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The Continuum Hypothesis

A basic reference is Godel'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 Godel'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 from some new axioms, but this turned out to be fallacious. (See Ellentuck, ``Godel's Square Axioms for the Continuum", Mathematische Annalen 1975.)

Maddy's ``Believing the Axioms", Journal of Symbolic Logic 1988 (in 2 parts) 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:

Cantor's reasons for believing CH aren't all that persuasive today.
Godel'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.
If GCH is true, this implies that 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.
Most of those who disbelieve CH think that the continuum is likely to have very large cardinality, rather than (as Godel 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."
There are also a few arguments in favour of CH, e.g. there's an argument that 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.
Most large cardinal axioms (asserting the existence of cardinals with various properties of hugeness: these are usually derived either from considering the hugeness of 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.
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 (e.g. Martin's axiom), which are ``maximality" principles (in the sense of (2) above), imply CH. The strongest (Martin's maximum) implies that . Of course the ``truth" or otherwise of all these axioms is controversial.
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 we have 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.

The integers are not only a bedrock, but also any finite number of power sets seem to be quite natural Intuitively are also natural which would point towards the fact that CH may be 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, CH is somewhat intuitively plausible. But after reading all this, it does seem that the weight of evidence tend to point the other way.

The following is from Bill Allen on Freiling's Axiom of Symmetry. This is a good one to run your intuitions by.

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 with . Define as . 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 . 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 union of countable sets, together with a single size set Y, the cardinality of X is also , so X is not all of R. Let a be in , 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.

Freiling's proof, 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.

The section above was excerpted from a posting from David Chalmers, of Indiana University.