More than a Decade in the Set-Theoretic Multiverse

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 models^[21] and large cardinals^[22] 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 physics^[24] 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):

The construction of V seems to be very aptly reflected (and intuitively motivated) by the iterative conception of sets, that is, the idea that all sets are formed in stages, through iterating the Power-Set and Union operations as far as possible.^[26]

Figure 1:

The cumulative hierarchy.

One natural question at this point concern the nature of V. Given the above construction, it is clear that V contains all sets. But is it a set? If it were a set, it would be the universal set. But this is impossible, as we know from Russell’s Paradox. Suppose that V is indeed the universal set, the set of all sets. Since the axioms of set theory that I presented in the last section state that no set can be member of itself, we can define the universal set as the set of all sets that are not member of themselves. But we said that V is the set of all sets. Thus, it cannot be member of itself. But if this is the case, then it satisfy the condition of being member of V, and thus it is indeed member of itself. But if it is member of itself, then it must not be member of itself, by definition. But this is a contradiction, since we derived that V is member of itself from the premise that it is not, and that it is not member of itself from the premise that it is a member of itself. Consequently, V is not a universal set.

Given this conclusion, there are two different ways to avoid the paradox. The first is to consider a principle of limitation of size, according to which V is too big to be a set and is thought to be a proper class instead.^[27] Classes are collection that can have members, but that cannot be member of anything else (as opposed to sets, that can have member and can be elements of other sets). Thus, we can have the class of all sets without incurring in Russell’s Paradox. However, in doing so, we also need to regiment the use of classes. This leads to development of what is known as “class theory”. In class theories, we add the possibility of quantifying over proper classes. To avoid the paradoxes and regiment the use of classes, some axioms are also added (mainly some version of a Class Comprehension Axiom). This gives rise to several different axiomatization of class theory. The two main examples of class theories are von Neumann-Bernays-Gödel class theory (NGB) and Morse-Kelley class theory (MK).^[28]

Both NBG and MK class theories are very powerful, but neither has been universally accepted by set theorists, for both mathematical and philosophical reasons. In particular, many set theorists are reluctant to accept the ontological commitment to classes. The problem lies in the fact that, as soon as quantification over classes is permitted, through Cantor’s diagonalization is possible to show that there is no Universal Class. Thus we are in a similar position as set theory and its antinomies: the supposed Universal Class cannot be a class, since it is too big. So we could assume the existence of “super-classes”, and then “super-super-classes”, and so on. However, there is little advantage in doing this, since in set theory we never need to quantify over classes.

The other way to avoid Russell’s Paradox is to limit the applicability of the Comprehension Schema. The axiom, in its full power, allows us to form a set from any stated property. This is needed to define the universal set as the set of all sets that are not member of themselves. It is possible however to restrict the Comprehension Schema in such a way as to avoid the paradoxes. This is the most common way to approach Russell’s Paradox. The restricted version of the Comprehension Schema is the Axiom Schema of Separation: it states that we can form a set of elements that satisfy a given property if and only if they are already elements of another set. In this way, we avoid the possibility of forming a universal set with a given property, since we need to refer every time to another set.

As I mentioned on p. 6, philosophically, there are two possible ways to approach the process of filling up the stages: potentialism and actualism. According to potentialism, the hierarchy is never completed, i.e. we cannot take it to be a completed entity and, for this reason, it never contains all sets. On the other hand, according to actualism the whole structure is already completed, i.e. it contains all possible sets. We can mix and match these two interpretations:

1. Radical potentialism: one can add both subsets (so adding to the width of the hierarchy) and ordinals (so one can always increase the height of the hierarchy);

2. Radical actualism: both the width and the height of the hierarchy are fixed;

3. Width potentialism/height actualism: one can always add new subsets, but not new ordinals (so the height of the hierarchy is fixed, while the width is not);

4. Height potentialism/width actualism: one can always add new ordinals, but not new subsets (so the width of the hierarchy is fixed, but its height isn’t).

The story so far has much to recommend: we have an axiomatic system, Z F C , strong enough to represent virtually every currently accepted mathematical theory and object, with a strong motivation behind it, the cumulative hierarchy. At first glance, then, it seems very natural to think of set theory as the theory of one single entity, the proper class V, whose members are the sets. Sadly, however, there is a fly in the ointment: Gödel proved that any system that interprets a modicum of arithmetic (and Z F C is certainly one such system) is incomplete, in the sense that it does not decide all sentences formulated in its language, such as, for instance, the consistency sentence of Z F C , i.e. a sentence to the effect that Z F C is consistent. Any system satisfying Gödel’s minimal conditions, can be seen to be in some sense inadequate. However, it was still believed that set theory was powerful enough to prove any other mathematical result (other than its own consistency). The blow against this belief was struck during the sixties: it was shown that the most important open question in set theory, the Continuum Hypothesis, cannot be solved within Z F C . Even worse, there are many mutually incompatible theories extending Z F C , that either validates CH or its negation (as we will see in more detail in the next section).

Moreover, it is not only such incompleteness problems that undermine the uniqueness of the universe of set theory V. It is also possible to define completely different universes that still satisfy Z F C . One of such different universes is the constructible universe L, first introduced by Gödel (1947). The main difference between V and L lies in what sets are we adding at each stage. While in V we are adding every possible set at each stage, in L we can only add sets that can be constructed from sets that already present at some previous stage. This gives us a much thinner universe, the first inner model of Z F C . Finally, since Z F C is a first-order theory, it is subject to the application of Löwenheim-Skolem Theorem: if a theory T has a model of infinite cardinality,^[29] then it has a model of each possible cardinality. Consequently, Z F C has not only uncountable models (and this seems intuitive, since it proves that the set of all real numbers is uncountable), but also countable models!

Now, while there is a widespread consensus that Z F C is justified by the iterative conception of sets and membership V, the same kind of intuitive justification does not automatically apply to extensions of Z F C , or to L. These different universes, on the contrary, seem to be justified only indirectly (extrinsically, to use Gödel’s terminology), that is, in view of their consequences.^[30]

2.3 The Continuum Hypothesis and Its Independence

The Continuum Hypothesis is probably the main reason behind the development of extensions of Z F C and of the multiverse itself. In this section, I discuss CH and also briefly present the current status of research on the matter.

The CH is a claim about the size of sets: it states that there is no set with more elements than the set of natural numbers N and fewer elements than the set of real numbers R . That is, since the cardinality of R , in symbols | R | , is 2 elevated to the cardinality of the natural numbers, 2 | N | , we have that | R | > 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:

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”.

5.1 The Philosophical View: Pluralism

Pluralism in set theory has long been advocated, and defended, by Hamkins (see especially J. Hamkins 2012). According to such a conception, there is no such thing as a single, true theory of sets instantiated by a single canonical model. Such a situation is made impossible by the independence phenomenon: any theory of sets can be extended in several, mutually incompatible ways, each one with its own collection of models. Likewise, its supposed canonical model can also be extended in incompatible directions. As Hamkins writes:

[T]he most prominent phenomenon in set theory has been the discovery of a shocking diversity of set-theoretic possibilities. Our most powerful set-theoretic tools, such as forcing, ultrapowers, and canonical inner models, are most naturally and directly understood as methods of constructing alternative set-theoretic universes. A large part of set theory over the past half-century has been about constructing as many different models of set theory as possible. [(J. Hamkins 2012, p. 418)]

Faced with such an embarrassment of riches, with countless universes and theories to choose from, it is impossible to pin down the one true theory of sets, with its associated canonical model; we can only settle for a plurality of theories, models and truths, without trying to determine the “truest” between them. According to Hamkins, all set theories and their models are part of the multiverse. This gives us the most variety of possible universes and theories. And, crucially, all such possible universes have equal status. That is, no universe is metaphysically privileged. Moreover, there is no criterion to select between universes, so every imaginable universe is part of this kind of multiverse. In Hamkins’ view, the main job of set theory is to deal with different kinds of constructions, which can verify, or falsify, the same set-theoretic claims (such as CH). Consequently, there is no reason to ban a particular universe: they are all perfectly legitimate model theoretic constructions. Such a view is clearly reminiscent of Poincaré’s reflections on geometry, quoted above. And, indeed, Hamkins explicitly states that his goal is to transform set theory in a “geometry-like” theory:

There is a very strong analogy between the multiverse view in set theory and the most commonly held views about the nature of geometry. […] At first, these alternative geometries were considered as curiosities, useful perhaps for independence results, for with them one can prove that the parallel postulate is not provable from the other axioms. In time, however, geometers gained experience in the alternative geometries, developing intuitions about what it is like to live in them, and gradually they accepted the alternatives as geometrically meaningful. Today, geometers have a deep understanding of the alternative geometries, which ar regarded as fully real and geometrical. The situation with set theory is the same. […] Like the initial reactions to non- Euclidean geometry, the universe view regards these alternative universes as not fully real, while granting their usefulness for proving independence results. Meanwhile, set theorists continued, like the geometers a century ago, to gain experience living in the alternative set-theoretic worlds, and the multiverse view now makes the same step in set theory that geometers ultimately made long ago, namely, to accept the alternative worlds as fully real. [(p.425 J. Hamkins 2012, p. 426)]

Such a view equates the independent statements of set theory to Euclid’s Fifth Postulate, or to the Commutative Axiom from group theory: they are not essential to the nature of what geometry or group theory is, but only one of the properties that a geometry or a group can have. In set-theoretic terms, this means that it doesn’t matter what the “real” truth value of the CH is, or what axioms we can add to Z F C to settle it. We have already settled it by knowing in which universes CH holds and in which ones it doesn’t. The satisfaction of CH (and the size of the continuum) becomes just one of the features of a universe: for example, we might have universes in which the continuum has size ℵ₁ (because CH here is satisfied), or ℵ₂ (because, in such a universe, PFA holds instead), and so on.

Philosophically speaking, this view is quite appealing and intuitive. It solves the main problem of independence in an elegant way, and it also opens up several new avenues of research. For example, the recent contributions to the modal theory of forcing or set-theoretic geology were both spearheaded by the pluralist idea.^[44] However, the question arises how to mathematically specify a multiverse framework – a unified, pluralist theory of sets. Indeed, pluralism in the foundations of mathematics can be instantiated in a number of radically different ways. Each one of these characterisations, i.e. each set-theoretic multiverse, is unique and comes with its own particular variety of pluralism.

5.2 The Mathematical Characterisation

From a philosophical point of view, we can classify the various types of multiverses by their commitment to the ontological existence of the universes. More precisely, we can recognise two main views:

1. a realist multiverse, committed to the full existence of the universes that form it (e.g. Mark Balaguer’s full-blooded Platonism^[45]);

2. an anti-realist multiverse, that isn’t committed to the platonic existence of the universes (see e.g. Shelah 2003).

1. the broad (or radical) multiverse, formed by all possible universes without any restriction (see e.g. J. Hamkins 2012);

2. a generic multiverse, where new universes are built using forcing; we can then have several variants of the generic multiverse, like Hugh Woodin’s set generic multiverse that uses the very strong Ω-logic as its background logic (W. H. Woodin 2011b), or Steel’s set generic multiverse, that supposes the existence of a core (Steel 2014);^[46]

3. a parallel multiverse, where all the universes instantiate a different powerset operation (Väänänen 2014);

4. the Hyperuniverse (HP), that is, the collection of all the countable transitive models of Z F C (see Arrigoni and S.-D. Friedman 2013).

At a general, informal level, a set-theoretic multiverse can be described as a collection of models of set theories. Given some consistent set theories T, T', ..., the models of T, T', ..., are part of the multiverse. Each model (intended in the usual way of a domain together with an interpretation function) represents a set-theoretic universe. Consequently, each consistent theory determines at least one universe in the multiverse. In the multiverse one can find models of more than one theory. For example, one multiverse can be defined as the collection of all countable transitive models of Z F C , thus in that multiverse one can find universes from the models of, say, Z F C , Z F C + ¬ C H , Z F C + C H , Z F C + V = L , etc. These models jointly capture all there is to know about sets. That is, each model is a legitimate universe of set theory and, as a consequence, there is no Single Universe. According to proponents of the multiverse, this lack of unity cannot be repaired in any way and set theory is precisely the study of all these alternative universes, in which the properties of sets can vary greatly from one to another.

Such a conception of the multiverse is semantic. There might be several, mutually incompatible models of Z F C . The set-theoretic multiverse is a collection of width extensions V[G] of V such that V[G] satisfies a statement φ, that’s indeterminate in V (CH is the prime example of such a φ). In most cases, these models are produced by the application of forcing. Thus, there is a clear connection between pluralism and potentialism: in order to be able to take the collection of all extensions of V, we need to assume that V is indeed extendible. And since these extensions are usually produced through forcing, and with forcing we add subsets to V, width potentialism is a needed assumption. It is also possible to assume height potentialism, and include in the multiverse also height extensions of V (albeit multiverses that do this are much rarer, and multiverses that include only height extensions haven’t been developed in the literature). Radical potentialism (that allows extensions both in height and width) is obviously also present in the several possible characterisations of the multiverse. Nevertheless, width potentialism remains the most common view. This is not to say that an actualist pluralism is in principle impossible. It would be perfectly fine to defend the view that there are several set-theoretic universes that cannot be extended neither in height nor in width. According to this view, one cannot use forcing to produce new extensions of the universe, thus one would need a procedure to generate these different universes. This hasn’t been really explored in the literature, but a possible way to approach such an actualist multiverse would be to define each universe as generated by different powerset operations (in a way similar to Väänänen 2014). On the other hand, actualism is much better connected to universism: if the universe is unique and the intended model of Z F C , then assuming that it is also un-extendible is a very natural step.

Truth in the set-theoretic multiverse is usually conceived in a supervaluationist way. First, truth is defined relatively to a universe: that is, a sentence φ will be true-in-U or false-in-U, where U is a universe of the multiverse. Then, define a notion of truth across the whole multiverse: a statement φ is True iff it is true-in-U for all U in the multiverse (and False if it’s false-in-U for all U in the multiverse). A sentence φ is determined if it has a truth value across the whole multiverse: so, if and only if for all U in the multiverse, φ is either true-in-U or false-in-U. If a statement is True (or False) then it is also determined in the multiverse (but not the other way around). For example, if we take the multiverse to be the collection of all models that settle CH, we have that CH is determined in the multiverse, since it will have a truth value in all universes of the multiverse. However, it will be neither True or False in the multiverse, since there will be both universes in which CH is true-in-U, and universes in which it’s false-in-U. On the other hand, if we take the multiverse to be the collection of all countable transitive models of Z F C , then CH will be neither True (or False), nor determined, since there will also be universes in the multiverse in which CH is not settled.