is an ambiguous branch on the tree of constructive mathematics. The deep results OF this theory depend on very serious applications OP classical logic, And the classical continuum, which is 0. rather obscure thing, is accepted without any comment, as a suitable domain of-‘deFi'nition For effective operations. Nevertheless, there is an analogy between recursion theory and the theory to be developed here; Many paradoxical results of elementary recursion theory are due to theFact that Functions and functionals are Finite objects, and, therefore, of the same type as natural numbers. Now, Functions From Baire space “in to We», being necessarily continuous, are determined by a se uence of neighbourhood Functions, and thus may be seen to be themselves members of Wm, Once more, we are in a. situation where Functions do not differ in type from their arguments and values. We also have to admit that, there is any elegance in these pages, it partly is due to modern recursion theory. For instance, the Following concept of many- one reducibility between subsets of ‘*’u) is starring A:<.B == Elfff‘ is a continuous Function From woo to “’w and Vo<[oieA2Pb<)eB] This so—ca.lled ”li\/o.oige—reducibility" was made the subject of classical study by some students of Addison's (cit Kechris and Moschovalg I978’ ht-_e9e. '98".). Their methods, however, are very far fl-om constructive, We introduce this concept in chapter 2, after a short gxpggitfon of‘ the principles of intuitionistic analysis. In the second part oi.‘ this thesis (chapters l0-l‘l) we turn to analytical sets, and the prcwpctive hierarchy. (cf. Note 3 on page 216). Analytical sets, being close relatives of good old "spreads", get a chapter of their own. it will be seen that the classical duality between analytical and co- analytical sets is severely damaged. (chapter 10). Some Famous results OF Souslin’s are partly rescued by Brouweirg bar theorem, which we will prpsent here under the name of Brouwer’s thesis.

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