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Re: [HM] Who invented the idea of an open set?
Posted:
Sep 20, 1999 8:37 PM
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I would like to elaborate a bit on the (apparent) strangeness
(experienced by Bill) of the statement
(1) "CH holds for closed sets".
I'm far from being an expert on these matters and hope that those
aficionados of descriptive set theory (or even "set theoretic real
analysis") may have more to say.
CH asserts that the cardinal card (IR) is the cardinal NEXT to card
(IN). Formulated in this way, it is not obvious how to connect CH with
statements like (1). However, CH can be stated in many different ways,
the easiest being the following:
(2) For every subset X of IR, CH (X),
where
CH (X) := IF card (X) > card (IN) THEN card (X) = card (IR).
It became well known (after Cohen's work) that one cannot prove CH (X)
for EVERY set X of real numbers on the base of the traditional axioms
of set theory (ZFC). However, long before that many mathematicians
tried to prove (or disprove) CH (X) for sets X of increasing
complexities:
(3) Cantor (1884) showed that CH (X) holds if X is closed. In short,
"CH holds for closed sets". This is essentially the content of the
famous Cantor-Bendixson theorem.
(4) Pavel Aleksandrov (1914) and Hausdorff (1916) established
independently that CH (X) is true for every Borel set X, so that "CH
holds for Borel sets". This astonishing result extends that of
Cantor's, since each closed set is a Borelian.
(5) Next came the Russian mathematicians Nikolai Luzin and Mikhail
Suslin. As a consequence of their combined work it was established,
inter alia, that "CH holds for analytic sets". (These are the
projections (continuous images) of Borel sets.) This result was
announced by Suslin (1917) and proved by Luzin and Sierpinski (1918,
1923). In the third edition of his Mengenlehre, Hausdorff puts things
in this way: "In a complete separable space [now usually called a
Polish space] every Suslin [analytic] set ÃÂ and thus, in particular,
every Borel set ÃÂ is either at most countable or of cardinality aleph
[of the continuum]." He then remarked that this result "is the most
comprehensive theorem on cardinality that we know" and that "For by
far the greatest number of point sets, the question as to their
cardinality, and with it the problem of the continuum, remain
unelucidated."
(6) How far can we go in this progression? In the fascinating
classification of sets of reals known as the projective hierarchy, the
analytic sets are those on the "box" sigma_1^1. As far as CH is
concerned, the real difficulties begin to appear on the class
sigma_2^1: it's impossible to go beyond without additional axioms.
Actually, Goedel (1938) showed the existence of models (under V = L)
in which CH fails on sigma_2^1. Later (1947) he insinuated a program
to supplement set theory "without arbitrariness by new axioms" (of
infinity) in order to settle the "real" truth value of CH. He was
partially right: if it is assumed that there exists a measurable
cardinal (Ulam, 1930) then one can prove that "CH holds for sigma_2^1
sets" as a consequence of work by Solovay (1969). But oddly enough, no
plausible axiom of infinity proposed so far is powerful enough to
decide CH. Many experts (as well as distinguished outsiders) have
argued that, contrary to the hopes of Hilbert and Goedel, CH is an
inherently vague statement.
For a recent historical survey see the article The mathematical
development of set theory from Cantor to Cohen, The Bulletin of
Symbolic Logic, vol. 2, March 1996, 1-71 (downloadable in ps format).
HYPOTHESES VS CONJECTURES
It would also be very interesting to discuss why some unsettled
mathematical statements are called "hypotheses" instead of
"conjectures". For example, some say that an hypothesis (in this
context) is a "plausible conjecture" while others (like Andr/e Weil)
say that it is simply "wishful thinking". Of course, we can say that
an unproved statement S becomes an "hypothesis" when many interesting
theorems of the form "if S is true, then ..." are known. (This is
really the case with CH and the Riemann hypothesis. But what about
many other famous "conjectures"?) By the way, in his essay On the
Infinite (1925) Hilbert refers to the question of whether CH is true
or not as the "famous problem of the continuum"; the word "hypothesis"
is not used. But two years later (The foundations of mathematics,
1927) he talks about "the proof or refutation of Cantor's continuum
hypothesis". I think that the use of "hypothesis" here became more
popular and well-established only after the 1934 monograph of
Sierpin/ski, Hypothe/se du continu.
Carlos Cesar
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