From: mareg@primrose.csv.warwick.ac.uk () Subject: Re: "How bad can 2^(Aleph_0) be?" history question Date: Mon, 4 Jun 2001 09:32:46 +0000 (UTC) Newsgroups: sci.math Summary: Natural sets of cardinality aleph-1 in mathematics In article <20010604000442.09149.00000890@nso-cv.aol.com>, kramsay@aol.commangled (Keith Ramsay) writes: >In article <5f6a0413.0106031620.7dbaea71@posting.google.com>, >sul01@altavista.com (John Sullivan) writes: >|For the past forty years I have been under the false impression that the >|aleph numbers are defined in terms of sets of all subsets, and whenever I >|have read about the continuum hypothesis, nothing until now has jolted me >|out of that false impression. This thread has been quite a revelation. > >It appears to be a common misimpression. George Gamov, the famous >physicist, wrote a book called _1,2,3 infinity_ which I thought was >mostly pretty good (although it's been awhile since I've looked at it) >but it makes this mistake too. > >Sets having cardinality 2^aleph_0 and 2^2^aleph_0 (also called >beth_1 and beth_2) appear more often in mathematics than sets of >cardinality aleph_1 or aleph_2, so in some ways that's the more >familiar sequence of cardinalities. Here is an example of a theorem in group theory that involves aleph_1. Let p and q be distinct primes and let G be an elementary abelian p-group (i.e. abelian p-group of exponent p). Does there there exist an elementary abelian q-group N and a nonsplit extension E of N by G? (That is, a group E having a normal subgroup N with E/N isomorphic to G, but with no subgroup C such that CN=E and C ^ N = 1.) The answer (surprisingly) is yes if and only if |G| = aleph_1. More generally, there exists a G-module N over the field of order q such that the cohomology group H^n(G,N) != 0 iff |G| = aleph_{n-1}. Derek Holt. ============================================================================== From: dlrenfro@gateway.net (Dave L. Renfro) Subject: Re: "How bad can 2^(Aleph_0) be?" history question Date: 5 Jun 2001 15:28:41 -0400 Newsgroups: sci.math Derek Holt [sci.math Mon, 4 Jun 2001 09:32:46 +0000 (UTC)] wrote, in part and quoting Keith Ramsay (04 Jun 2001 04:04:42 GMT): >> Sets having cardinality 2^aleph_0 and 2^2^aleph_0 (also called >> beth_1 and beth_2) appear more often in mathematics than sets of >> cardinality aleph_1 or aleph_2, so in some ways that's the more >> familiar sequence of cardinalities. > > Here is an example of a theorem in group theory that involves > aleph_1. There are well-known examples in descriptive set theory. Of course, the Borel hierarchy for the Borel sets has (ordinal) length aleph_1. But there's a better example that grew out of attempts to prove CH. In trying to prove CH, early researchers focused on nicely definable classes of sets when a proof for arbitrary sets wasn't forthcoming. Recall the Borel hierarchy --->>> closed ---> F_sigma ---> F_sigma-delta ---> ... open ---> G_delta ---> G_delta-sigma ---> ... We say that a class of sets satisfies CH (or the sets individually satisfy CH) if each of the sets has countable cardinality or cardinality c. A stronger property, in the sense that for some models of ZFC a class of sets can satisfy CH but not satisfy what I'm about to describe, is that every uncountable set in the collection contains a nonempty perfect subset. The fact that this is actually a stronger property was proved by Godel. Godel showed in 1938 that every subset of the reals can satisfy CH and yet this perfect set property can fail for some uncountable projective sets (see below). [More precisely, Godel showed that the axiom of constructibility implies there exists an uncountable co-analytic set with no nonempty perfect subset.] Many of the results I mention below for CH were actually proved by verifying this stronger perfect set property. One could argue that Cantor showed open sets satisfy CH in 1873-74 when he introduced cardinality, since it is easy to see that every open interval has the same cardinality as the reals. Around 1882 Cantor proved that closed sets satisfy CH. It immediately follows that the F_sigma sets also satisfy CH. In 1903 W. H. Young proved that the G_delta sets satisfy CH. From this it immediately follows that the G_delta-sigma sets also satisfy CH. Nothing further was known until Hausdorff and Aleksandrov (independently) proved that every Borel set satisfies CH, around 1915-16. In 1916 Suslin found an error in Lebesgue's influential 1905 paper on Borel sets (and on various definability issues). Lebesgue claimed that the projection of any Borel set is a Borel set while proving something else. [The statement Lebesgue gave an incorrect proof of wound up being correct, though. This was shown by Lusin in 1917.] Suslin introduced a wider class of sets obtained by looking at all continuous images of Borel sets. [It is equivalent to just consider projections from R^2 into R of G_delta sets in R^2.] I think Suslin originally called them A-sets (for Aleksandrov, I believe), but after a while they became known as analytic sets. Suslin proved in 1916 that the analytic sets satisfy CH. In 1925 (by this time, at least) Lusin introduced the hierarchy of projective sets. Let 'A', 'C', and 'P' represent analytic, complements of, and projections of (equivalently in the present context, continuous images of). Then the projective hierarchy is A ---> PCA ---> PCPCA ---> ... CA ---> CPCA ---> CPCPCA ---> ... Note: This can be continued through the countable ordinals by taking unions at the limit ordinals. Even if you do this up to aleph_1 (the ordinal), you're still quite a bit away from considering ALL subsets of the reals, since you're still only looking at c subsets of the reals. [The proof is just like the proof that there are c Borel sets.] The CA sets are also called the co-analytic sets. Souslin (1916, I think) showed that the CA sets are either countable, have cardinality aleph_1, or have cardinality c. [What he actually proved is that every CA set is a union of aleph_1 many Borel sets, from which the CH property immediately follows.] Around 1925-26, Lusin and Sierpinski proved that PCA sets can have cardinality < aleph_1, = aleph_1, or = c. [For a proof, see around pp. 518-520 of Jech's "Set Theory", 1978.] I know that it is consistent in ZFC for CPCA sets to have a variety of cardinalities, but I don't know which ones are possible. Probably every aleph_n for finite n's can arise, but I don't know for sure if even this is the case. CH can be extended to the higher projective classes with certain (not-obviously-connected-to-CH axioms). For example, Solovay proved in 1969 that if there exists a measurable cardinal, then the PCA sets satisfy CH. [In fact, he showed they have the stronger perfect set property I mentioned earlier.] Martin showed sometime in the 1970's that the existence of a measurable cardinal implies that every PCPCA set has cardinality < aleph_1, = aleph_1, = aleph_2, or c. Woodin showed in 1984 that if there exists a supercompact cardinal, then every projective set has CH. [The perfect subset property, and some other nice properties, in fact.] Dave L. Renfro