Self-similarity in the Foundations Paul K. Gorbow Thesis submitted for the degree of Ph.D. in Logic, defended on June 14, 2018. Supervisors: Ali Enayat (primary) Peter LeFanu Lumsdaine (secondary) Zachiri McKenzie (secondary) University of Gothenburg Department of Philosophy, Linguistics, and Theory of Science Box 200, 405 30 GOTEBORG,¨ Sweden arXiv:1806.11310v1 [math.LO] 29 Jun 2018 2 Contents 1 Introduction 5 1.1 Introductiontoageneralaudience . ..... 5 1.2 Introduction for logicians . .. 7 2 Tour of the theories considered 11 2.1 PowerKripke-Plateksettheory . .... 11 2.2 Stratifiedsettheory ................................ .. 13 2.3 Categorical semantics and algebraic set theory . ....... 17 3 Motivation 19 3.1 Motivation behind research on embeddings between models of set theory. 19 3.2 Motivation behind stratified algebraic set theory . ...... 20 4 Logic, set theory and non-standard models 23 4.1 Basiclogicandmodeltheory ............................ 23 4.2 Ordertheoryandcategorytheory. ...... 26 4.3 PowerKripke-Plateksettheory . .... 28 4.4 First-order logic and partial satisfaction relations internal to KPP ........ 32 4.5 Zermelo-Fraenkel set theory and G¨odel-Bernays class theory............ 36 4.6 Non-standardmodelsofsettheory . ..... 38 5 Embeddings between models of set theory 47 5.1 Iterated ultrapowers with special self-embeddings . ......... 47 5.2 Embeddingsbetweenmodelsofsettheory . ..... 57 5.3 Characterizations.................................. .. 66 6 Stratified set theory and categorical semantics 73 6.1 Stratifiedsettheoryandclasstheory . ...... 73 6.2 Categoricalsemantics ............................... .. 77 7 Stratified algebraic set theory 85 7.1 Stratifiedcategoriesofclasses . ..... 85 7.2 Interpretation of the Set-theories in the Cat-theories ................ 90 7.3 ThesubtoposofstronglyCantorianobjects . ....... 99 8 Where to go from here? 103 8.1 Category theoretic approach to embeddings between models of settheory . 103 8.2 Directions for further research on stratified algebraic set theory .......... 104 3 4 CONTENTS Chapter 1 Introduction 1.1 Introduction to a general audience Zooming in on a fern, one may be struck by how its parts resemble the whole, and zooming in further, how the parts of the parts again exhibit a similar structure. Likewise, when learning about atoms, solar systems and galaxies, one may naturally be struck by the analogy between a typical spiral galaxy and the accretion disc of an infant solar system, as well as the analogy between how the planets stand to the sun and how the electrons stand to the atomic nucleus. These analogies are far from perfect, but nonetheless inevitable. In the mathematical realm, idealized forms of this phenomenon have become well known to the broad public through the theory of fractals, a field from which a rich variety of beautiful images has spun off, to the delight of the human mind. The term self-similarity has come to be used quite broadly for this phenomenon. More formal mathematical umbrella-terms are endomorphism, the slightly narrower self-embedding and the even more narrow automorphism. In either case, the words morphism and embedding are largely left open to be defined as appropriate for the domain of study. Generally, an endomorphism exhibits a part with similar structure as the whole, but where some details of the structure may be lost; with a self-embedding the part and whole have the same structure; and with an automorphism no proper part is involved, but the structure of the whole emerges in several different ways on the same whole. Since the whole is in a trivial sense a part of itself, any automorphism is also a self-embedding, and moreover any self-embedding is an endomorphism. In some fields, e.g. the field of the present monograph, so much structure needs to be preserved that the notions of endomorphism and self-embedding coincide. On the other hand, for finite structures the notions of self-embedding and automorphism end up coinciding. Let us consider a circular clock from 0 to 12 as an example. The clock has only an hour-dial, and this dial takes just one step forward every hour, never passing in between two numbers on the clock: We have 0 (considered equal to 12) at the top of the clock. If one does addition and subtraction on this clock, then we have 2+3 = 5, 5+8 = 1 and 5 − 8 = 9, for example. It turns out that this mathematical structure (a finite group called C12) has a few self-embeddings. For example, notice that the even numbers of the clock exhibit a similar structure to that of the whole clock (this substructure is a subgroup of C12 called C6). Let us define a function f from the clock to itself by f (x)= x + x, with the addition done in C12, of course. So f (0) = 0, f (3) = 6 and f (7) = 2, for example. This is an endomorphism of C12, which reveals that C12 has C6 as a similar part of itself. The essential criterion that f passes to attain the status of endomorphism, is that f (x + y)= f (x)+ f (y), for all x, y in C12. The function f is neither a self-embedding 5 6 CHAPTER 1. INTRODUCTION nor an automorphism, because C6 is not the same as C12 (half of the details of the structure were left out). However, it turns out the function g(x)= x + x + x + x + x is an automorphism. For if you renumber your clock by this function g, then you will go around the clock with the numbers 0 at the top, and then 5, 10, 3, 8, 1, 6, 11, 4, 9, 2, 7, and finally come back again to 0. If you then just change the mechanics so that the hour-dial takes 5 steps each hour instead of just 1, then this clock will also show the time correctly, stepping first from 0 to 1, then from 1 to 2, and so on. Thus, the same structure as the original structure, is present on the whole in a very different way. In this monograph, structures for the foundations of mathematics are studied, particularly structures satisfying axioms of set theory. By a key theorem of Tennenbaum from the 1960:s, these structures are so complex that it is impossible to devise a theoretical algorithm or computer program to describe them, even if it were allowed to run for infinite time. Remarkably, the existence of these complex structures follows from rather innocent looking axioms of set theory. In particular, this monograph studies structures of set theory and structures of category theory of such extraordinary complexity. To give a hunch of the kind of structures studied, here follows a sketch of a construction of a non-standard model of arithmetic: In arithmetic we have a first order predicate language with numerals 0, 1, 2, 3,... denoting the standard natural numbers, and we have the operations of + and × as well as the relation < for comparing size. We now add a name c to this language. There is no way of showing that c<t for any numeral t = 0, 1, 2,... in the original language, because this name may be assigned to denote any numeral. This means that it is consistent that 0 < c, 1 < c, 2 < c, and so on. Taking all these sentences together, one finds that it is consistent with arithmetic that c is larger than every standard natural number. G¨odel’s Completeness Theorem says that for any consistent set of sentences, there is a structure that satisfies those sentences. So there is a structure satisfying all the sentences of arithmetic, which still has numbers greater than all of the standard natural numbers 0, 1, 2,... By Tennenbaum’s Theorem, this non-standard structure is not algorithmically describable. Thus, we find ourselves in a situation where we can show the existence of structures which are so complex that our prospects for describing them are severely limited. Nonetheless, a rich mathematical theory has emerged from the study of these structures, especially for those structures of smallest possible infinite size (countably infinite structures). Chapters 4 and 5 of this monograph are concerned with such structures of set theory. A theorem of Friedman from the 1970:s states roughly that every countable non-standard structure of the conventional theory of arithmetic, known as Peano Arithmetic (PA), has a proper self-embedding. Friedman also proved this for a fragment of Zermelo-Fraenkel set theory (ZF), the conventional set theory. So the phenomenon of self-similarity is abundant in the theory of non-standard foundational structures. In Section 5.2, theorems along these lines are refined and generalized for the setting of a fairly weak fragment of ZF. ZF axiomatizes a hierarchical conception of sets. Each set can be assigned a rank, such that if x is a member of y, then the rank of x is less than the rank of y. Thus it makes sense to ask whether a structure of ZF can be extended to a larger structure, such that new sets are only added at higher level of this rank-hierarchy. Such an extension is called a rank-end-extension. In Section 5.1 it is shown that each of a certain class of structures of ZF can be rank-end-extended to a structure with a non-trivial automorphism such that the sets that are not moved by the automorphism are precisely the sets in the original structure. This constitutes a set-theoretic generalization of an arithmetic result by Gaifman. In Section 5.3, the Friedman- and Gaifman-style theorems are combined in various ways to obtain several new theorems about non-standard countable structures of set theory. In Chapters 6 and 7 we turn to category theoretic foundations of mathematics. While set 1.2. INTRODUCTION FOR LOGICIANS 7 theory is concerned with the relation of membership, category theory is concerned with transfor- mations.