Truth in a World of Post Truth: 3

In classical set theory, the distinction between sets and classes is crucial. This can be confusing at first, especially because in standard Zermelo–Fraenkel set theory (ZFC), the most commonly used foundation for mathematics, only sets officially exist in the theory. 

A set is a collection of objects that can be elements of other sets. For example, the set {1,2,3} contains three elements and can itself be an element of another set. A class, on the other hand, is a collection that is defined by a property, like ‘the class of all sets’ or ‘the class of all ordinals.’ Some of these collections are too large to be sets without causing paradoxes. For instance, if we tried to collect all sets into a single set, we’d run into contradictions like Russell’s Paradox. So we treat these very large collections as proper classes, which cannot be members of any other collection. 

In ZFC, this distinction is informal, classes are just a way of speaking outside the formal theory. But other systems of set theory have been created to formalise the concept of classes alongside sets. Two of the most important are von Neumann–Bernays–Gödel (NBG) and Morse-Kelley (MK) set theory. In NBG, we expand the language of set theory to explicitly include both sets and classes. Sets are defined as classes that are members of other classes. Proper classes, by contrast, are too big to be members of any class, including themselves. The advantage of NBG is that we can now talk about things like ‘the class of all sets’ or ‘the class of all ordinals’ within the formal system without contradiction. 

Morse-Kelley set theory goes even further. While NBG only allows formulas with limited quantification over classes, MK allows full quantification over both sets and classes, making it a more powerful system, though also less conservative in its assumptions. All of these systems, ZFC, NBG, and MK, still operate within what we call the universe of set theory, usually denoted by V. This universe is imagined as a vast hierarchy of sets, built up in levels, where each level contains sets made out of the sets from earlier levels. This hierarchy is cumulative: we begin with the empty set, then form sets from the empty set, then form sets from those, and so on. At any point in this hierarchy, we can define certain collections that are not sets but still describable and these are the classes. 

Now comes the pluralist or multiverse view, most famously promoted by the logician Joel David Hamkins. This view challenges the idea that there is one true universe of sets. Instead, it suggests there are many different legitimate set-theoretic universes. Each one is a self-contained world with its own rules, structures, and hierarchies. In one universe, the Continum Hypothesis might be true; in another, false. Both are valid from the pluralist perspective. 

The Continuum Hypothesis is a famous question in mathematics about the nature of infinity. To understand it, you need to know that infinities can come in different sises. The smallest infinity is the sise of the set of natural numbers, 1, 2, 3, and so on, which are called aleph-null, written ℵ₀. This set is infinite, but it's the kind of infinity you can count, even if you'd never finish. The real numbers, on the other hand, include not just the whole numbers but also all the decimals, fractions, and irrational numbers like √2 and π. There are vastly more of these. In fact, Cantor showed that you can't match them up one-to-one with the natural numbers, no matter how you try. So the real numbers are a bigger infinity. We call their sise the cardinality of the continuum, written as 2^ℵ₀. 

The Continuum Hypothesis asks whether there's a sise of infinity in between these two: is there a set bigger than the natural numbers but smaller than the real numbers? If the answer is yes, then there’s some new kind of infinity in the middle. If the answer is no, then the real numbers are the next step up from the natural numbers, with no in-between. Mathematicians worked for decades on this question. In the 20th century, Kurt Gödel and Paul Cohen showed that the Continuum Hypothesis can't be proved or disproved using the standard rules of set theory, known as ZFC. This means the question is independent of those rules. So, if ZFC is consistent, then you can create a perfectly valid mathematical universe in which the Continuum Hypothesis is true, and another equally valid one in which it’s false. 

This discovery is important because it shows that not all mathematical questions have just one right answer. Some truths depend on which universe of mathematics you’re working in. This is what Joel Hamkins calls the multiverse view: instead of believing there is only one absolute mathematical universe, we can think of there being many different universes, each with its own sets and its own version of truth. The Continuum Hypothesis is a gateway into the deeper questions of mathematics. It teaches that mathematical truth is sometimes relative to the system or framework you're using. It also shows how logic, philosophy, and mathematics interact at the highest levels. And most strikingly, it reveals that even in mathematics, some questions are undecidable, not because we don’t know the answer yet, but because there is no single answer at all. 

So within any single universe (say, one model of ZFC), we can make the usual distinction between sets and classes. But from the perspective of the multiverse, this distinction is relative. What looks like a proper class in one universe might be seen as a set from a larger universe. For example, suppose we are working in a universe V, where the collection of all sets forms a proper class. From within V, this collection is too large to be a set. But now imagine we ‘zoom out’ to a bigger universe V′, in which V itself is a set. In V the collection that was a proper class, in V′, V is now just an ordinary set. This relativity is key to understanding the pluralist perspective. It doesn’t deny the set/class distinction which still makes sense and operates within each universe. But it shows that the boundary between sets and classes is not fixed across all of mathematics. Instead, it depends on which universe you are working in. This undermines the idea of an absolute universe of sets, and instead encourages us to see mathematics as taking place across a landscape of multiple universes, each with its own internal logic. In this multiverse framework, we no longer try to locate one final, correct universe of sets. Instead, we explore and compare different universes, understanding how they relate, how one might extend into another, and how our familiar concepts, like ‘set’ and ‘class’, shift depending on the perspective we adopt. The pluralist view thus offers a flexible and dynamic picture of mathematical reality, one that is especially valuable when trying to understand deep issues in the foundations of mathematics. 

This pluralist multiverse view in set theory offers a compelling lens through which to revisit Kant’s antinomies discussed in my last two notes and it may, in fact, illuminate where Kant’s critical project could have gone had he had access to modern developments in logic and mathematics. Kant’s antinomies, especially the first antinomy of pure reason (the world has a beginning in time / the world has no beginning in time), arise when the mind tries to apply pure concepts of reason (Ideas) to the totality of appearances, attempting to think the world as a completed totality (a “set” of all appearances, so to speak). The contradictions that emerge aren’t empirical but stem from the structure of reason itself. For Kant, both the thesis and antithesis of the antinomies can be logically defended, but they lead to contradiction when reason tries to transcend the limits of possible experience. His resolution is transcendental idealism: the world as a totality is not an object of experience and thus not a legitimate object of knowledge. 

However, this is where a Hamkins-style pluralism could deepen the story. Instead of concluding that reason necessarily runs into contradiction when thinking totalities, we might follow the Hamkins model and say: reason is generating multiple internally coherent models of the world, each consistent, but mutually incompatible when taken together. In Kant’s terms, the Ideas of Reason (e.g., the cosmos as a whole, or the totality of causal chains) are not false, but they do not refer to a single metaphysical reality. Rather, they are regulative, guiding us toward different conceptual frameworks or “universes”, each with its own logic and coherence. The antinomies then are not symptoms of error, but symptoms of a monist assumption: the assumption that there must be one total, absolute framework in which all contradictions must be resolved. 

Hamkins' pluralism reframes this. It suggests that these antinomies arise because we are collapsing multiple consistent systems into one. The contradiction appears only if we insist on a single ontological framework, just as in set theory, the Continuum Hypothesis appears paradoxical only if we assume there must be a single answer. But if we allow for multiple frameworks, each grounded in its own axioms (or in Kant’s terms, its own synthetic a priori structures), then the contradiction dissolves into a divergence of perspectives. In this way, Hamkins offers a post-Kantian upgrade: rather than shutting down metaphysical inquiry as unknowable, we shift toward a multiversal epistemology, where multiple internally valid perspectives exist. 

This could reframe the Kantian antinomies not as breakdowns of reason, but as the creative branching of rational systems, each forming a different consistent universe of thought. It also repositions the Ideas of Reason as generative templates for constructing conceptual universes, not as illusions or overreaches. Finally, it avoids Priest’s radical solution of paraconsistent logic, which Kant would have likely rejected as violating the principle of non-contradiction, by maintaining classical logic within each system. Thus, Hamkins remains closer to Kant's own epistemological conservatism, while still offering a more dynamic, less monolithic vision of reason. In sum, Hamkins’ multiverse shows where Kant could have gone. He could have replaced the stark either/or of thesis and antithesis with a pluralist logic of perspectives, retaining the rigor of reason while embracing its capacity to generate multiple, non-reducible, but internally coherent worlds.