OK, Cohen has constructed a model in which both ZFC and ~CH are true. Isn't this model an answer to the continuum problem? Hasn't he showed that it is indeed possible to construct a set with cardinality between that of the integers and that of the reals? Why is it still not considered sufficient to settle CH? Why is one model not enough? Why for all models? In other words, why do we have to answer whether "ZFC |- CH" instead of just "CH" itself?
[Math] Why is Cohen’s result insufficient to settle CH
lo.logicset-theory
Related Solutions
The answer for the second question is no. Truss proved in [1] that if we repeat Solovay's construction from a limit cardinal $\kappa$, we obtain a model in which the following properties:
- Countable unions of countable sets of real numbers are countable;
- Every well-orderable set of real numbers is countable;
- Every uncountable set of reals has a perfect subset;
- DC holds iff $\omega_1$ is regular iff $\kappa$ is inaccessible in the ground model;
- Every set of real numbers is Borel.
This shows that it is possible to have $CH+\aleph_1\nleq2^{\aleph_0}+\operatorname{cf}(\omega_1)=\omega$. However it does not answer the original (first) question.
Bibliography:
- Truss, John, Models of set theory containing many perfect sets. Ann. Math. Logic 7 (1974), 197–219.
How kind of you to take an interest in my paper. Please see also my blog post about the dream solution and the arxiv entry for the paper.
First, I shall make a quibble, and then I'll address your question at the end.
The quibble is that your quotation from the paper is not accurate. The full paragraph from the paper reads:
I have argued, then, that there will be no dream solution of the continuum hypothesis. Let me now go somewhat beyond this claim and issue a challenge to those who propose to solve the continuum problem by some other means. My challenge to anyone who proposes to give a particular, definite answer to CH is that they must not only argue for their preferred answer, mustering whatever philosophical or intuitive support for their answer as they can, but also they must explain away the illusion of our experience with the contrary hypothesis. Only by doing so will they overcome the response I have described, rejection of the argument from extensive experience of the contrary. Before we will be able to accept CH as true, we must come to know that our experience of the $\neg$CH worlds was somehow flawed; we must come to see our experience in those lands as illusory. It is insufficient to present a beautiful landscape, a shining city on a hill, for we are widely traveled and know that it is not the only one.
The difference is that it should say "extensive experience of the contrary" rather than "extensive evidence of the contrary", a difference that affects the meaning, since the point is that we have experience in both the CH and in the $\neg$CH worlds. In particular, there is a symmetry here, and I hope it was clear that implicitly include your variation as part of my intended meaning.
Now, let me consider your final question, which is very good.
- Can one make the view showing that either Cohen reals are illusory, or that the ability to add sufficient number of Cohen reals so as to make not-CH true is illusory, coherent?
I take the answer to be yes, these views are made coherent by what I have called the universe view in my paper The set-theoretic multiverse, from which the dream solution paper is adapted. The universe view is the view I am arguing against, and although I have attacked the universe view for being mistaken, I do not attack it as incoherent. The question is whether the alternative set-theoretic universes that we seem to have discovered via forcing and other methods exist as legitimate concepts of set or not. I have argued at length that they do. But the opposing universe view is that no, there is just one absolute background concept of set, and the purpose of set theory is to discover what is true there. This seems to be a perfectly coherent view. It is a view advanced explicitly by Daniel Isaacson, who I quote extensively in my dream solution paper, and also by Donald Martin, in his paper "Multiple universes of sets and indeterminism in set theory", Topoi 20, 5--16, 2001, among others.
Criticizing my argument, Peter Koellner has emphasized that one can view my account of the naturalist account of forcing, rather than providing evidence that forcing extensions are real, instead as the desired explanation of the illusion of forcing extensions of $V$. And perhaps this criticism is the detailed answer to your question. That is, Koellner argues that the details of the proof of the naturalist account of forcing is how one explains away the illusion of forcing. So that would seem to be a coherent view. My reply to that argument, in my multiverse paper, is that such an account of forcing seems fundamentally crippling to our mathematical intuition, if we must regard all talk of actual forcing extensions of $V$ as ever-more-fantastical simulations of the extensions inside $V$, something like the writings of the exotic-travelogue writer who never actually ventures west of sixth avenue, or the absurdity of the mathematician who insists that yes, the real numbers exist with a full Platonic existence, but the complex numbers do not; they must be simulated inside the reals, such as with ordered pairs. The multiverse perspective makes a philosophically simple position, taking the existence of the forcing extensions at face value, while nurturing a robust use of forcing that will ultimately aid our set-theoretical understanding.
Finally, let me say that I agree completely with Andrej's point about geometry, and I discuss this analogy in section 4 of my multiverse paper.
Best Answer
I am sympathetic to this question, which often arises for those first learning of Cohen's theorem, and I don't think it is an idle question. I recall my sophomore undergradatue self being confused about it when I first studied the set-theoretic independence phenomenon. And I think that Carl is right, that this particular issue is not addressed on the other CH questions.
I view the question as arising from the following line of thought: Cohen proved that it is possible that CH fails. Thus, it is possible that there is a set of reals whose cardinality is strictly between the integers and the continuum. But if we can decribe how such a set can exist, then haven't we actually described a set of such intermediate cardinality? That is, doesn't this mean that CH is simply false?
This line of thinking may be alluring, but it is wrong. The reason it is wrong, as Gerhard explains in his comment, is that it doesn't appreciate the role of models, or what might be called the set-theoretic background. What Cohen did was to show that if ZFC is consistent, then so is ZFC + ¬CH. (In contrast, Goedel proved that if ZFC is consistent, then so is ZFC + CH.) Thus, Cohen's intermediate-cardinality set has the property that it is intermediate in cardinality in the model that Cohen describes, with respect to that set-theoretic background, but it will not be intermediate-in-cardinality with respect to other set-theoretic backgrounds. The property of being intermediate in cardinality is dependent on the set-theoretic background in which this property is considered. For example, a set $X$ is uncountable if there is no function from the natural numbers onto it. But perhaps there is no such function mapping onto $X$ in a model of set theory $M$, but there is a larger model of set theory $N$, still having $X$, but for which now there IS a function mapping the natural numbers onto $X$. In fact, this very situation follows from Cohen's forcing method: any set can be made countable in a forcing extension.
Thus, whether a set forms a counterexample to CH cannot be observed looking only at that set---one must consider the set-theoretic universe in which the set is considered, and the possible bijective functions that might witness its countability or not. The very same set of reals can be countable in some models of set theory and bijective with the continuum in others.