Why doesn't the independence of the continuum hypothesis immediately imply that ZFC is unsatisfactory?

Question

I am risking a "possible duplicate of... " here, in particular with respect to this question, this question, and this question. Nevertheless here goes.

I am going to keep this question as simple as possible:

First note two facts:

1. It has been proven that CH is independent of ZFC.

2. ZFC is intended as a foundational system of mathematics.

Secondly: the independence of CH from ZFC has (if I understand it correctly) led many people to state that "CH is neither true nor false".

My response is: It seems to me that the independence of CH from ZFC cannot be "blamed" on CH, but should be ZFC's responsibility, for the simple reason that CH never had any pretention of being in any way related to the ZFC axioms, while the ZFC axioms claim to be a foundation of mathematics.

Therefore if ZFC and CH are independent, this says absolutely nothing about the truth-status of CH, and merely implies that ZFC is inadequate. ZFC is merely one possible axiomatizations that some smart people came up with a long time ago, so why don't we take ZFC with a grain of salt. Why do people disagree with my argument, and instead claim that the independence of ZFC and CH implies that CH is "neither true nor false"?

Answer 1

It is true that, if $\mathsf{ZFC}$ is consistent, $\mathsf{CH}$ is undecidable within it. This is just one example of a more general fact: any consistent recursively enumerable first-order theory at least as strong as Peano arithmetic contains a statement undecidable in that theory. This is the first of Gödel's incompleteness theorems; the second gives an example, namely a statement calling the theory consistent.

The real question isn't whether $\mathsf{ZFC}$ has known limitations of this kind; of course it does. The question is which other statements we should add as axioms. The $\mathsf{C}$ in $\mathsf{ZFC}$ is the axiom of choice, which is itself undecidable in $\mathsf{ZF}$. The history of set theory has seen much broader support in favour of adding $\mathsf{AC}$ than in favour of subsequently adding $\mathsf{CH}$.

Why? Well, let's look at some of the differences:

• Although $\mathsf{AC}$ was initially much more controversial than it is today, it has come to enjoy broad support for the simple reason that, although it has some counter-intuitive consequences such as Zermelo's well-ordering theorem, its negation has "even worse" consequences such as trichotomy violation.
• Not only is $\beth_1=\aleph_1$ (i.e. the $\mathsf{CH}$) undecidable in $\mathsf{ZFC}$; so is $\beth_1=\aleph_n$ for any positive integer $n$. Why would we adopt the first as an axiom? By contrast, $\mathsf{AC}$ doesn't have an infinite family of obvious counterparts that feel equally feasible.
• One good thing about the $\mathsf{CH}$ is that it's a special case of a more general idea that looks nice, the $\mathsf{GCH}$ ($\beth_\alpha=\aleph_\alpha$ for all ordinals $\alpha$). There is some interest in adding $\mathsf{GCH}$ to $\mathsf{ZFC}$, but just $\mathsf{CH}$ on its own? That's an unpopular compromise between $\mathsf{ZFC}$ (which is weak enough for some people's tastes) and $\mathsf{ZFC+GCH}$ (which is strong enough for some other people's tastes).
• Lastly, outside of set theory $\aleph_1$ usually doesn't even come up as a concept, and so $\mathsf{CH}$ has no obvious benefit when founding any other areas of mathematics. (There are exceptions, e.g. nonstandard analysis uses $\mathsf{CH}$ to consider real infinitesimals.) By contrast, $\mathsf{AC}$ has uses all over the place, e.g. proving every vector space has a basis.

Answer 2

There are two issues that I see with your question.

The first and foremost is using the terms "true" and "false" without fully understanding them (granted, many mathematicians do this as well).¹ We have a good idea what is $\Bbb N$ and what is $\Bbb R$. So it is easy to talk about truth in analysis as relative to $\Bbb R$ in some rich enough language that encapsulates the real numbers as we understand them; and truth in number theory is just truth in $\Bbb N$ as a model of PA.

Set theory, however, is far less intuitive. This can be easily seen by the many counter-intuitive results in set theory (from Banach–Tarski to the Division Paradox) that follow from the fact that our intuition is simply not very good with infinite objects that have little structure.

So as we do not have a clear and uniform intuition as to what are sets like, not in the same way we have about the natural numbers or even the real numbers, it is hard—even, impossible—to have a canonical model of set theory where we can evaluate every statement and decide whether or not it is true.²

As we have no canonical way to determine truth and false, there is no canonical way to decide if the Continuum Hypothesis, or many many other statements are "true" or "false". And this opens the door to approaches other than Platonism.

For a Platonist, every statement should be either true or false. That is also a naive approach to mathematics. Sure. It's like that in real life, to some extent. Either you are a doctor, or you're not. Either the sun will engulf the earth as it expands to a red giant, or it won't (for one of many reasons). Things in our life tend to have certain absoluteness to them,³ so we try to apply these principles to mathematics as well.

But mathematics has nothing to do with our reality. Especially set theory, or anything which deals with the infinite, really. So why should there be a canonical universe for truth and false? If anything, set theoretic research shows that one the plethora of mathematical universes with very different theories holding true in them, can all be very interesting.

Okay. So there is no true and false here. What about the second issue?

 

The second issue is that it is actually a good thing that ZFC is not a complete theory. It is a good thing that our foundation doesn't tell us all the answers.

Foundational theories are not there to "give us all the answers", they are there "to formalize our arguments into a mathematical context". ZFC does that magnificently for the most part.

There is a lot of research into "removing unneeded hypothesis" from mathematics. You prove something under assumption that a function is analytic. But maybe proving it under smoothness is enough? Maybe just continuous? Maybe just measurable? Maybe any function?

You want foundations that are strong enough to support your work, but not too strong that they make a lot of unneeded assumptions for you. And while I agree, axioms like $V=L$ are sexy, and they resolve a lot of questions (e.g. CH and things like the Suslin Hypothesis), for the most part, mathematics works just peachy without them.

Not only that, once you start putting in set theoretic "decisions" into your foundations, your axiomatization of the concept of "set" will invariably have to become technical. ZFC is simple, it is elegant. Putting CH into it will involve long and technical statements about functions, about cardinals, about so much more. This will muddy the waters, and for what? Yes, CH has consequences in analysis and in general, but are those enough to get CH "canonical"?

The answer seems negative. Not because set theorists don't care, but because there have been little to no pressure from the "working mathematician" side regarding adding new axioms to set theory. And again, this is not without reason.

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Footnotes.

1. Truth is always relative to a fixed structure, in a lot of cases we just have some tacit agreement about the structure.

2. Even in the case of $\Bbb N$ not everyone would agree. For example, "Supercompact cardinals are consistent with ZFC" is a statement about the natural numbers. Some mathematician would say it is true, others would disagree.

3. They really don't, though.

Answer 3

There are classical arguments, which have historically been seen as compelling, which are intended to argue that the ZFC axioms are all accurate statements about the intuitive concept of a pure, well-founded set.

In principle, it would be possible for us to take some new axioms, in addition to ZFC. The addition of these new axioms could, in principle, allow us to prove statements such as CH. In particular, of course, we could just take CH itself as an axiom, in addition to ZFC.

The issue is that, for systems like ZFC, many people like to have a justification for why the axioms should be assumed. We can take any statement as an axiom, just for the sake of reasoning, but for specific systems such as Peano arithmetic, Euclidean geometry, or ZFC, we like to see a "reason" why each axiom is accepted.

Looking at CH in particular, it seems hard to justify why we would take CH (or its negation) as a new axiom of ZFC. Doing so feels like missing the point - it feels like we are arbitrarily taking sides. And some other, "more justified" axiom might come along later but go in the opposite direction from the arbitrary choice we made.

There has been a significant amount of published discussion on this question, which is more philosophical than mathematical. There are some people working in set theory who do feel that new axioms might be discovered that would resolve CH. If a sufficient justification for these axioms could be given, perhaps a large number of mathematicians would view this as resolving the issue of CH. But others in set theory are more pessimistic. One argument they give is that, because we understand so well the way in which CH is independent from ZFC, if some new axiom were to decide the CH question, we could apply the techniques developed for CH to the new axiom, which might cast doubt on accepting the new axiom on equal footing with the rest of ZFC.

Finally, there is a linguistic triviality: some people misuse the word "true" to mean "provable", so if they say "In ZFC, CH is neither true nor false" they only mean "In ZFC, CH is neither provable nor disprovable". In any particular model of ZFC, of course, either CH is true or CH is false. So any claim that "CH is neither true nor false" has to be read in so other way besides talking about truth in some particular model.

Answer 4

One aspect of your question that seems to have been ignored by most of the other answers (I believe that Carl Mummert touches on the idea) and comments here is that it is not just due to the fact that $ZFC$ does not decide $CH$ that leads some mathematicians, logicians, and philosophers to believe that $CH$ does not have a determinate truth value. It is also the fact that $CH$ has proved incredibly resilient in it's ability to elude decidability when in the presence of large cardinal axioms. So in response to your question "why do people disagree with my argument" the answer is "because your premise that they do so solely on the justification that $CH$ is independent of $ZFC$ (the second "fact" you mention should be noted) is a false assumption." They do believe that the 'responsibility lies with $ZFC$', but the massive amount of work that has happened since the '60s in extending $ZFC$ with larger axioms of infinity have still left $CH$ undecidable.

After Gödel proved that $L$ is a model of $(ZFC+CH)$ and before Cohen had invented forcing, Gödel was still skeptical that $CH$ would turn out to be decided by $ZFC$. In his 1947 essay What is Cantor's Continuum Problem? he states this belief explicitly in Section 4 Some observations about the question: In what sense and in which direction may a solution of the continuum problem be expected?:

So from either point of view, if in addition one takes into account what was said in Section 2, it may be conjectured that the continuum problem cannot be solved on the basis of the axioms set up so far, but, on the other hand, may be solved with the help of some new axiom which would state or imply something about the definability of sets.

The latter half of this conjecture has already been verified; namely, the concept of definability mentioned in footnote 17 (which itself is definable in axiomatic set theory) makes it possible to derive, in axiomatic set theory, the generalize continuum hypothesis from the axiom that every set is definable in this sense.

In a footnote he then goes on to say:

On the other hand, from an axiom in some sense opposite to this one, the negation of Cantors conjecture could perhaps be derived. I am thinking of an axiom which (similar to Hilbert's completeness axiom in geometry) would state some maximum property of the system of all sets, whereas axiom A [his axiom $(V=L)$] states a minimum property.

So, even in 1947, Gödel understood that his $(V=L)$ is a minimal statement about the nature of $V$ and that there could in principle be a maximal statement about the nature of $V$ where there are more sets than just the definable ones (meaning $L$) which he believed would result in more subsets of $\mathbb{R}$ and therefore that $CH$ would be false. Under the necessary consistency assumptions, if $(ZFC + Minimal)$ implies $CH$ and $(ZFC + Maximal)$ implies $\lnot CH$, then $CH$ is undecidable in $ZFC$. This is what Gödel believed would be the case.

Gödel went on to suggest that one approach to solving the problem would be to assume stronger axioms of infinity, which is referred to as his program for large cardinals. Peter Koellner writes in his article The Continuum Hypothesis for the Stanford Encyclopedia of Philosophy:

Gödel's program for large cardinal axioms proved to be remarkably successful. Over the course of the next 30 years it was shown that large cardinal axioms settle many of the questions that were shown to be independent during the era of independence. However, CH was left untouched. The situation turned out to be rather ironic since in the end it was shown (in a sense that can be made precise) that although the standard large cardinal axioms effectively settle all question of complexity strictly below that of CH, they cannot (by results of Levy and Solovay and others) settle CH itself. Thus, in choosing CH as a test case for his program, Gödel put his finger precisely on the point where it fails. It is for this reason that CH continues to play a central role in the search for new axioms. [Emphasis mine]

Not only is Koellner explicit about large cardinal axioms being unable to give us a definite resolution to $CH$, he implicitly states that there is still a search for new axioms to extend $ZFC$. In other words, $ZFC$ is not where the buck stops in terms of asking whether or not $CH$ has a definite truth value.

There are many different schools of thought about whether or not $CH$ has a definite truth value, but like with any realism vs. anti-realism question they can be placed into two camps: realism (it does) or anti-realism (it does not). The terminology that Koellner uses in the above article, as well as two other SEP articles Large Cardinals and Determinacy and Independence and Large Cardinals, dealing with the same sort of topics, are non-pluralism and pluralism. From the latter:

The main question that arises in light of the independence results is whether one can justify new axioms that settle the statements left undecided by the standard axioms. There are two views. On the first view, the answer is taken to be negative and one embraces a radical form of pluralism in which one has a plethora of equally legitimate extensions of the standard axioms. On the second view, the answer is taken (at least in part) to be affirmative, and the results simply indicate that ZFC is too weak to capture the mathematical truths.

Pluralism is the view that there are many different universes of sets, there are many different formalizations and axioms that describe these different universes, and all of them are equally equivalent in terms of mathematical truth. Pluralism is the anti-realist view, it states that in some universes $CH$ is true, in other's it is false, and that is all that can be said of the matter. Each system, under this view, is equally privileged to make the claim about $CH$. What I believe is probably the quintessential explication of pluralism is Joel Hamkin's paper The Set Theoretic Multiverse. Consider the abstract:

The multiverse view in set theory, introduced and argued for in this article, is the view that there are many distinct concepts of set, each instantiated in a corresponding set-theoretic universe. The universe view, in contrast, asserts that there is an absolute background set concept, with a corresponding absolute set-theoretic universe in which every set-theoretic question has a definite answer. The multiverse position, I argue, explains our experience with the enormous range of set-theoretic possibilities, a phenomenon that challenges the universe view. In particular, I argue that the continuum hypothesis is settled on the multiverse view by our extensive knowledge about how it behaves in the multiverse, and as a result it can no longer be settled in the manner formerly hoped for.

The non-pluralist view, or in Hamkin's terminology, the universe view, is that there is one conception of set, and therefore questions such as $CH$ do have a definite answer. Gödel himself was of this view, owing to the fact that he was a mathematical platonist. Gödel believed that mathematical objects are real, they are abstract objects that exist, and therefore any question about their nature has a determinate truth value. He believed, therefore, that $CH$ being independent of $ZFC$ just means that we need new axioms. Hugh Woodin's contemporary work (1, 2 on the large cardinal program is part of an attempt to create $L$ like models for every large cardinal axiom. His (relatively) recent proposal is that there is an ultimate enlargement of L once one reaches an $L$ like supercompact cardinal, leading to the axiom that $V=Ultimate$ $L$ which would give a non-pluralistic decidability of $CH$. From his article Strong Axioms of Infinity and the Search for $V$:

Godel’s Axiom of Constructibility, ¨$V = L$, provides a conception of the Universe of Sets which is perfectly concise modulo only large cardinal axioms which are strong axioms of infinity. However the axiom $V = L$ limits the large cardinal axioms which can hold and so the axiom is false. The Inner Model Program which seeks generalizations which are compatible with large cardinal axioms has been extremely successful, but incremental, and therefore by its very nature unable to yield an ultimate enlargement of $L$. The situation has now changed dramatically and there is, for the first time, a genuine prospect for the construction of an ultimate enlargement of $L$.

Whether or not one accepts pluralism or non-pluralism, they are still prompted to confront the fact that $ZFC$ is not where the buck stops. It's true that almost all of contemporary mathematics can be decided by $ZFC$, but it has never been a secret that there are many results which are not decided (See Finite functions and the necessary use of large cardinals by Harvey Friedman and of course all work related to Gödel's incompleteness theorems).

In summation, the main thrust of my answer is that your presupposition that it is because $CH$ is independent of $ZFC$ that people who understand the issues believe that $CH$ has no definite truth value is false. $ZFC$ was proposed as a foundational system because the logicians working at the time (starting from the foundational crisis up to whenever we want to decide that set theory became it's own distinguished branch of mathematics, probably in the '60s after Cohen) believed that the $ZFC$ axioms did capture everything we wanted to capture about the intuitive notion of set. At least they believed this until we had concrete independence results that showed the opposite. So the idea that $ZFC$ perfectly captures our idea of set turned out to not be true and the search for new axioms is an implicit motivation in a lot of current set theory (although, of course, there are set theorists who do not care about foundations and are doing the work for it's own sake).

All of this has come about due to a vast amount of philosophical, logical, and mathematical thinking about the results of set theory and the pluralist or multiverse view (the view you claim to have issue with in your question) did not arise simply because $CH$ is independent of $ZFC$.

Answer 5

One reason that some philosophically minded logicians have argued that, in the OP's words, "CH is neither true nor false" is that they have come to doubt whether CH is in fact a clear claim with a fully determinate content. In a sense (though maybe this isn't the best way of putting the point) it is a vague claim, and so neither determinately true nor false for that reason.

How can this sort of idea even be a runner? The worry in fact comes down to this: is the concept of the totality of arbitrary subsets of a given infinite set definite? And the suspicion is that a hundred years of work in the vicinity (including those independence results) casts some doubt on whether we do have a definite conception in play here.

I'm reporting, not endorsing. But a very distinguished proponent of this sort of view is the late Solomon Feferman. There's e.g. a relevant essay of his (accessible in both senses!) here: Is the Continuum Hypothesis a definite mathematical problem? And there's commentary by Peter Koellner here.

Answer 6

"CH is neither true nor false" is a philosophical claim. What to make of it depends on philosophy of mathematics issues which is probably outside of the scope of MSE.

Should we "blame" ZFC for leaving CH independent? You seem to argue that being a "foundation of mathematics" means being able to decide every mathematical statement. Then, if you consider CH a mathematical statement, ZFC is certainly not a "foundation" in that sense. But modern mathematicians will tell you they have no such pretensions about what a "foundation" should be doing.