One central notion in mathematics is cardinality (I’ve given an intuitive of cardinality in this post). In particular, there are different types of infinity, with some being larger than others. The smallest infinity is the cardinality of the natural numbers. We say that any set with such a cardinality is countable, since its elements can be put in a one to one correspondence with the natural numbers.
What, then, is the next largest cardinality? This is a natural question to ask if we are tying to get a handle on the different cardinals out there. One natural hypothesis is that the cardinality of the real numbers is the next largest cardinality. Georg Cantor hypothesized exactly this back in 1878. We call this the continuum hypothesis (CH). Precisely, we might state it as:
(CH) There is no set whose cardinality is strictly between that of the naturals and the real numbers.
This might seem natural for a few reasons. For example, we are familiar with the real numbers, we generate real numbers by using integers to form rationals and then completing them in some way, and we know that the cardinality of the real numbers is larger than that of the naturals.
A central questions of mathematics, and in particular set theory, has been whether or not CH is true (it actually has interesting connections to machine learning frameworks). For a while there was hope that we could give a definitive proof of CH or its negation. The idea was that, just as we can prove 5+7=12 and other mathematical theorems, we might be able to prove CH as a theorem of set theory.
However, this was not to be the case. We now know that CH is independent of the standard axioms of set theory, known as the Zermelo–Fraenkel-Choice (ZFC) axioms. This means that the standard axioms are consistent with both CH and its negation. For example, take Harry Potter. There are some things in Harry Potter that are pinned down by the book. For example, that Harry has a scar in the shape of a lightning bolt on his forehead. However, there are many things that the book does not pin down. For example, the novels do not establish whether or not there is someone living in magical London with the name “Ignatius Smith”. In other words, both the statement “There is an Ignatius Smith who lives in magical London” and “It is not the case that there is an Ignatius Smith who lives in London” are perfectly consistent with Harry Potter. It is this sense in which both CH and its negation are consistent with the standard ZFC axioms.
This brings us to the paper by Joel Hamkins I will take a look at today.
***The original paper can be found here.***
Hamkins explores whether or not we are to have a dream solution to the continuum hypothesis. By “dream solution” Hamkins means
One by which we settle the continuum hypothesis (CH) on the basis of a new fundamental principle of set theory, a missing axioms widely regarded as true, which determines the truth value of CH.p. 135
This is a dream solution because the resolution of CH falls out naturally from the adoption of a new axiom. To draw a parallel in the Harry Potter case, suppose we cared about the Ignatius Smith hypothesis. Furthermore, suppose that J.K. Rowling wrote a new Harry Potter book, and in it there was information that implied with certainty that there was an Ignatius Smith living in magical London, but without explicitly saying it. We have good reason to accept the new book—it was written by J.K. Rowling, it is a compelling story, it adds to our understanding of the Harry Potter world, etc.—and it settles the Ignatius hypothesis. This would be a dream solution to the Ignatius Smith hypothesis. This is parallel to the case in set theory, except our reasons for accepting a new axiom would be different (to say exactly what they are in a complicated affair which is beyond the scope of this post).
Hamkins argues that we cannot attain a dream solution to CH. As I understand it, his argument is mainly arguing from a sociology of mathematics perspective. He writes
I claim that our extensive experience in the set-theoretic worlds in which CH is true and others in which CH is false prevents us from looking upon any statement settling CH as being a natural set-theoretic truth. We simply have had too much experience by now with the contrary situation. Even if set-theorists initially find a proposed new principle to be a natural truth for sets, nevertheless once it is learned that the principle settles CH, then this preliminary judgment will evaporate in the face of deep experience with the contrary, and set-theorists will look upon the statement merely as an intriguing generalization or curious formulation of CH or ¬CH, rather than as a new truth. In short, success in the second step of the dream solution inevitably undermines success in the first step.p. 135-136
In order to understand Hamkins’ argument we have to get a grip on what he means by experience in a set-theoretic world. To this I introduce the idea of a model. I will not give the formal details; instead I will provide an intuition. A model of some set of axioms is a structure that satisfies those axioms. You can think of the axioms as constraints. If we think about the running Harry Potter example, a model of the Harry Potter novels would be something like a fully specified universe that satisfies all of the content in the novels. In each universe/model there would be a character named Harry Potter and a character named Draco Malfoy who often do not see eye to eye, for example. However, there would be some models in which Ignatius Smith was living in magical London, and some in which he wasn’t. This is because the constraints from the novels don’t pin this down. In our previous terminology, this is because the Ignatius Smith hypothesis was independent of the Harry Potter novels.
When mathematicians do work to show that something is or is not independent of ZFC they often try to explicitly construct models of ZFC. If they can construct one model in which some statement is true, and one in which it isn’t, then this shows that the statement is independent of ZFC. Mathematicians have done this with CH; they have explicitly constructed models where CH holds and models where CH fails.
This is what Hamkins means when he says that mathematicians have “extensive experience” in certain set-theoretic worlds. By constructing and playing around models in which CH was true and models in which CH is false, they have learned the terrain and feel of both. Instead of some determinant fact, it feels more like a switch you can turn on and off. Hamkins summarizes the situation like this:
It is for this reason that the dream solution has become impossible. Our situation with CH is not merely that CH is formally independent and we have no additional knowledge about whether it is true or not. Rather, we have an informed, deep understanding of how it could be that CH is true and how it could be that CH fails. We know how to build the CH- and ¬CH-worlds from one another. Set-theorists today grew up in these worlds, comparing them and moving from one to another while controlling other subtle features about them. Consequently, if someone were to present a new set-theoretic principle Φ and prove that it implies ¬CH, say, then we could no longer look upon Φ as manifestly true for sets. To do so would negate our experience in the CH worlds, which we found to be perfectly set-theoretic. It would be like someone proposing a principle implying that only Brooklyn really exists, whereas we already know about Manhattan and the other boroughs. And similarly if Φ were to imply CH. We are simply too familiar with universes exhibiting both sides of CH for us ever to accept as a natural set-theoretic truth a principle that is false in some of them. So success in the second step of the dream solution fatally undermines success in the first step.p. 139
This is the heart of it for Hamkins. If we have some plausible principle Φ on the table, as soon as it is the case that the set-theorists learn that it settles CH, it will no longer seem plausible to them. Hamkins gives some examples of this happening with certain axioms candidates in the paper, and they are worth reading. I myself don’t know enough about the practice of mathematics to know if this is an accurate characterization of the dispositions of mathematicians, but given Hamkins’ background as a mathematician and the examples he gives I have certainly shifted my credence towards the hypothesis that a dream solution to the continuum hypothesis is unattainable.