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Subject: sci.math FAQ: The Continuum Hypothesis
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Archive-Name: sci-math-faq/AC/ContinuumHyp
Last-modified: December 8, 1994
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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 C = aleph_2 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:

    1. Cantor's reasons for believing CH aren't all that persuasive
       today.
    2. 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 not 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 not GCH.
    4. Most of those who disbelieve CH think that the continuum is likely
       to have very large cardinality, rather than aleph_2 (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."
    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 not CH. And
       CH is much more likely to settle various outstanding results than
       is not 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 (e.g. Martin's axiom), which are
       ``maximality" principles (in the sense of (2) above), imply not
       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 not 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 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.

   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.




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