More than a Decade in the Set-Theoretic Multiverse

1 Introduction The importance of set theory can hardly be overestimated: from its first development by Georg Cantor and Ernst Zermelo, to the most recent results, set theory has always been seen as providing a foundation for mathematics. And this is for an important reason: it is possible to model and develop all of mathematics in set theory. Set theory is usually identified with its standard axiomatisation, Zermelo-Fraenkel with Choice ( Z F C ) , which is often taken to formally articulate our intuitive conception of sets and membership. The seminal work of Zermelo first suggested a "cumulative" conception of sets, according to which sets are constructed in stages and form a v-shaped structure, the universe of set theory V. The cumulative hierarchy V was defined in such a way that encompassed our intuitive notion of sets and membership, while avoiding to fall pray to the set-theoretic paradoxes that plagued the early development of set theory. However, from the sixties onward it became clear that some propositions in the language of set theory cannot be proved from the canonical axioms of Z F C – the Continuum Hypothesis (CH), a central set-theoretic claim, being a case point.1 The independence2 of CH from Z F C , i.e. the fact that Z F C doesn't prove neither CH nor ¬CH, strongly suggested that the usual set theory was inadequate as it stood. Whence the quest for finding an adequate extension of Z F C , usually referred to as Gödel's Programme, started: new axioms were proposed, but none of these new axiomatisations were stably accepted as "the" axiomatisation of set theory. The reason is that several such new axioms are mutually exclusive: choosing one implies that all the results proved under an incompatible axiom can no longer be proven. This led to competing set theories, and to the necessity to develop a formal methodology to compare the axiom candidates. At present, none of these theories is accepted as the "new" set theory. The independence phenomenon is both pervasive and inevitable, something which has led some set-theorists and philosophers to support a pluralist conception of sets. In recent years, such a conception has emerged and progressively gained prominence in the debate on the foundations of mathematics. More precisely, the independence phenomenon has manifested itself through a proliferation of models of set theory. To wit, in order to show that statements such as CH are independent of set theory, models have been constructed which settle the truth of these statements in multiple, mutually incompatible ways. According to pluralism, despite the relatively early emergence of a proper class-sized3 structure (V, ∈) encompassing all sets, which may naturally be seen as playing the role of "single universe",4 set theory is not the theory of a single set-theoretic structure, but rather of multiple set-theoretic structures. Such a pluralist view is then mathematically instantiated by the set-theoretic multiverse.5 Opposed to this view is universism, the thesis that there exists only one universe of set theory, that instantiates the "One True Set Theory". Adding new axioms to Z F C will eventually produce an axiomatisation that settles all the open (and independent) questions, thus pinning down which one is the "true" set-theoretic universe, the canonical and intended model of this new axiomatization of set theory.6 To anticipate (making use of technical notions that will be introduced in due course), there are several kinds of set-theoretic models, but the following ones are the most relevant for the multiversist's purposes: set-models in V (e.g., countable transitive models), inner models, and outer models. Inner models are transitive subclasses (M, ∈) of V which arise from somehow restricting the Power-Set operation (Gödel's constructible universe L here is the standard example). Outer models are models obtained through forcing. On the standard interpretation of forcing, one starts with a countable ground model M and extends it to a model MG which contains a generic filter G7 over a poset8 P ∈ M . By contrast, on a pluralist interpretation, outer models are seen as proper classes9 VG, which should, in turn, be seen as extensions of a ground universe. By "outer model" here I mean a model obtained through set-forcing, class-forcing, hyperclass-forcing (i.e. forcing applied to sets, classes with set conditions and classes with class conditions respectively) and, in general, any model-theoretic technique able to produce width extensions of V (intuitively, extensions resulting from adding "new" subsets to V). Since the claim that it is possible to extend V is at least philosophically problematic,10 a face value interpretation of VG is formally barred. However, there are ways to make sense of such an interpretation of forcing.11 These set-theoretic models involve different possible attitudes towards the extendibility of V. Actualism is the view that V is a fully determinate, inextensible object, whereas potentialism is the view that either the height (the length of the sequence of the ordinals) or the width (the extent of the Power-Set operation) of V are open-ended (or both), and thus extendable.12 Radical actualism may be seen as a different way of expressing the universist view, since, clearly, universists must hold that there is a fully determinate entity that is referred to by the axioms. On the other hand, ontological pluralism prima facie appears to be more in line with a radical potentialist understanding of set-theoretic reality, although this is not inevitable.13 A pluralistic conception of sets can be mathematically characterised in different ways, which give rise to different set-theoretic multiverses. At one extreme we have Hamkins' broad (or radical) multiverse, in which all universes are on a par: all models of any (consistent) set theory are legitimate – all provide an acceptable interpretation of the membership relation. At the other end, we have the set-generic multiverse, in which all universes are produced by a single, "core" universe and they all witness the same basic theory.14 One could also define the universes of the multiverse using a strong logic, like, for example, Friedman's Hyperuniverse (see S.-D. Friedman 2016), or define them as different cumulative hierarchies arising from several power set relations with varying definitions (see Väänänen 2014). Thus, rejecting the standard universist conception of sets, or any conception of sets according to which there is a single universe of sets, by no means settles the question concerning the nature and justification of the multiverse. In the rest of this introduction I will introduce some background notions and context behind this special issue. I begin by presenting the classic set theory Z F C (Section 2). I then describe its axioms (Section 2.1) and the corresponding set-theoretic universe V generated from them (Section 2.2). In my next step, I discuss independence results and the main reason behind the need of extending Z F C (Section 2.3). Then, I will first explain the first proposals of extending Z F C (Section 3). After that, I move on to introduce the two main side of the recent debate in the foundations of mathematics: universism (Section 4) and the multiverse conception of sets (Section 5). Finally, an outline of the contributions included in this special issue (Section 6) concludes the introduction. 2 From ZFC to the Multiverse Conception of Sets In this section, I outline the success story that is Z F C and classical set theory. First of all, I introduce Z F C , its axioms (Section 2.1), and its underlying conception of sets, viz. the iterative conception (Section 2.2). Section 2.3 then discusses the main problems afflicting Z F C and its underlying conception of sets: the so-called independence propositions, i.e. propositions that are logically independent from Z F C , such as the Continuum Hypothesis. 2.1 Zermelo-Fraenkel with Choice: The Basic Set Theory Set theory is a mathematical theory devoted to the study of sets from a formal point of view. It is the main research field for anyone interested in the foundations of mathematics, that is, for anyone interested in the logical and philosophical foundation of mathematics.15 Historically, set theory was first developed by Cantor and Richard Dedekind in the late 19th century, while trying to formalise the concepts of set of points and set of reals.16 The linguistic and formal conventions (e.g. the quantifiers) developed in the same period by Gottlob Frege,17 and the notation and the syntax developed by Giuseppe Peano,18 were soon incorporated in the theory, thus making possible a first axiomatization (by Zermelo (1908), further developed by Abraham Fraenkel (1922)). This first axiomatic system (ZF) was later enhanced by Paul Bernays, Thoralf Skolem, Johann von Neumann and Gödel, with the addition, among other things, of the Axiom of Choice (AC), thus yielding the standard set theory Zermelo-Fraenkel with Choice, Z F C .19 After this first period, characterised by an optimistic development of what was considered the foundation of mathematics, Gödel (1931) first (with the incompleteness theorems) and Cohen (1964) after him (with the proof that the Continuum Hypothesis, CH, is not provable in Z F C ) took away much of the optimism that accompanied the initial development of Z F C : set theory, as axiomatised by Z F C , was clearly not enough to uniquely pin down the structure of sets (as the Peano Axioms are thought to be pinning down the structure of the natural numbers). More recently, the further development of forcing,20 and the investigation of inner models21 and large cardinals22 gave new strength to Z F C . Consequently, at the beginning of the 21st century, there was the widespread hope that set theory, as axiomatised by Z F C , can hope again to become "the" foundation for mathematics.23 The main point of interest of set theory is the fact that, from a very simple collection of axioms, it is possible to formalise the whole mathematics, from abstract algebra to chaos theory. With the axioms of Z F C it is possible to model all of the known mathematics. Indeed, in a similar way to scientific reductionism, according to which every science is in the end reducible to physics24 Z F C is usually taken to be a partial, but accurate, description of a universe of sets that goes from the empty set to infinite sets: the cumulative hierarchy. 2.2 The Universe of ZFC: The Cumulative Hierarchy The intended universe of set theory (the universe of all sets) is usually referred to as the cumulative hierarchy, or von Neumann's hierarchy. So, how is the cumulative hierarchy built? First, we consider only one set: the empty set. This is the first level of the hierarchy.25 Then, every new finite level is constructed using the power-set operation to form a new level with all the subsets of all the sets of the previous level. When we reach a limit level, we take the union of all the previous no-limit levels. In this way, we can prove for any set that it is in a level of the hierarchy. Thus, the universe of sets V consists of (ordinal-indexed) levels, recursively defined as follows (also, cfr. Figure 1): V 0 = ∅ ; V α + 1 = P ( V α ) ; V λ = ⋃ V α ( for all α 2 | N | . And since | N | = ℵ 0 , we have that | R | = 2 ℵ 0 . In other words, CH is the claim that there is no cardinality between ℵ0 (the cardinality of N ) and 2 ℵ 0 , the cardinality of R . Thus: 2 ℵ 0 = ℵ 1 . Despite its simplicity, this statement has important consequences: if CH is false, then we can build a set S with no bijection with the natural numbers, but only with an injection into the real numbers. This means that some element (actually, an infinite amount of them) of S will always be left out and, at the same time, we would have the usual behavior between S and R (with an infinite amount of real numbers that are not elements of S), thus getting that the cardinality of S is strictly bigger than the one of the natural numbers and strictly less than the one the reals: | N | < | S | < | R | . And yet, given the iterative process of the cumulative hierarchy we know that, using only the operations used to build up V, there is no way a set such as S could be built in the first place. More generally, CH is fundamental for our understanding of infinite sets and of their arithmetical properties (this is especially true of the so-called Generalized Continuum Hypothesis, GCH, concerning sets of arbitrary size). In particular, the GCH neatly defines the exponentiation between cardinals: since the GCH states that 2 ℵ α = ℵ α + 1 , exponentiation becomes a trivial matter. On the other hand, without the GCH the general problem of the evaluation of κ λ , where κ and λ are two infinite cardinals, is still open, and we can only define the most basic properties of exponentiation.31 The Continuum Hypothesis is probably the most important and famous open problem in set theory. It was first conjectured by Cantor, in the late 19th century, and was the first of David Hilbert's Millennium Problems. The importance of the CH was immediately understood in the mathematical community, and a lot of effort was put in trying to prove it from Z F C .32 However, it soon became clear that the challenge was impossible: first Gödel proved that ¬CH cannot be proved from Z F C , and then in 1963 Cohen showed how to build a model of Z F C where the CH was true and a model of Z F C where the CH was instead false,33 thus proving that the CH was independent of the axiom of Z F C . From that moment, set theory underwent a deep transformation: the main effort was in building more of these models (universes), each different from the others, and in enhancing the canonical set of axioms Z F C so that CH would become provable in it. So what could be a solution to the continuum problem? What would be needed is either a proof of CH or of its negation. Since no such proof can be obtained in Z F C , Gödel (1947) shifted the focus to adding new axioms that imply the CH, thus proving it, or that imply that the CH is false. This is what is known as Gödel's Programme: find an axiom A, that becomes universally accepted (by being either self-evident, thus intrinsically justified, or with such good consequences that we cannot avoid including it, thus being extrinsically justified), such that Z F C + A ⊢ C H or Z F C + A ⊢ ¬ C H . Since 1947, a large number of candidate extra axioms have been proposed, many of which are mutually incompatible.34 The upshot, then, is a number of extensions of Z F C , many of which are mutually incompatible. The hard moral to draw from independence, then, is that universists who believe that there is a single, determinate universe of sets (a "single V") may just be wrong: although V clearly reflects the Z F C axioms, multiple V's, so to speak, can be seen as reflecting alternative extensions of Z F C and as satisfying set-theoretic statements left undecided by Z F C , such as CH. The independence of CH is then a problem because we don't know whether CH is true in V or not. The plethora of extensions of Z F C that settle CH all paint a very different picture of what the cumulative hierarchy looks like. Which one should we take to be the "true" one? This presents us with a choice: electing one of these theories as "the one" set theory, thus choosing a precise strengthening of V and scraping all the other theories (and universes), or building a multiverse, where all these theories are on the same "plane". 3 A Natural Extension of ZFC: Large Cardinals The first axiom candidates to extend Z F C and, hopefully, settle the independent question of CH were the Large Cardinal Hypotheses (LCs for short).35 These axioms state the existence of a particular large cardinal or of an entire class of them. A large cardinal is a transfinite cardinal number that cannot be reached by any set-theoretic operation. Their size is larger than any other cardinal number that can be defined in Z F C . Moreover, their existence cannot be proved from Z F C . Felix Hausdorff (1908) introduced the first large cardinals, the weakly inaccessible cardinals. A cardinal is weakly inaccessible iff it is a regular weak limit cardinal, i.e. iff it is equal to its cofinality36 and if it's neither a successor cardinal37 nor zero. After this first seminal definition, a vast number of new large cardinals were introduced. They are usually categorised in several different types accordingly to their properties. The most general classification divides them in "small" large cardinals (i.e. the large cardinals compatible with V = L, from the weakly inaccessible up to measurable cardinals) and "large" large cardinals (i.e. the large cardinals incompatible with V = L, from measurable upward). Other possible classification includes "combinatorial" large cardinals (between inaccessible and weakly compact), "Ramsey-like" cardinals (between weakly compact and measurable), and the "extender" large cardinals (between measurable and superstrong). The most intriguing property of LCs is that they form a linearly ordered hierarchy in which the larger large cardinals prove the existence of the smaller ones. For example, a weakly compact cardinal38 proves the existence of a strongly inaccessible cardinal,39 and the existence of that weakly compact cardinal is proved by the existence of a Q-indescribable cardinal,40 and so on (see Figure 2). However, there is a limit of how high we can go in this hierarchy: if we keep defining stronger and stronger large cardinals we end up contradicting the Axiom of Choice. The first large cardinal that does that is the Reinhardt cardinal, the first of the choiceless cardinals.41 Figure 2:The hierarchy of LCs, from A. Kanamori (2003). This hierarchy of LCs can also be seen as a hierarchy of extensions of Z F C , each of which is the result of adding a LC Hypothesis to Z F C . Given two LC Hypotheses, H1 and H2, if H1 ⇒ H2 then Z F C + H 1 ⇒ C o n ( Z F C + H 2 ) . Thus LCs can also be used to measure the consistency strength of statements and theories. This in turn gives us a hierarchy of theories, linearly ordered by their consistency strength. Gödel himself, when introducing its famous Programme to settle the CH (Gödel 1947), thought that large cardinals were a legitimate addition to Z F C – one that would eventually help deciding CH. That is, the hope was that there was a large cardinals α such that its corresponding axiom "there exists a large cardinal α", if added to Z F C , would prove CH (or its negation). Alas, this wasn't possible. As proved by Lévy and Solovay (1967), large cardinals don't settle the CH in neither direction. Robert Solovay's proof, together with Cohen's forcing method, meant the end of Gödel's Programme. Even though other axioms were proposed in the following years (e.g. Forcing Axioms and the Determinacy axioms42), none of them has been generally accepted, unlike LCs, many of which are now seen to be legitimate extensions of Z F C . So we are now left with several possible axioms that can extend Z F C , that are in most cases mutually incompatible. For example, the full Axiom of Determinacy (AD) is incompatible with AC (see Mycielski et al. 1971). So how to conceive of set theory in light of the independence phenomena? There are two main conflicting views, universism and pluralism, which I now briefly outline. 4 Universism Universism is the thesis that there is only one set-theoretic universe, namely, V. This universe is considered the intended model of Z F C and set theory (similarly, N is considered the intended models of PA and arithmetic), as opposed to all the others models, the non-canonical models, such as, for instance, the constructible universe L. The main philosophical attitude which universists may adopt to substantiate their view may be called Gödelian Platonism. According to this, in "constructing" V, we have been guided by a special form of intuition about a mind-independent realm of objects. Our mental vision, of course, is not perfect - or else, according to Gödelian platonists themselves, we'd have a completely determinate conception of V, which is something that, even by the Gödelian platonist's lights, we certainly lack. Still, Gödelian platonists typically argue, we can expect our set-theoretic intuition to become sharper in time, so as to provide us with a more exhaustive description of the realm of sets. On such a view, non-canonical models of set theory are not epistemologically on a par with "real V". The main problem with Gödelian Platonism, however, is that it doesn't seem to have the resources to tell us which of all the possible, mutually incompatible extensions of Z F C is true of the intuitive "concept of set". Indeed, Gödel thought that stronger set-theoretic axioms, based on such a concept, and possibly also extrinsically justified, would ultimately fill in the gaps in our knowledge of V (cf. Gödel 1964, pp. 476