Measurement in Set Theory

2.1 Interpreting Independence Results

Those who believe there to be a single concept of iterative set described by the axioms might explain that when doing set theory they are studying the one true universe of sets, V, described by the axioms of set theory. This view is called the universe (or Universist) view and is, historically, the default position of set theorists.

Another possibility is to take the models produced by forcing constructions to be an argument against the existence of a single, true universe of set theory. One can even adopt the perspective first put forward by Joel David Hamkins that there is not one concept of iterative set but many, each one instantiated by an equally valid universe of sets (Hamkins 2012). From this perspective, set theorists study a multiverse of universes that satisfy some or all of the axioms of set theory, no one universe fundamentally more correct or real than any other. Hamkins calls this perspective Set Theoretic Pluralism. Hamkins’s view is not the only conception of a multiverse of sets; others, including Hugh Woodin, John Steel, and Sy David Friedman have articulated multiverse theories less permissive than that of Hamkins. (Adopting a multiverse view does not commit one to a realist ontology. See, for example, S.-D. Friedman). Woodin’s Generic Multiverse is so restrictive it seems likely that Woodin actually ascribes to the universe view sometimes called Monist Platonist, according to which independent statements of set theory do have truth values and there does exist a distinguished universe V.

In addition to universe and multiverse views of set theory, a third possibility is that, in some essential way, we do not know what we are doing when we do set theory because the concept of iterative set is underdefined. Sol Feferman argues that “the continuum is neither a definite mathematical problem nor a definite logical problem” (Feferman n.d.) — and Michael Rathjen has proven Feferman’s conjecture that the continuum hypothesis is indefinite in the context of a particular semi-intuitionistic set theory (Rathjen 2016). Under Feferman’s view, set theory might not be a suitable foundation for mathematics.^[4]

In the sequel we look at the study of set theory as measurements. Under that view one can consider the abundance of models of set theory to be an artifact of our measurements rather than an attribute of the universe itself; one can also take the various universes as independently existing, real (or formal) structures. Our interpretation of certain functions as measurements is thus compatible with the universe view without forbidding a multiverse interpretation of set theory.

3.1 Measurement Types and Scales

In this section we introduce types and scales of measurement. Types classify the nature of an act of measurement while scales classify the relational structures produced by the data.

Any act of measurement is an assignment of value from one relational structure to another. Measurements are either fundamental or not fundamental according to their measurement type. Measurements that can be obtained directly from the object of interest, like measurement of length of an object by a ruler, are called fundamental. In contrast, measurement of temperature is not fundamental because we cannot measure the temperature from the temperature itself. We must rely on an intermediary, such as the rate of expansion of a metal.

Measurement can be further classified by the nature of the data it generates, called the measurement scale. Although there is not complete agreement on the possible scales, we will give here the four most standard scales discussed in any introduction to statistics.

A nominal scale assigns labels (names) to objects in a consistent way. An example might be labeling each animal on a farm with the name of their species (e.g., horse, cow, dog, etc.).

In an ordinal scale, only the ordering on labels of the data is important: 3 is less than 4, or disagree is less than neutral. Ordinal scales are used for self-reports of satisfaction with anything from a taco to a visit to the Grand Canyon. In these situations, one person might consider a given taco to be a 2 while another person considers it to be a 4. Furthermore, the distance between 3 and 4 might not be the same as the distance between 1 and 2 or between 4 and 5, even for the same taco consumer. Averaging measurements of ordinal scales, although common practice, does not carry meaning about the objects being ordered.^[5]

In contrast, the distance between the values of an interval scale is fixed and constant between values and therefore is meaningful. This is why, for example, we can convert between temperatures in Celsius and those in Fahrenheit by the familiar formula 9 5 C + 32 = F . Notice that we needed to add 32; that is because, while the interval between values is fixed, the zero-point of the scales is arbitrary.^[6]

Ratio scales, like those for measuring length, volume, and area have a non-arbitrary zero point, and one can translate from inches to centimeters by multiplying by a ratio derived from the identity 1 inch = 2.54 centimeters.

What can reasonably be done with the numbers we obtain as measurements further differentiates measurements of length, temperature, and internal states. Measurement of length is additive; the sum of two measured lengths corresponds to the concatenation of the objects being measured. Measurement of temperature is not additive: 100° Fahrenheit is not “twice” 50° Fahrenheit, and summing ordinal measurements is not well-defined.

3.2 What can be Measured? And How?

Measurement is of sufficient general and mathematical interest to have attracted the efforts of Aristotle, Euclid, Gottfried Wilhelm Leibniz, Henri Poincaré, Bertrand Russell, Hermann von Hemholz, Patrick Suppes, and many others. Mathematicians have applied mathematical thinking to clarify what could or should be meant by a measurement, to offer axiomatizations for measurement, and to make explicit the underlying claims of measurement systems. In Section 4 we propose to reverse this effort and apply the theory of measurement to offer a possible interpretation of independence phenomena in set theory.

Because we are making the unconventional assertion that we can and do measure abstract, mathematical objects, we offer the reader a very brief account of the history and philosophy of measurement as they relate to the questions of what can be measured and how. For a more detailed treatment, see, e.g., (Tal 2020).

The idea that some categories are more amenable than others to numerical description is at least as old as Aristotle. In Categories (Barnes and Aristotle 2014) Aristotle took there to be quantities such as numbers, lines, and surfaces that could be compared by equality or inequality, but not by degrees, while qualities like justice, health, hotness, coolness, and paleness admit comparison by degrees, but not by equality. Even velocity, which we now routinely quantify, was considered a quality: Without a way to measure it in terms of quantities, speed could only be more fast or less fast, but two speeds could not be equal.

The notion of when one quantity measures another is explicit in Euclid: “a magnitude — such as a line, a surface or a solid — measures another when the latter is a whole multiple of the former” (Euclid (1908) Book V, def 1 & 2). Clearly, the measurement scale Euclid has in mind is a fundamental, additive, ratio scale.

The great fourteenth-century French philosopher and mathematician Nicole Oresme challenged many Aristotelian ideas, including his dichotomy between quantities and qualities. Using a geometric argument, Oresme described changes in velocity geometrically, demonstrating that velocity could be interpreted as a quality and as a quantity (Grant 1972; Sylla 1971; Tal 2020). In so doing, Oresme showed that at least some qualities could be meaningfully discussed as quantities.

Leibniz took this idea further: His “principle of continuity” claims that all natural changes are produced by degrees. It says that “nothing jumps in nature and that one thing cannot pass from one state to another without passing through all the other states that can be conceived of between them” (d’Alembert and Glaus 2008). Leibniz’s law of continuity seems to have influenced Kant, who parceled magnitudes into extensive and intensive. Length is the prototypical example of an extensive magnitude — one “wherein the presentation of the parts makes possible (and hence necessarily precedes) the presentation of the whole” (Kant 1998, p.234, A163/B203).

Just as Oresme’s work with velocity subverted Aristotle’s clear distinction between quality and quantity, the scientific developments of the nineteenth century undermined Kant’s distinction between intensive and extensive. Temperature, color, and even electrical conductance and resistance were broken down into pieces by theoretical and technological advancements, in spite of their not being composed of spacial or temporal parts; suddenly temperature was a question of velocity and color was one of wavelength.

Two more recently developed perspectives on measurement are the Representational Theory of Measurement (RTM) advanced by Patrick Suppes (and others) and model-based accounts of measurement. The essential idea of RTM is this: “A measurement scale is a many-to-one mapping — a homomorphism — from an empirical to a numerical relational structure, and measurement is the construction of scales” (Tal 2020). RTM was a response to the widespread effort to import ideas of measurement from the physical sciences into psychology. In their chapter in the Stevens’ Handbook of Experimental Psychology (Luce and Suppes 2002, p. 1), Luce and Suppes explain their motivation:

…by the 1940s a number of physicists and philosophers of physics concluded […] correctly, that the classical measurement models were for the most part unsuited to psychological phenomena. But they also concluded, incorrectly, that no scientifically sound psychological measurement is possible at all. In part, the theory of representational measurement was the response of some psychologists and other social scientists who were fairly well trained in the necessary physics and mathematics to understand how to modify in substantial ways the classical models of physical measurement to be better suited to psychological issues.

Model-based accounts of measurement emphasize that there are two levels at which a measurement is determined. The first is the actual process of interacting with the environment through measuring devices, and the second is the abstract framework — the model — in which we interpret the results of the first. “Model” in this context is used in the general mathematical and scientific sense rather than the logical one.

Model-based accounts of measurement recognize that one’s choice of model and modeling assumptions impact what we observe. They also suggest that the act of measurement does not read off an independently existent property from an object, but instead provides information that increases our knowledge about the object (Mari, Wilson, and Maul 2023).

Our discussions of measurement in set theory in Sections 4 and 6 emphasize RTM and model-based accounts of measurement.

4.1 Cardinality

Colloquially, one refers to the cardinality of a set as the “size of the set,” and we intuitively expect this type of measure to share properties with other measurements of size like length, area, and volume. For two sets A and B to have the same size means that there exists a bijection between A and B. The bijection pairs off the elements from one set with elements of the other in such as way that no element is left over from either set. A bijection between sets A and B demonstrates that the given sets have the same size, but it does not allow us to say the size of either set is something or another.

Naturally, if there is a bijection between sets A and B and between sets B and C, there is also a bijection between sets A and C. Because of this transitivity, the sizes of sets form equivalence classes. One could choose an arbitrary representative from each equivalence class to denote the size of the sets in that class, but a better generalization of counting is to use a distinguished class of objects, the cardinal numbers, as the representatives. A cardinal number is typically defined as an initial ordinal. For completeness, we briefly define ordinals and cardinals now.

In everyday life we count by enumerating: When we point at a plate of cookies and count one, two, three we are identifying a first, second, and third cookie. We consider the smallest (positive) natural number that has not yet been used, one, and assign it to the first cookie. To proceed we again assign the smallest natural number that has not yet been used, two, to the second cookie. Repeating this process until we run out of cookies, we say that the number of cookies is the last number we assigned.

For the natural numbers, a number’s position when counting (its place in the ordering) and its size (the number of elements it can count) coincide. For infinite sets, this is no longer the case, and we need to distinguish between an element’s position in the list of ordinals and its size. We consider ordinals first.

To extend the idea of assigning the “least numeral” to infinite sets, we need a linearly ordered class of sets to use as labels, and that class needs to have the property that every nonempty subset has a least element. The ordinals form exactly such a class.

There are many equivalent definitions that characterize when a set X is an ordinal; we’ll use the von Neumann ordinals. Identify 0 with the empty set, ∅. The set containing the empty set {∅} is 1, the set containing 0 and 1, namely {∅, {∅}}, is 2, and so on. Then the successor of an ordinal α, denoted by α + 1, is α ∪ {α}; whenever we have a set X of ordinals such that for every α in X, the successor of α is also in X, the first ordinal greater than every member of X is the limit ordinal ∪ X. We denote by ω the least ordinal greater than every (finite) natural number.

Cardinality can be described as a map from the class of all sets to the class of all cardinal numbers where the cardinality of a set A, denoted by |A|, is the (unique) least ordinal in the equivalence class of sets bijective with A. That initial ordinal is a cardinal.

Although each ordinal has one and only one cardinality, many ordinals can have the same cardinality; the cardinality map from the ordinals onto the cardinals is not one-to-one. For a measure of size, that’s not an alarming property. Many shoe boxes have the same length; many ordinals have the same cardinality.

Another property shared by cardinality and other measures of size such as length, area, and volume is additivity. Without loss of generality, assume sets A and B are disjoint, |A| = α, and |B| = β. Suppose we concatenate A and B by forming the union |A ∪ B|. If the cardinalities of both A and B are finite, then the cardinality of the their union is the sum of their cardinalities: |A ∪ B| = α + β = |A| + |B|. Now suppose α, the size of A, is less than or equal to β, the size of B, and that either α or β is infinite. Then the cardinality of their union |A ∪ B| is again equal to the sum of their cardinalities, α + β, which in this case is equal to β. Therefore, cardinality is additive. From here, however, the parallels with length become weaker.

In the context of forcing over a given ground model, both the ground model and the forcing extensions use their own sets to determine the size of a given set X. Consider a forcing that collapses cardinals and a set Y that is uncountable in the ground model M and countable in the extension M[G]. If the size of Y was not ℵ₁ in M, then the other uncountable cardinals smaller than |Y| in M are also collapsed to ℵ₀ in M[G]; there is no rational scaling factor, or indeed anything analogous, that can recover in M[G] the fact that those sets had different cardinalities in M. This suggests that each measurement of cardinality is meaningful only in the model in which it takes place.

4.2 Order Type

The cardinality of a set depends only on how many elements are in the set, not on any internal structure it might have. One could, however, reasonably wish to classify internal structure of sets as well. Intuitively, the order type of a set describes the properties of its order relation, and two sets have the same order type if and only if there is an order-preserving bijection between them. Specifically, two ordered sets A and B have the same order type if and only if there exists an order-preserving bijection f from A to B such that for any two elements x and y in A, if x ≤ y holds in A, then f(x) ≤ f(y) holds in B.

Although the order type of more general sets is of mathematical interest, we will restrict our conversation to well-ordered sets. A well-ordered set is a totally ordered set in which every non-empty subset has a least element. Preservation of order and bijection are transitive, so the order types of well ordered sets form equivalence classes. Every equivalence class of well-ordered sets has a unique ordinal number representative. Call that ordinal α the order type of the well-ordered set X.

In the physical universe of finite objects, the ideas can be arranged in a line of seven ordered objects and contains seven objects coincide, so there is no distinction between order type and cardinality. For infinite sets, the two ideas are distinct. Adding finitely many elements to an infinite set doesn’t change the cardinality of the set, but it can change its order type. Suppose we construct a set of order type ω + 1 by adding a maximal element to a set of order type ω; the cardinality of the new set is still ℵ₀, but we can no longer construct an order-preserving isomorphism between ω and ω + 1. On the other hand, if we take ω and add one element at the start of the ordered list, we get 1 + ω. There is an order-preserving bijection between 1 + ω and ω that maps the least element of ω to the least element of 1 + ω. This noncommutativity holds in general, with sums of increasing numbers of ordinals having an increasing number of possible values.^[7]

4.3 Cofinality

Cofinality describes another aspect of the structure of an ordered set A. Given a subset B of A, say informally that B is cofinal in A if B contains elements as large as any element of A. That is, for every element a in A, there is an element b in B such that a ≤ b. The cardinality of the least such set B is the cofinality of A. Another description (equivalent if we allow the Axiom of Choice) is to say that the cofinality of a set A is the least ordinal β such that there is a function f from β to A with cofinal image. Specifically, for every a in A, there is an x in β with a ≤ f(x). In the sequel we will restrict our discussion to cofinalities of ordinals.

The cofinality of A is at most the cardinality of A, and many sets and many ordinals have cofinalities strictly smaller than their cardinalities. For example, ω _ω has cardinality ℵ _ω but confinality ω, as witnessed by the function f from ω to ω _ω mapping n to ω _n . For cardinals, the relative sizes of the cardinality and the cofinality of the set identify an important property: A cardinal κ is regular if it is equal to its own cofinality; otherwise it is singular.

Cohen’s method of forcing shows that ZFC is consistent with 2 ℵ 0 = ℵ n for any positive natural number n; note that these are all regular cardinals. Indeed, cofinality considerations offer one of the few limits on the possibilities for the continuum. While cofinality can be changed by forcing, with sets of uncountable cofinality in the ground model having countable cofinality in the forcing extension, the continuum 2 ℵ 0 cannot be forced to be the singular cardinal ℵ _ω .

4.4 Properties of Measurements of Cardinality, Cofinality, and Order Type

In our everyday speech, we talk about the cardinality, the order type, and the cofinality of a set in a way that suggests that we expect a given set to have exactly one measurement of each of these attributes. However, properties witnessed by functions are vulnerable to changes by forcing, suggesting that these measurements are not fundamental.

Recall that fundamental measurements can be read from the object itself without an intermediary. Initially one might expect that it is possible to measure the cardinality, order type, and cofinality directly from a given set and that these measurements would be resilient to changes is measurement tools and observers in ways similar to the measurement of length of ordinary objects under ordinary circumstances: Except in extreme conditions, the length of a metal rod is unchanged by its environment, and the measurement of that length is independent of the environment and measurement system (up to the tolerance of the instruments and the skill of the people operating them).

The properties of sets discussed in this section do not demonstrate such resilience. Within a given model a set has one and only one cardinality, order type, and cofinality (even if we don’t know what they are), but those values might be different in an inner or outer model. In scientific measurement, we expect that if we are using two different proxies for temperature (such as the expansion of two different metals), we will still arrive at the same measurement. If we didn’t, we would reject one or both of the proxies. On the other hand, we expect the distance between two points to vary with (but not within) the geometry in which we've chosen to measure it. In general we have a sense of which geometry is appropriate for a given question, and we consider a correctly computed measurement in a different geometry (called a measurement-model below) to be incorrect or inapplicable. Here set theory is unlike geometry, where we have highly attuned intuition based on millennia of cumulative embodied experience.

In the broader scientific community, one tries to find models that fit phenomena of interest. These models, even when excellent for their specific application, are not always compatible with one another. General relativity and quantum mechanics are a paradigmatic example: Both seem correct where they are intended to apply, and neither can be meaningfully interpreted in the other. In this sense, we cannot say that either is a model of physical universe in its entirety.

One way to accommodate a Universist or Monist Platonist perspective is to view the models of set theory as scientific models — approximations of the true universe of sets that we perhaps don’t yet (and may never) have the knowledge to measure. Such a stance would permit that the study of models of set theory is itself of interest without demanding a commitment to the ontological status of said models. It would also allow one to adopt new axioms that settle questions like the continuum hypothesis without rejecting the diversity of models of set theory. The array of multiverse views can also be articulated in this framework.

Mathematicians have thousands of years of experience with addition and multiplication of natural numbers. On the strength of that experience and its correspondence to the ordinary world, we are confident that we can discern the standard model of arithmetic from the nonstandard ones. The existence of nonstandard models of the theory of arithmetic does not necessitate a multiverse of models of arithmetic on the natural numbers, all of them equally ontologically real. Instead, one can (and some do) accept the uncountably many non-isomorphic nonstandard models of arithmetic as consequences of the machinery, as mathematically interesting in their own right, and as useful for understanding the limits of what can be specified in the formal language in which we do mathematics.

Before proceeding, we make a small remark on the word “measureable” in set theory. Descriptive set theory explores the deep connections between measures in analysis that generalize the protypical additive ratio measures of the physical world, like those of length and area, and various descriptions of the size of a set. Two different measures might classify the same set with different values. For example, one measure might classify an interval of real numbers with its length while another measure might classify it with the cardinality of the set of points in the interval. Measurable cardinals are those that admit a two-valued measure with certain properties; essentially they classify all subsets as either large (measure 1) or small (measure 0). Though one could consider such a classification a measurement, attempting such an account is not within the scope of this paper, and the possibility of such an account does not alter the philosophical observations about measurement in set theory made in the coming sections.

In the sequel we will call the perspective of a set theorist (including but not limited to the axioms, models, and tools such as forcings they choose to use) their measurement-model. By this we mean the model (in the sense of scientific model) they are using to conduct their measurements. Our basic contention is that the properties of sets that we measure are not independent of the measurement-model in which they take place. They depend on the instrumentation (e.g., axiom system and models) used to measure them, which are themselves a reflection of the relative position of the observer.

6.1 What Qualifies as a Concept of Set?

The mathematical universes created by forcing (and other model construction techniques) do agree, by and large, on classical mathematics. Hamkins argues that, “since one expects to find all the familiar classical mathematical objects and structures inside any one of the universes in the multiverse” (Hamkins 2012), set theory can still offer a realist foundation for classical mathematics — even if we accept all of these universes as equally extant, real, and valid. Though they all exist, Hamkins notes, “we may prefer some of the universes in the multiverse to others, and there is no obligation to consider them all as somehow equal” (Hamkins 2012).

The expansive multiverse view of Hamkins is maximally accommodating of mathematical practice. It asserts, in some sense, that if a mathematical model feels real — namely, if it was constructed using what we recognize as valid mathematical methods — then it is real. Under this view, there can be no classical resolution to the continuum hypothesis of the sort longed for by David Hilbert (1902). The value of the continuum is the spectrum of possibilities it satisfies in all the possible worlds satisfying the hypotheses of set theory and nothing more.

And what are these universes of set theory? Hamkins suggests that they are instantiations of different concepts of set, remarking in his groundbreaking paper The Set-Theoretic Multiverse (Hamkins 2012) that for the purposes of that article he “shall simply identify a set concept with the model of set theory to which it gives rise.”

Although Hamkins does not provide an argument in favor of this interpretation in that paper, we give one against it here and suggest it is more plausible that the multitude of models generated by forcing arguments are an artifact of mathematical tools we use to measure sets than that they constitute so many distinct concepts of set. We will subsequently consider what coherence, if any, it is reasonable to ask of said models.

Mathematicians and philosophers have grappled with identifying different concepts of set and their consequences for 150 years. In his recent book Conceptions of Set and the Foundations of Mathematics, for example, philosopher Luca Incurvati illustrates how different conceptions of set give rise to mathematically and philosophically discernible distinctions (Incurvati 2020).

To diagnose the origins of the paradoxes of naïve set theory and distinguish the concept of iterative set was significant mathematical and philosophical work. For many working set theorists, these are the only two concepts of set they know. What, then, would it mean for every model of set theory to be its own instantiation of a concept of set? It seems unlikely that each is an identifiable concept of set in the same way as the naïve and iterative concepts of set. Furthermore, there is no evidence that this is the case. There are not, for example, some small number of the universes of set theory that do give rise to distinct concepts of set in this sense and thereby justify the possibility that they might all do so.

Much more likely is the possibility that Hamkins has something different in mind when he suggests that these models can be taken as instantiations of different concepts of set. The question is: what does he mean by concept of set? The Wittgensteinian idea that a concept of set is what a concept of set does could be interpreted as a kind of formalism rather than a realist account of the inevitable consequences of the concept of iterative set. Formalism is clearly not what Hamkins has in mind, however, especially as he offers set theoretic pluralism as a realist foundation for mathematics. We proceed under the assumption that there is not sufficient evidence to consider the models of set theory to be models of distinct concepts of iterative set. In what follows, we consider them to be models in the scientific sense: approximations in which we can perform and analyze measurements.

6.2 The Multiverse as a Foundation for Mathematics

Perhaps all we really need for a multiverse to provide a foundation for mathematics is agreement on the Π 1 0 sentences that describe what Steel calls “concrete mathematics” (Steel 2014) and protection of some sort against forming questions that appear mathematical but are actually gibberish. One could consider that a prerequisite to saying the same thing is using the same language; Steel’s multiverse project addresses these syntactic issues.

Steel works to build a unifying language and a unifying framework to “maximize interpretative power” and provide a common ground for the development of all current and future mathematics (Steel 2014, p. 2). Rather than be content with agreement on classical mathematics, Steel proposes to build a framework that can serve all the universes in the multiverse using the large cardinal hierarchy. Whether we lose expressive power when moving from the language of set theory to the multiverse language and whether there is a unique world called a core (similar to a universe) that must be contained in every other world in the multiverse are important questions.

A Monist Platonist might view the forcing extensions as (scientific) models of a measurement giving information about a single true universe of set theory, V. To make the (scientific) modeling assumption that convergence of those models accords with proximity to truth in V is akin to positing the existence of a core of the multiverse, namely that universe onto which they are converging.

Hugh Woodin’s generic multiverse is the collection of possible universes of sets generated by closure of any given model of set theory under generic extensions and refinements (Woodin 2011). A sentence is true in the generic multiverse if and only if it holds in every universe of the multiverse generated by forcing over V.^[9] Then, assuming a proper class of Woodin cardinals and that the Ω-Conjecture is true, Woodin argues that the generic multiverse position reduces truth about the universe to truth in a small fragment of the universe (Woodin 2011, p. 2).

For very interesting technical developments beyond the scope of this paper, we refer the interested reader to the work of Gunter Fuchs, Hamkins, and Jonas Reitz in set theoretic geology (Fuchs, Hamkin, and Reitz 2015), to Toshimichi Usuba’s recent result that the existence of an extendible cardinal^[10] implies that the mantle is a ground model of V (Usuba 2019), and to Gabriel Goldberg’s proof that the assumption of an extendible cardinal in Usuba’s result cannot be weakened (Goldberg 2021).^[11] The aforementioned body of work speaks directly to the possible existence of a core in the sense of Steel, but there is not yet a generally agreed upon interpretation of the results, or of Usuba’s theorem in particular. Some, such as Steel, see the results as evidence of a core but have certain reservations (for details, see Steel (2024)), others, Hugh Woodin in particular, consider Usuba’s result definitive proof that the generic multiverse does have a core (Woodin 2023).

6.3 Moderate Multiverse Views

Other multiverse views fall somewhere between those of set theoretic pluralism and the generic multiverse.

Peter Koellner, who refers to the multiverse view as the pluralist view and the universe view as non-pluralist view, posits that one can be a non-pluralist with respect to one theory and pluralist with respect to another. For example, citing a view he ascribes to Feferman, he asserts that “many people would agree that a Π 1 0 statement like the Reimann Hypothesis has a determinate truth value but CH does not” (Koellner 2013). He describes a hierarchy of positions one might take, being non-pluralist up to a certain theory in the interpretability hierarchy and a pluralist beyond it. For example, one could be a non-pluralist about ZFC but a pluralist about CH.

Neil Barton has written extensively on various forms of potentialism (which we will define shortly), on the question of whether there is a single, maximal universe of sets or a multiverse of sets, and on challenges posed to the universe view by the seeming mathematical reality of forcing extensions, especially class forcing extensions of V. He also offers ways a Universist might interpret certain forcing extensions in the context of forcing over a countable transitive model of ZFC (Barton 2020).

Caroline Antos, Sy-David Friedman, Radek Honzik, and Claudio Ternullo describe possible positions on the multiverse in terms of whether one considers it possible to add anything to V (Antos et al. 2015). They call a person that considers V to be an actual (maximal) object that cannot be modified an actualist. According to this position, any perceived changes to V (such as extension by forcing) are actually occurring inside V. A potentialist position asserts that V can never be fully fixed, even though there are unchangeable portions of V. These positions are not absolute: A person could be an actualist with respect to the height of the universe and a maximalist with respect to width, for example. Height actualism is the position that the height of V is fixed. Height potentialism asserts that it is possible to add new ordinals to V. Width actualism, similarly is the position that no new subsets can be added to V, while width potentialism allows that new subsets can always be added. They argue for a hybrid position that is actualist in width but potentialist in height, attributing that basic idea to Zermelo (1930). That position serves the development of the Hyperuniverse Programme (see Arrigoni and Friedman (2013)).

The multiverse conceptions mentioned here share an expectation that all universes in the multiverse have some fragment in common; they differ on the extent and nature of that overlap. For a more robust understanding of the set theoretic pluralism put forth by Hamkins, see Hamkins (2012); for Steel’s multiverse, see Steel (2014), Maddy and Meadows (2020), and Steel (2024); for more on Woodin’s generic multiverse, see Woodin (2011); for an account of arguments Woodin has made against the generic multiverse, see Meadows (2021). For a general discussion of set theory as a foundation of mathematics in light of multiverse arguments, see Maddy (2017).

We propose that interpreting the plethora of models of set theory as scientific models of the concept of set is compatible with the universe view without forbidding a multiverse interpretation of set theory.