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Självständiga Arbeten I Matematik SJÄLVSTÄNDIGA ARBETEN I MATEMATIK MATEMATISKA INSTITUTIONEN, STOCKHOLMS UNIVERSITET Richard´s paradox: Impredicativity and innity coinciding av Per Helders 2020 - No 45 MATEMATISKA INSTITUTIONEN, STOCKHOLMS UNIVERSITET, 106 91 STOCKHOLM Richard´s paradox: Impredicativity and innity coinciding Per Helders Självständigt arbete i matematik 15 högskolepoäng, grundnivå Handledare: Rikard Bögvad 2020 Richard´s paradox: Impredicativity and infinity coinciding Per Helders, MM6004, Stockholm University HT 2020 Department of Mathematics, 15 p Supervisor: Rikard Bögvad ”All generalizations are wrong” Tage Danielsson Many thanks to Lena, who found many typos and errors when reading through and checking my English, and who also has been most patient with me during this period. 1 Richard´s paradox: Impredicativity and infinity coinciding Contents ABSTRACT........................................................................................................................... 3 INTRODUCTION................................................................................................................... 3 Impredicativity and infinity...................................................................................................... 3 Actual infinity and potential infinity.......................................................................................... 4 BACKGROUND.................................................................................................................... 5 Zeno´s paradox of motion...................................................................................................... 5 Torricelli´s paradox................................................................................................................. 7 Hilbert´s paradox.................................................................................................................... 9 Russell´s paradox.................................................................................................................. 11 Variables and domains.......................................................................................................... 15 Axioms.................................................................................................................................. 16 Impredicative definitions........................................................................................................ 17 RICHARD´S PARADOX....................................................................................................... 18 Diagonalization...................................................................................................................... 18 Presentation of Richard´s paradox........................................................................................ 19 Diagonalization again - impredicativity and infinity coinciding............................................. 25 Denumerability - one-to-one correspondence..................................................................... 28 Richard and Cantor............................................................................................................... 29 Transfinite numbers............................................................................................................... 30 Cantor and God..................................................................................................................... 33 Some other voices................................................................................................................. 35 CONCLUSIONS................................................................................................................... 38 Summary of conclusions....................................................................................................... 38 Scienticifity............................................................................................................................. 39 Ontology of mathematical objects - Constructivism and intuitionism.................................. 39 CONSEQUENCES - EXTENSIONS OF THE CONCLUSIONS......................................... 42 Intuitionistic logic vs classical logic - useability of mathematical propositions..................... 42 Epilogue: Tertium Non Datur - proof by contradicition....................................................... 47 References.............................................................................................................................51 APPENDICES...................................................................................................................... 53 App. I: Original formulation of Richard´s paradox (French)................................................... 53 App. II: Page 15 from manuscript from G Cantor to Acta Mathematica 1885 (German)........55 App. III: Page 123 from Acta Mathematica vol. 7, 1885 (printing of Cantor´s manuscript).......56 2 Abstract The principal purpose of this paper is to, on the basis of Richard´s paradox, study how mathematicians over time have discussed some paradoxical reasonings which are more or less connected to that paradox. Hence, we will look upon these questions from a partly historical perspective, but also treat the question why some of these paradoxical reason- ings are still around and discussed. However, when studying this issue, an additional issue emerges, and with that issue also an additional purpose. This additional purpose is to study, with mathematical/ logical tools, the issue of the split or discrepancy between classical and constructive/ intuitionistic mathematics. This split is so closely related to some of the paradoxical reasonings mentioned above, that it can be explained in terms of the principal questions raised, and, furthermore, the location and the nature of this split can be illuminated by the principal results. Therefore, we will make a synoptic account, both historically and in terms of content, of the split mentioned. Doing this, we will initially focus on two mathematical topics that have been, and still are, regarded as problematic. It is inside the framework of these two topics that answers have been sought for concerning problems revealed by some paradoxical reasoning. The two topics are: 1. Impredicativity (circularity) 2. Quantification over infinite sets We will see that Richard´s paradox, maybe more than any other, reveals problems belon- ging to these two topics, not in the sense of problems as being part of the paradox, but rather as being part of the mathematical reasoning from the time of its presentation and up until today. Indeed, Richard´s paradox is the event where these two topics coincide in a way that is very useful for an analysis of them and of the contradictory reasonings connected to them (i.e. is there a common denominator?), but also, as we will show, calls for caution to carefully distinguish between these two topics when they coincide. Additionally as a third topic, the analysis will also have impact on what ontological sta- tus we assert to different kinds of mathematical objects. INTRODUCTION Impredicativity and infinity Generally speaking, paradoxes express some kind of contradiction. Sometimes this contradiction is contained within the paradox, e.g. Zeno´s paradoxes, but more recent paradoxes often reveal a contradiction that has its origin not in any flawed reasoning in the paradox itself, but in an external reasoning, e.g. Russell´s paradox. Apart from Richard´s paradox, we will briefly describe four other paradoxes: Zeno´s paradox of motion, Torricelli´s paradox, Hilbert´s paradox and Russell´s paradox. The purpose of describing them here is that they all contribute, in different ways, to create a background consisting of observations regarding impredicativity and infinity from 3 which we will examine Richard´s paradox and the conclusions that may be drawn from it. However, initially we will make the two notions above more precise. The word im- predicativity is derived from ”predicativity”, which is a piece of terminology first appear- ing in works of Bertrand Russell1. The meaning of this word has changed slightly over time, but it is still closely related to circularity. Thus, as an example, we use the adjective impredicative on definitions that have elements of their definiendum in their definiens, i.e. in order to define a word, they use that very word in a circular way in the definition. We will already here establish that this can in some contexts cause the possibility of contradictions. How and why this is the case will be treated below (see below e.g. under Russell´s paradox). Less easy to make precise is the concept of infinity. However, it seems easy to define. We can for instance write the progression (a 1, 2, 3, ...) and we feel comfortable with this expression of the infinity, and that a is a linguistic entity that has an instantiation of the infinity as its reference. It is more difficult≔ when it comes to the properties of the infinite, as well as the meaning and the consequences of the three dots in the end of the expression. This has led to a dichotomy in terms of some of these properties. The main dividing line in this discussion is whether the infinity is to be regarded as some kind of complete set or not, i.e. is the infinity in terms of numbers possible to treat as a com- pleted entity with a given cardinality or not? The two sides of this dichotomy is called the actual infinite and the potential infinite respectively, and they go back to Aristotle. Actual infinity
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