# In What Way are Set Theorists' 'Experiences' in the CH Worlds Flawed, if Any?

This is in regards to Joel David Hamkins' new paper "IS THE DREAM SOLUTION OF THE CONTINUUM HYPOTHESIS ATTAINABLE?" (look under title in arXiv). I quote from the last paragraph of his paper:

"My challenge to anyone who proposes to give a particular, definite answer to CH is that they must not only argue for their particular 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 evidence of the contrary. Before we will be able to accept CH as true, we must come to know that our experience of the not-CH worlds was somehow flawed; we must come to see our experience in those lands as illusory."

Let me make a slight variation in the last sentence of his I quoted:

Before we will be able to accept not-CH as true, we must come to know that our experience of the CH worlds was somehow flawed; we must come to see our experience in those lands as illusory.

Since the goal of set theory (at least from Hamkins' perspective of the orthodox view (the set-theoretical universe as unique--it is the universe of all sets, "The set-Theoretical Multiverse: a model-theoretic philosophy of set theory")) is to have V (for ZFC, for example) to contain all possible sets short of inconsistency, it would seem that from this perspective that the CH worlds are already flawed and that to defend CH against not-CH one would have to say that the existence of 'Cohen reals' in the not-CH worlds is somehow illusory (or at least the belief that one can add sufficient number of Cohen reals to make CH false from the Naturalist View of Forcing perspective). 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?

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Isn't this rather argumentative? –  Charles Matthews Mar 21 '12 at 10:18
Depends on what you mean by "argumentative". We all agree that the question cannot be settled the on purely "axiomatic/deductive" playground. The interesting thing, however, is that while AC is commonly (but not universally) accepted because it enables us to prove many useful down to Earth things that would be undecidable otherwise, and rejected by some because it brings up a few clear monsters like Banach-Tarski, the implications of CH or its negation in the "everyday math" are much less transparent. So, the question is "can we get something our intuition revolts against from negating CH?" –  fedja Mar 21 '12 at 10:47
Link to "Is the dream solution to the continuum hypothesis attainable?" by Joel David Hamkins: arxiv.org/abs/1203.4026 –  Pierre-Yves Gaillard Mar 21 '12 at 14:41
I suggest the math-philosophy tag. –  Joel David Hamkins Mar 22 '12 at 8:19

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.

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Let me also apologize to those who object to having mathematical philosophy on MO, who I hope will allow it. The truth is that much of the current work in the philosophy of set theory involves deeply philosophical issues surrounding some of the most technical methods and ideas, such as forcing and large cardinals, and the subject requires experts in both realms, mathematics and philosophy, to progress. –  Joel David Hamkins Mar 21 '12 at 21:49
Some of us actually wish there were lots more mathematical philosophy (at a proper 'research' level) on MO! –  Jacques Carette Mar 21 '12 at 23:31
@ Prof. Hamkins: Sorry to have misquoted you. I would still be interested in getting a sense of the experience set theorists' have in the CH and GCH worlds. Is there a survey article which covers that experience? If so, please let me know. Also, have you any idea how the notion of set theory being the foundations of mathematics will survive in the set-theoretic multiverse? Should it? –  Thomas Benjamin Mar 22 '12 at 19:39
No problem---don't worry about it. As for getting experience in the CH, GCH and $\neg$CH worlds, I would recommend any of the standard treatments of forcing and iterated forcing. My sense of the future for viewing set theory as a foundation of mathematics is that this will continue as it has, in that set theorists will continue to explore the diversity of models of set theory, while quietly abandoning the sometimes-heard remarks concerning what is "really" true. –  Joel David Hamkins Mar 22 '12 at 20:09

I find it useful to make an analogy between what set theorists and geometers do.

A geometer has mastered the study of models of a certain theory known as "affine geometry". He eats models of geometry for breakfast, he can imagine them, and has developed an intuition about them. Questions about Euclid's fifth postulate were historically important for the development of geometry, but nowadays we understand the situation very well. We have outgrown questions such as "is the fifth postulate really true" and "are non-euclidean geometries just an illusion?".

A set theorist has mastered the study of models of a certain theory knowns as "ZFC". He eats models of sets for breakfast, he can imagine them, and has developed an intuition about them. Questions about Cantor's Hypothesis were historically important for the development of set theory, but nowadays we understand the situation very well. Why then have we not outgrown questions such as "is CH really true" and "are models of not-CH just an illusion?"

Whatever the reasons, the best thing a set theorist can do is write a paper in which he explains as carefully as he can to the semi-, pseudo- and quasi-experts that in fact there is a wealth of models of ZFC, just like there is a wealth of geometries. The models of ZFC are not superior or inferior according to whether they validate Cantor's hypthesis, just like the geometries are not superior or inferior according to whether they validate the fifth postulate.

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Very nice analogy. –  Spiro Karigiannis Mar 21 '12 at 17:12
I have questions regarding the analogy between set theory and geometry, primarily the following: We know that in some cases we can represent alternatives to Euclid's fifth postulate as holding in spaces having curvature. The notion of curvature in spaces provides a larger context in which it makes sense to speak of the equality of alternative geometries. What is the larger context, if any, that allows one to speak of the equality of models of set theory coherently? This isn't to say that set theorists can't explore every possible model of set theory-- –  Thomas Benjamin Mar 21 '12 at 18:53
@Hwitz: I wonder what you would have said in the 16th century version of Mathoverflow. –  Andrej Bauer Mar 21 '12 at 20:35
@Benjamin: if set-theorists have a strong intuition about the set-theoretic multiverse, but they are at the present unable to describe it with rigorous mathematical methods, then we should all rejoice to have something new and grand to work on. And of course, since there is a multiverse, the idea that mathematics is founded upon one tiny little undetermined point of the universe becomes silly. The new view of foundations will simply have to be more relativistic in nature. –  Andrej Bauer Mar 21 '12 at 20:38
Perhaps I should mention that I am not a set theorist and I do not eat models of classical set theory for breakfast, but I know precisely what Joel is getting at because I do eat models of intuitionistic mathematics for breakfast. That's a larger multiverse than Joel's. –  Andrej Bauer Mar 22 '12 at 10:10

The part of the question that I would like to address is

. . . it would seem that from this perspective that the CH worlds are already flawed and that to defend CH against not-CH one would have to say that the existence of 'Cohen reals' in the not-CH worlds is somehow illusory

It might seem like since there are a variety of universes, that some of them must be flawed. In particular that since there are universes which are models of $CH$ and $\neg CH$ that some of these universes must be flawed. The accepted answer to this question has shown that it is coherent to resolve this difficulty in thinking by calling conflicting universes or sets illusions. However, one can overcome this apparent cognitive difficulty by living locally in the multiverse. For the purpose of understanding how to see through the proper perspective in the set-theoretic multiverse, let me create some characters, if you will tolerate, which live in the multiverse. Suppose in the multiverse there are travelers and inhabitants. Travelers like to travel far and wide and often, while inhabitants like to stay home. Let's go to a universe $V$ which models $GCH$. Inhabitants in this universe know that the continuum hypothesis is true. However, they also know that there are many travelers coming to visit their universe. An inhabitant meets a traveler who has just arrived from another universe $W$ where the continuum is $\aleph_{10}$. The traveler sees that in universe $V$ the continuum is $\aleph_1$, and so accepts that the continuum is now a different size because she is in a new place. The inhabitant is having a more difficult time understanding since he has not traveled beyond his native universe. The traveler tells him that in universe $W$ the size of the real numbers is $\aleph_{10}$, but he doesn't understand because the size of the real numbers is most definitely $\aleph_1$, and he has no "experience of the contrary". So, the traveler decides to take him with her to another universe $X$ where the continuum is $\aleph_2$. Via a generic filter and forcing relation, they travel to universe $X$ where the inhabitant of universe $V$ can now see with his own eyes that in this world the continuum is indeed the second uncountable cardinal! So, truth in the multiverse depends on location. This way of understanding is discussed in detail in Hamkins' paper on the set-theoretic multiverse, and in the paper regarding a dream solution to CH which is quoted below:

Part of my goal in the multiverse article was to tease apart two often-blurred aspects of set-theoretic Platonism, namely, to separate the claim that the set-theoretic universe has a real mathematical existence from the claim that it is unique.

Second, I would like to respond to Andrej's perspective that all universes are created equal. I think that there are places in the multiverse which are more pleasant than others depending on one's preferences. For example, the inhabitants of a universe where $GCH$ holds might believe that their universe is the best since there are so many travelers who have a lay-over there. I like this type of universe very much and indeed in my experience with forcing, it is very helpful (in order to count in the ground model) to have the $GCH$ hold. I have also been to a universe where Martin's Axiom holds and I really liked doing mathematics in that universe. However, some regions of the multiverse may be less appealing, say a universe without the axiom of choice.

Finally, I would like to address the second to last paragraph of Joel's response. He says that it is crippling to have to consider the rich set-theoretic multiverse as a mere simulation, an illusion we experience in the universe. I agree, but I think it is important to discuss and distiguish the difference between the dreams of the universe and the reality of the multiverse. In every universe, there are the classes of names of other universes with a variety of sizes of the continuum, for example. But these classes of names are not themselves the universes to which they point since if they were then the universe which dreams of them would not be coherent. It is only in the presensce of a generic filter that this dream class can become a real universe. What I am saying is that even if someone holds the view that the multiverse is really an illusion experienced in the universe then that person could not be talking about the actual multiverse, but only a reflection.

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Very nice story, especially, "Via a generic filter and forcing relation, they travel to universe $X$ where the inhabitant of universe $V$ can now see with his own eyes that in this world the continuum is indeed the second uncountable cardinal!" Do I smell a sci-fi novel brewing? –  Quinn Culver Mar 22 '12 at 22:20
@Erin: I have a question regarding the following: "In every universe, there are classes of names of other universes [forcing extensions?] with a variety of sizes of the continuum, for example. But these classes of names are not themselves the universes to which they point since if they were then the universe which dreams of them would not be coherent [I presume because then one would end up with an inconsistent theory]. It is only in the presence of a generic filter [a consistent set of sets?] that this dream class can be can become a real universe." –  Thomas Benjamin Mar 25 '12 at 3:51
Here is my question: Consider, for example, the formal language of the integers which contains following rule for constructing 'names', (a)^(1/n) where a and n are integers and n is not equal to 0. With this rule one can come up with the following names (-1)^(1/2), (2)^(1/2). Obviously (-1)^(1/2) names nothing in the reals and (2)^(1/2) names nothing in the rationals although according to the rule (a)^(1/n) where a, n are integers and n not= 0 these are perfectly well-formed names. Is this the sense in which you are speaking about –  Thomas Benjamin Mar 25 '12 at 4:04
"In every universe, there are classes of names of other universes"? Now consider the formal language for L, the constructable universe. Are you, in analogy with my aforementioned simple examples (if I am not stretching this analogy too far, of course) saying that the formal language of L contains the names of all possible Cohen reals, random reals, reals that collapse cardinals, essentially all possible forcing extensions of L? If so, assuming that the formal language of L is the exact syntactic replica of the universe L then what are these names without referents in L? Gaps –  Thomas Benjamin Mar 25 '12 at 4:17
(as per the formal definition of gap in the theory of linear orders)? Since it seems incoherent to say that reals that collapse cardinals and reals that don't collapse cardinals exist on a single 'real line', it would seem reasonable to say that each type of real in this case exists on a separate 'branch' of a tree that represents a partial order. If that is so then the set of names in the formal language of L would seem to represent the maximal power set (the set of all possible sets definable in the formal language of L). Is this what you are trying to say, or have I misunderstood? –  Thomas Benjamin Mar 25 '12 at 4:33