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Extremal Axioms Extremal axioms Jerzy Pogonowski Extremal axioms Logical, mathematical and cognitive aspects Poznań 2019 scientific committee Jerzy Brzeziński, Agnieszka Cybal-Michalska, Zbigniew Drozdowicz (chair of the committee), Rafał Drozdowski, Piotr Orlik, Jacek Sójka reviewer Prof. dr hab. Jan Woleński First edition cover design Robert Domurat cover photo Przemysław Filipowiak english supervision Jonathan Weber editors Jerzy Pogonowski, Michał Staniszewski c Copyright by the Social Science and Humanities Publishers AMU 2019 c Copyright by Jerzy Pogonowski 2019 Publication supported by the National Science Center research grant 2015/17/B/HS1/02232 ISBN 978-83-64902-78-9 ISBN 978-83-7589-084-6 social science and humanities publishers adam mickiewicz university in poznań 60-568 Poznań, ul. Szamarzewskiego 89c www.wnsh.amu.edu.pl, [email protected], tel. (61) 829 22 54 wydawnictwo fundacji humaniora 60-682 Poznań, ul. Biegańskiego 30A www.funhum.home.amu.edu.pl, [email protected], tel. 519 340 555 printed by: Drukarnia Scriptor Gniezno Contents Preface 9 Part I Logical aspects 13 Chapter 1 Mathematical theories and their models 15 1.1 Theories in polymathematics and monomathematics . 16 1.2 Types of models and their comparison . 20 1.3 Classification and representation theorems . 32 1.4 Which mathematical objects are standard? . 35 Chapter 2 Historical remarks concerning extremal axioms 43 2.1 Origin of the notion of isomorphism . 43 2.2 The notions of completeness . 46 2.3 Extremal axioms: first formulations . 49 2.4 The work of Carnap and Bachmann . 63 2.5 Further developments . 71 Chapter 3 The expressive power of logic and limitative theorems 73 3.1 Expressive versus deductive power of logic . 73 3.2 Metalogic and metamathematics . 76 3.3 Limitative theorems . 84 3.4 Abstract logics and Lindström’s theorems . 88 3.5 Examples . 90 Chapter 4 Categoricity and completeness results in model theory 103 4.1 Goals of model theory . 103 4.2 Examples of categoricity and completeness results . 107 4.3 Ultraproducts . 108 4.4 Definability . 111 4.5 A few words about types . 113 4.6 Special models . 116 4.7 Classifying theories . 118 Part II Mathematical aspects 123 Chapter 5 The axiom of completeness in geometry, algebra and analysis 125 5.1 Geometry . 127 5.2 Algebra and analysis . 137 5.3 Axiom of continuity and its equivalents . 142 5.4 Generalizations and isomorphism theorems . 144 5.5 Degrees of infinity, pantachies and gaps . 152 5.6 Infinitesimals and non-Archimedean structures . 154 5.7 Continua in topology . 161 Chapter 6 The axiom of induction in arithmetic 163 6.1 A few historical remarks . 163 6.2 Definitions of finiteness . 171 6.3 First-order arithmetic . 173 6.4 Second-order arithmetic . 177 6.5 Non-standard models of arithmetic . 178 6.6 Transfinite induction in set theory . 183 Chapter 7 Two types of extremal axioms in set theory 187 7.1 Introductory remarks . 187 7.2 Zermelo: two axiomatizations of set theory . 192 7.3 Axioms of restriction . 198 7.4 Large cardinal axioms . 213 7.5 Sentences independent from the axioms . 219 Part III Cognitive aspects 221 Chapter 8 Mathematical intuition 223 8.1 Philosophical remarks . 223 8.2 Research practice . 232 8.3 Teaching practice . 250 A final word 267 Bibliography 271 Author index 293 Subject index 301 Preface The term extremal axiom was introduced by Carnap and Bachmann in their article Über Extremalaxiome (Carnap and Bachmann 1936). Ax- ioms of this sort ascribe either maximal or minimal property to models of a theory. Examples considered by Carnap and Bachmann included: the completeness axiom in Hilbert’s system of geometry, the induction axiom in arithmetic and Fraenkel’s axiom of restriction in set theory. The completeness axiom (replaced later by the continuity axiom) expresses the condition of maximality of the geometric universe: that universe can- not be expanded without violating the other axioms of the system. The axiom of induction is an axiom of minimality: it expresses the idea that standard natural numbers form a minimal set satisfying the other ax- ioms of the system. Fraenkel’s axiom of restriction says that the only existing sets are those whose existence can be proved from the axioms of set theory. There are further axioms which can be considered extremal in the above sense. Gödel’s axiom of constructibility and Suszko’s axiom of canonicity are minimal axioms in set theory; as in the case of Fraenkel’s axiom, they also express the idea that the universe of sets should be as narrow as possible. On the other hand, axioms of the existence of large cardinal numbers are maximal axioms; they express the idea that the universe of sets should be as rich as possible. Extremal axioms are related to the notion of an intended model. The intended model of a mathematical theory is a structure that has usually been investigated for a long time and about which we have collected es- sential knowledge supported also by suitable mathematical intuitions. As examples of such structures one can take for example, natural, rational or real numbers, the universe of Euclidean geometry and perhaps also the universe of all sets considered in Cantorian set theory. Given such a structure one tries to build a theory of it, ultimately an axiomatic one. It may happen that one can prove that a theory in question characterizes 10 Preface the intended model in a unique way (up to isomorphism or elementary equivalence). On the other hand, such theories as group theory or gen- eral topology do not have one intended model; they are thought of as theories concerning a wide class of structures. We feel obliged to explain the reasons for collecting the material that follows within a single monograph. There are numerous publications de- voted to particular problems mentioned above. However, a synthetic ap- proach to them seems to be rare. Besides the original paper by Carnap and Bachmann, there exist only relatively few works devoted to the ex- tremal axioms in general. Those worth mentioning include: Awodey and Reck 2002a, 2002b, Hintikka 1986, 1991, Schiemer 2010a, 2010b, 2012, 2013, Schiemer and Reck 2013, Schiemer, Zach and Reck 2015, Tarski 1940. In our opinion, the following topics are relevant with respect to the issue of extremal axioms: 1. Revolutionary changes in mathematics of the 19th century. This concerns above all: (a) The rise of modern algebra understood as the investigation of arbitrary structures (domains with operations and relations) rather than – as before – looking for solutions of algebraic equations. (b) Discovery of systems of geometry different from the Euclidean geometry known from Euclid’s Elements and hitherto consid- ered the true system of geometry. The proposal of thinking about geometries as determined by invariants of transforma- tions. 2. The rise of mathematical logic. The codification of the languages of logic (type theory, first-order logic, second-order logic, etc.) made it possible to talk about mathematical structures in a precise way. 3. Attempts at axiomatic characterization of fundamental types of mathematical systems. The axiomatic method previously (before the 19th century) present only in Euclid’s system of geometry has become widespread in other mathematical domains. 4. Attempts at unique characterization of chosen mathematical struc- tures. In these cases, where mathematical research was focused on the properties of specific a priori intended models, the question Preface 11 arose of a possibility of a unique (with respect to structural or semantic properties) characterization of such models. 5. The emergence of metalogic. About one hundred years ago investi- gations, and the existence of several systems of logic was admitted. As a consequence, questions naturally arose concerning the com- parison of these systems, their general properties, and so on. 6. Limitative results in logic and the foundations of mathematics. Metalogical reflection very soon brought important results showing the possibilities and limitations of particular logical systems, above all concerning the famous incompleteness results. It became evi- dent that certain methodological ideals can not be achieved simul- taneously – for instance, “good” deductive properties and a great “expressive power” are in conflict, in a precisely defined sense. 7. Philosophical reflection on mathematical intuition as a major factor in the context of discovery. Our main research goal is modest; we admit that the material covered by the book is to a large extent known to specialists. However, we believe that presenting the origins of extremal axioms and the development of research on them could be of some value to the reader who is interested in mathematical cognition. We make some use of the material contained in: Pogonowski 2011, 2016, 2017, 2018a, 2018b. The book consists of three parts focusing in turn on logical, mathe- matical and cognitive aspects of extremal axioms. Part I: Logical aspects. We begin here with a discussion con- cerning the relations between mathematical theories and their models. Then we present remarks about the origin of extremal axioms as well as the origin of certain metalogical properties, notably those of categoric- ity and completeness. We recall some famous limitative theorems which show possibilities and limitations in the unique characterization of mod- els. The notion of the expressive power of logic is useful in this respect. The final chapter of this part gives examples of important results from model theory related to the properties of categoricity and completeness. Part II: Mathematical aspects. Here we discuss the role of par- ticular extremal axioms and selected properties of theories based on ex- tremal axioms. First, we present results related to the continuity axiom in geometry, algebra and analysis. Then we recall the role played by the induction axiom in arithmetic. Finally, we discuss two types of extremal 12 Preface axioms in set theory, namely the axioms of restriction and the axioms of the existence of large cardinal numbers.
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