Thursday, October 08, 2020

Revisiting the Continuum Hypothesis

I have been thinking about CH lately for two reasons

1) I reread the article

Hilbert's First Problem: The Continuum Hypothesis by Donald Martin from Proceedings of Symposia  in Pure Mathematics: Mathematical developments arising from Hilbert Problems. 1976. (For a book review of the symposia and, The Honor Class, also about Hilbert's problems, see here.)

The article takes the point of view that CH CAN have an answer. He discusses large cardinals (why assuming they exist is plausible, but alas, that assumption does not seem to resolve CH) and Projective Det.  (why assuming it is true is plausible, but alas, that assumption does not seem to resolve CH).

(A set A \subseteq {0,1}^omega is DETERMINED if either Alice or Bob has a winning strategy in the following non-fun game: they alternate picking bits a_1, b_1, a_2, b_2, ... with Alice going first. If a_1 b_1 a_2 b_2... IS IN A then Alice wins, IF NOT then Bob wins. Martin showed that all Borel sets are determined. Proj Det is the statement that all projections of Borel sets are determined. AD is the axiom that ALL sets A are determined. It contradicts AC.)

But what really inspired this post is the last paragraph:

Throughout the latter part of my discussion, I have been assuming a naive and uncritical attitude towards CH. While this is in fact my attitude, I by no means wish to dismiss the opposite viewpoint.  Those that argue that the concept of set is not sufficiently clear to fix the truth-value of CH have a position that is at present difficult to assail. As long as no new axiom is found which decides CH, their case will continue to grow stronger, and our assertions that the meaning of CH is clear will sound more and more empty.

2) Scott Aaronson mentioned in a blog post (see here) that  he has read and understood the proof that CH is independent of set theory.

SO, this seemed like a good time to revisit thoughts on CH.

 I took a very short poll, just two people, about CH: Stephen Fenner (in a perfect world he would be a set theorists) and Scott Aaronson (having JUST read the proof that CH is ind.  he has thought about it recently).

Here are some thoughts of theirs and mine

1) All three of us are Platonists with regard to the Naturals (I was surprised to find recently that there are people who are not!) but not with regard to the reals.  So we would be OKAY with having CH have no answer.

2) All three of us  agree that it would be nice if SOME axiom was both

a) Intuitively appealing or aesthetically appealing ,  and

b) resolved CH.

I always thought that (a) would be the hard part-- or at least getting everyone (not sure who we are talking about) to AGREE on a new axiom. But even getting an axiom to resolve CH seems hard.  Large cardinals don't seem to do it, and various forms of Determinacy don't seem to do it.

Scott reminded me of Freiling's Axiom of Symmetry (see here) which IS intuitive and DOES resolve CH (its false) though there are problems with it--- a minor variant   of it contradicts AC (I am QUITE FINE with that since AC implies Banach-Tarski which Darling says shows `Math is broken'.)

Stephen recalled some of Hugh Woodin's opinions of CH, but Hugh seems to have changed his mind from NOT(CH): 2^{aleph_0} = aleph_2, to CH:  2^{aleph_0} = aleph_1.(See here.)

3) All three of would be okay with V=L, though note that this would put many set theorists out of work. All the math that applies to the real world would still be intact.  I wonder if in an alternative history the reaction to Russell's paradox would be a formulation of set theory where V=L. We would KNOW that CH is true, KNOW that AC is true. We would know a lot about L but less about forcing.

4) Which Geometry is true: Euclidian, Riemannian, others? This is now regarded as a silly question: Right Tool, Right Job! If you build a bridge use Euclid. If you are doing astronomy use Riemann. Might Set Theory go the same way? It would be AWESOME if Scott Aaronson found some quantum thing where assuming 2^{aleph_0} = aleph_2 was the right way to model it.

5) If I was more plugged into the set theory community I might do a poll of set theorists, about CH. Actually, someone sort-of already has. Penelope Maddy has two excellent and readable articles where she studies what set theorists believe and why.

Believing The Axioms Ihere

Believing The Axioms IIhere

Those articles were written in 1988. I wonder if they need an update.


  1. Quite rightly, bullet point #4 above illustrates that the parallel postulate is only true or false in a specified model. So shouldn't it be true that CH is either true or false in the specified model called "the cumulative hierarchy?" Or is the cumulative hierarchy "underdetermined" in some way? Model theory makes no allowances for semantic underdetermination because every sentence is either true or false in a given model. Is this somehow philosophically wrong or naive? If so, how? And this touches on bullet point #1. Are you guys not Platonist regarding the cumulative hierarchy? I grant you that ZFC is not quite as "intuitively" elementary as PA but it feels close, right? (I am aware, of course, that mathematically ZFC and PA are NOT close in power.) Is it not the case that both PA and ZFC both "indicate" canonical models in spite of the existence of non-standard models, i.e. the natural numbers and the cumulative hierarchy, and that every sentence should indeed either be true or false in the canonical model?

    As an aside, if "Platonism" makes you uncomfortable at cocktail parties you may wish to switch to a less metaphysical mathematical realism. One such can be found described quite well in Tragesser's book "Husserl and Realism in Logic and Mathematics."

    1. You're 100% right. Disputing that CH has a truth value in the cumulative hierarchy requires disputing that P(P(ℕ)) is well-defined (i.e. claiming that it is undetermined), or something even sillier. It's postmodern mathematics, best suited for (a) contrarians, (b) philosophers, and (c) mathematicians who need to justify to granting agencies their work redoing the foundations of mathematics. I'm sad to say that groups (a) and (c) include a few extremely smart and knowledgeable people. They are few, but they are loud, and their story is titillating for a broad audience, so it persists.

  2. Regarding your proposal for how to settle CH by finding a new axiom that is intuitively appealing and yet still settles CH, I argue in my paper "Is the dream solution of the continuum hypothesis attainable?" ( that this is impossible. The essence of my argument is that for us to learn that an axiom candidate decides CH will automatically undermine any attempt to take it as natural or intuitive, because of our prior extensive familiarity (via forcing and so on) with set-theoretic worlds having the opposite outcome.

    1. Does your argument rule out the possibility of finding a new, natural axiom (presumably based on new phenomenological insights) which is specifically applicable to the cumulative hierarchy (rather than ALL set-theoretic worlds)?

    2. It does not. Hamkins is overstating his position with the word "impossible" in the above comment. An excerpt from the paper better classifies his argument as a prediction ("the entire episode" refers to his exposition of the reception of Freiling's Axiom):

      "The entire episode bears out the pattern of response I predict for any attempted use of the dream solution template, namely, a rejection of the new axiom from a perspective of deep mathematical experience with the contrary."

      To be clear, his prediction might be correct. That seems to be the mood of the times. But that might have more to do with humility and practicality than anything deep; it seems unwise to devote a lot of energy to solving a problem that the greatest geniuses in recent times failed to solve.

      Hamkins ends his paper with the following (note there is an implied vice-versa after "flawed"):

      "Before we will be able to accept 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. 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."

      That is eloquent but misleading. A dream solution need not undermine the reality of nonstandard models. Nor the usefulness or elegance of nonstandard models.

      A dream solution could even reinforce the usefulness or elegance of nonstandard models; it needn't appear as "a beautiful landscape, a shining city on a hill." It is possible there will be a dream solution proving |ℝ| = Aleph_42 in the cumulative hierarchy (equivalently, in the universe of third-order arithmetic). I would be as shocked as everyone else, but it wouldn't be entirely without precedent; to me, a physics novice, Newtonian mechanics is more of a "beautiful landscape, a shining city on a hill" than general relativity, quantum mechanics, and string theory. Perhaps the universe of iterated powersets is, surprisingly, as weird as physical reality? We are stuck with physical reality, but if the cumulative hierarchy turns out to be ugly and annoying enough to human mathematicians, we could always redefine "the standard model" of set theory, without any intellectual dishonesty.