Newsgroups: sci.math
From: [email protected] (David Chalmers)
Subject: Continuum Hypothesis - Summary
Date: Wed, 13 Mar 91 21:29:47 GMT
Thanks to all the people who responded to my enquiry about the status of the Continuum Hypothesis. This is a really fascinating subject, which I could waste far too much time on. The following is a summary of some aspects of the feeling I got for the problems. This will be old-hat to set theorists, and no doubt there are a couple of embarrassing misunderstandings, but it might be of some interest to non-professionals.
A basic reference is Gödel's "What is Cantor's Continuum Problem?", from 1947 with a 1963 supplement, reprinted in Benacerraf and Putnam's collection Philosophy of Mathematics. This outlines Gödel's generally anti-CH views, giving some "implausible" consequences of CH.
"I believe that adding up all that has been said one has good reason to suspect that the role of the continuum problem in set theory will be to lead to the discovery of new axioms which will make it possible to disprove Cantor's conjecture."
At one stage he believed he had a proof that C = aleph_2 from some new axioms, but this turned out to be fallacious. (See Ellentuck, "Gödel's Square Axioms for the Continuum", Mathematische Annalen 1975.)
Two people pointed me to a recent paper by Maddy: "Believing the Axioms", Journal of Symbolic Logic 1988 (in 2 parts). This is an extremely interesting paper and a lot of fun to read. A bonus is that it gives a non-set-theorist who knows the basics a good feeling for a lot of issues in contemporary set theory.
Most of the first part is devoted to "plausible arguments" for or against CH: how it stands relative to both other possible axioms and to various set-theoretic "rules of thumb". One gets the feeling that the weight of the arguments is against CH, although Maddy says that many "younger members" of the set-theoretic community are becoming more sympathetic to CH than their elders. There's far too much here for me to be able to go into it in much detail.
Some highlights from Maddy's discussion, also incorporating a few things that other people sent me:
(1) Cantor's reasons for believing CH aren't all that persuasive today.
(2) Gödel's proof of the consistency of CH shows that CH follows from ZFC plus the Axiom of Constructibility (V=L, roughly that the set-theoretic universe = the constructible universe). However, most set-theorists seem to find Constructiblity implausible and much too restrictive. It's an example of a "minimizing" principle, which tends to cut down on the number of sets admitted to one's universe. Apparently "maximizing" principles meet with much more sympathy from set theorists. Such principles are more compatible with ~CH than with CH.
(3) If GCH is true, this implies that aleph_0 has certain unique properties: e.g. that it's that cardinal before which GCH is false and after which it is true. Some would like to believe that the set-theoretic universe is more "uniform" (homogeneous) than that, without this kind of singular occurrence. Such a "uniformity" principle tends to imply ~GCH.
(4) Most of those who disbelieve CH think that the continuum is likely to have very large cardinality, rather than aleph_2 (as Gödel seems to have suggested). Even Cohen, a professed formalist, argues that the power set operation is a strong operation that should yield sets much larger than those reached quickly by stepping forward through the ordinals:
"This point of view regards C as an incredibly rich set given to us by a bold new axiom, which can never be approached by any piecemeal process of construction."
(5) There are also a few arguments in favour of CH, e.g. there's an argument that not-CH is restrictive (in the sense of (2) above). Also, CH is much easier to force (Cohen's method) than ~CH. And CH is much more likely to settle various outstanding results than is ~CH, which tends to be neutral on these results.
(6) Most large cardinal axioms (asserting the existence of cardinals with various properties of hugeness: these are usually derived either from considering the hugeness of aleph_0 compared to the finite cardinals and applying uniformity, or from considering the hugeness of V (the set-theoretic universe) relative to all sets and applying "reflection") don't seem to settle CH one way or the other.
(7) Various other axioms have some bearing. Axioms of determinacy restrict the class of sets of reals that might be counterexamples to CH. Various forcing axioms which are "maximality" principles (in the sense of (2) above), imply ~CH. The strongest (Martin's maximum) implies that C = aleph_2. Of course the "truth" or otherwise of all these axioms is controversial.
(8) Freiling's principle about "throwing darts at the real line" is a seemingly very plausible principle, not involving large cardinals at all, from which ~CH immediately follows. Freiling's paper (JSL 1986) is a good read. More on this at the end of this message.
Of course I've conspicuously avoided saying anything about whether it's even reasonable to suppose that CH has a determinate truth-value. Formalists will argue that we may choose to make it come out whichever way we want, depending on the system we work in. On the other hand, the mere fact of its independence from ZFC shouldn't immediately lead us to this conclusion - this would be assigning ZFC a privileged status which it hasn't necessarily earned. Indeed, Maddy points out that various axioms within ZFC (notably the Axiom of Choice, and also Replacement) were adopted for extrinsic reasons (e.g. "usefulness") as well as for "intrinsic" reasons (e.g. "intuitiveness"). Further axioms, from which CH might be settled, might well be adopted for such reasons.
One set-theorist correspondent said that set-theorists themselves are very loathe to talk about "truth" or "falsity" of such claims. (They're prepared to concede that 2+2=4 is true, but as soon as you move beyond the integers trouble starts. e.g. most were wary even of suggesting that the Riemann Hypothesis necessarily has a determinate truth-value.) On the other hand, Maddy's contemporaries discussed in her paper seemed quite happy to speculate about the "truth" or "falsity" of CH.
Personally, not only do I see the integers as bedrock, but I'm also prepared to take any finite number of power sets before I have any problems. At least that far, I'm a diehard Platonist. I'm considerably less sure whether to be a diehard Platonist about the paradise of ridiculously big cardinals. But my intuition is happy with reals and sets of reals, so as a naive non-set-theorist I'm sticking to the belief that CH is determinate one way or the other. As one correspondent suggested, the question of the determinateness of CH is perhaps the single best way to separate the Platonists from the formalists.
And is it true or false? Well, I'd always found CH somewhat intuitively plausible. But after reading all this, it does seem that the weight of evidence tend to point the other way. On the other hand, that's only a second-order approximation based on limited knowledge - more subtle and sophisticated arguments (third-order approximations) might begin to push things the other way. Still, if I had to lay money on it (God, of course, knows the right answer and could settle the bet), I'd feel safer with the money on ~CH.
I've enclosed (with permission) a brief but enlightening message from Bill Allen on Freiling's Axiom of Symmetry. This is a good one to run your intuitions by.
Many thanks to Bill Allen, James Cummings, David Feldman, Torkel Franzen, Calvin Ostrum, Keith Ramsay, and Peter Suber. I'd be very interested to hear more about this subject.
--Dave Chalmers.
Date: Fri, 8 Mar 91 01:11:37 -0800
From: William C. Allen ([email protected])
To: [email protected]
Subject: Re: Status of the Continuum Hypothesis
Organization: UCLA Mathematics Department
I have a colleague, Chris Freiling, here at UCLA, who, a few years ago proved that a certain plausible assertion is equivalent to the negation of continuum hypothesis (CH) over ZFC. The idea is something like this:
Let A be the set of functions mapping Real Numbers into countable sets of Real Numbers. Given a function f in A, and some arbitrary real numbers x and y, we see that x is in f(y) with probability 0, i.e. x is not in f(y) with probability 1. Similarly, y is not in f(x) with probability 1. Let AX be the axiom which states
"for every f in A, there exist x and y such that x is not in f(y) and y is not in f(x)"
The intuitive justification for AX is that we can find the x and y by choosing them at random.
In ZFC, AX = not CH.
proof:
If CH holds, then well-order R as r_0, r_1, .... , r_x, ... with x < aleph_1. Define f(r_x) as {r_y : y <= x}. Then f is a function which witnesses the falsity of AX.
If CH fails, then let f be some member of A. Let Y be a subset of R of cardinality aleph_1. Then Y is a proper subset. Let X be the union of all the sets f(y) with y in Y, together with Y. Then, as X is an aleph_1 union of countable sets, together with a single aleph_1 size set Y, the cardinality of X is also aleph_1, so X is not all of R. Let a be in R \ X, so that a is not in f(y) for any y in Y. Since f(a) is countable, there has to be some b in Y such that b is not in f(a). Thus we have shown that there must exist a and b such that a is not in f(b) and b is not in f(a). So AX holds. -|
I like Freiling's proof, since it does not invoke large cardinals or intense infinitary combinatorics to make the point that CH implies counter-intuitive propositions. Freiling has also pointed out that the natural extension of AX is AXL (notation mine), where AXL is AX with the notion of countable replaced by Lebesgue Measure zero. Freiling has established some interesting Fubini-type theorems using AXL.
See "Axioms of Symmetry: Throwing Darts at the Real Line", by Freiling, Journal of Symbolic Logic, 51, pages 190-200. An extension of this work appears in "Some properties of large filters", by Freiling and Payne, in the JSL, LIII, pages 1027-1035, but Chris tells me he's not as fond of the latter paper as he is the former.
--Bill Allen
--UCLA
From: [email protected]
Date: Thu, 28 Mar 91 15:03 EST
Subject: Continuum Hypothesis
I just came across your posting about CH and found it quite interesting. A few comments:
One set-theorist correspondent said that set-theorists themselves are very loathe to talk about "truth" or "falsity" of such claims. (They're prepared to concede that 2+2=4 is true, but as soon as you move beyond the integers trouble starts. e.g. most were wary even of suggesting that the Riemann Hypothesis necessarily has a determinate truth-value.) On the other hand, Maddy's contemporaries discussed in her paper seemed quite happy to speculate about the "truth" or "falsity" of CH.
Many set theorists are loathe to talk about CH being "true" or "false", but I'm not sure how many would agree that trouble starts as soon as you move beyond the integers. I recently did a small survey (only about 10 people) at a set theory conference. I had a list of 10 statements, and for each one you had to choose among 5 choices:
The statements were:
One or two people were Platonists the whole way-- restricted their responses to the first three possibilities. Many were Platonists through the Riemann Hypothesis, and then switched to "neither true nor false". I was one of the few who became uncertain once I got beyond the integers. (Remarkably, one person turned in an answer sheet with many answers filled in and crossed out, and wrote "can't decide". In the end, the only thing he was willing to commit himself to was that 2+2=4 is true!)
Personally, not only do I see the integers as bedrock, but I'm also prepared to take any finite number of power sets before I have any problems. At least that far, I'm a diehard Platonist. I'm considerably less sure whether to be a diehard Platonist about the paradise of ridiculously big cardinals. But my intuition is happy with reals and sets of reals, so as a naive non-set-theorist I'm sticking to the belief that CH is determinate one way or the other. As one correspondent suggested, the question of the determinateness of CH is perhaps the single best way to separate the Platonists from the formalists.
Suppose we let X_0=the set of natural numbers, and let X_{n+1}=the power set of X_n. Then you seem to be willing to accept all of these sets as "existing" in some determinate sense. What about their union? What about the power set of their union? If you won't accept these, aren't you drawing a pretty arbitrary line? If you do accept them, then where do you stop? As for using the determinateness of CH as a test to separate Platonists from formalists: I think it's important to recognize that Platonism isn't an all-or-nothing affair. This may be a good test for a certain level of Platonism. I fail--I don't think CH has a determinate truth value. But some people (like intuitionists) don't even think Fermat's last theorem has a determinate truth value. This might be a good test for a lower level of Platonism, and I pass this test--I can't conceive of Fermat's theorem not having a determinate truth value. (By the way, on my survey everyone said Fermat's theorem was either true or false but they didn't know which, except for the guy who only would commit himself to 2+2=4.)
Dan Velleman
Dept. of Mathematics Amherst College
[email protected]