Vaught's conjecture for superstable theories of finite rank (Q952488)

Vaught's Conjecture, in spite of its apparently narrow and focussed statement, has been one of the central questions of abstract model theory for over forty-five years. This paper represents the most significant (and complicated) progress on the general question since the mid seventies. A good overview of the state of work on Vaught's Conjecture (mostly for theories of modules) as of about 2000 can be found in [\textit{V. A. Puninskaya}, J. Math. Sci., New York 109, No. 3, 1649--1668 (2002; Zbl 1020.03033)]; an excellent, much earlier, survey emphasizing why Vaught's Conjecture should be considered to be a significant question can be found in [\textit{D. Lascar}, J. Symb. Log. 50, 973--982 (1985; Zbl 0593.03015)]. \textit{R. L. Vaught} [``Denumerable models of complete theories'', in: Infinitistic Methods, Proc. Symp. Foundations Math., Warsaw 1959, 303--321 (1961; Zbl 0113.24302)] conjectured that a countable complete first-order theory has either countably many or \(2^{\aleph_{0}}\) many countable models (up to isomorphism). A fairly straightforward argument of \textit{M. Morley} [J. Symb. Log. 35, 14--18 (1970; Zbl 0196.01002)] shows that a countable complete theory has countably many, \(\aleph_{1}\) many, or \(2^{\aleph_{0}}\) many countable models, so in a certain sense the question boils down to whether first-order logic can distinguish between \(\aleph_{1}\) and \(2^{\aleph_{0}}\), lending the problem some purely foundational interest. The early history of the problem involved solutions (often technically very complicated) for special kinds of complete theories (linear orders, trees; A. Miller showed [\textit{J. R. Steel}, Lect. Notes Math. 689, 193--208 (1978; Zbl 0403.03027)] that Vaught's Conjecture for theories of partial order was equivalent to the full conjecture). There has also been an extended investigation of the conjecture for various algebraic theories. In particular, the fruitful development of the model theory of modules from the mid-1970s led to many results confirming Vaught's Conjecture for particular classes of theories of modules, for which, see the paper of Puninskaya referenced above. This proved to be a key part of Buechler's investigations reported here. Meanwhile, on the purely model-theoretic side, the main significant result was that of \textit{S. Shelah, L. Harrington} and \textit{M. Makkai} [Isr. J. Math. 49, 259--280 (1984; Zbl 0584.03021)], who proved that Vaught's Conjecture holds for \(\omega\)-stable theories, with a proof that was already technically quite deep and complicated. The next natural step would be to consider superstable theories; and it is this task which has occupied the author's attention for some time. In a series of papers in the late 1980s, the author [Lect. Notes Math. 1292, 32--71 (1987; Zbl 0655.03022); J. Symb. Log. 53, No. 2, 625--635 (1988; Zbl 0665.03020); J. Symb. Log. 53, No. 3, 975--979 (1988; Zbl 0665.03021)] proved Vaught's Conjecture for a wide class of weakly minimal theories; following \textit{L. Newelski}'s [Fundam. Math. 134, No. 2, 143--155 (1990; Zbl 0716.03024)] proof of Saffe's Conjecture, these results in particular gave Vaught's Conjecture for superstable theories of \(U\)-rank 1. The present paper, proving Vaught's Conjecture for superstable theories of finite \(U\)-rank, has been anticipated for some time (a preprint of related material appeared as early as 1993). The long delay in publication is explained by the extreme technical complexity of the work. Here one must praise the author for his efforts. This paper is very carefully organized into nine, mostly short, sections; the background for each section is carefully explained; the connections between the sections are carefully delineated; and, most importantly, at the end of Section 1, following a detailed exposition of all the terminology and techniques necessary to the proof, there is a carefully written informal outline of how the proof works. The general feeling has been that theories which are ``uncomplicated'' in some model-theoretic sense, such as the ones under consideration here, should have some sort of structure theory associated with their models; and a good structure theory should at least allow the counting of isomorphism types of models (and, one would hope, to do a lot more of interest). That is what Buechler does here. The key result is the Structure Theorem: Let \(T\) be a countable superstable theory of finite \(U\)-rank and with less than continuum many countable models. Then for each countable model \(M\) of \(T\) there is a finite \(A\subset M\) and \(J\subset M\) such that \(M\) is prime over\(A\cup J\), \(J\) is \(A\)-independent, and \(\{\text{stp}(a/A):a\in J\}\) is finite. From this, Vaught's conjecture follows immediately. For the remainder of this discussion, the theory being considered is countable, superstable with finite \(U\)-rank, and with less than continuum many countable models. Sections 2--5 are devoted to establishing the properties of irreducible elements (Theorem 2.1). [\(a\in M\) is \textit{reducible} if there are \(b_{0},b_{1}\in\text{acl}\{a\}\) such that \(\text{tp}(a/b_{0}b_{1})\) is isolated and \(a\notin\text{acl}(b_{i})\), \(i=0,1\).] It is here that a great deal of the technical complexity occurs; the author's meticulous attention to detail and organization is an essential aid in following these arguments. It is by using irreducible elements that the sets of the Structure Theorem are constructed. The other major point is the construction of the ``structure group''; essentially a reduction of the general case to the case of ``generalized modules'': abelian structures in the sense of \textit{E. R. Fisher} [Lect. Notes Math. 616, 270--322 (1977; Zbl 0414.18001)]. One of the early ``special case'' confirmations of Vaught's Conjecture was that of \textit{J. T. Baldwin} and \textit{R. N. McKenzie} [Algebra Univers. 15, 359--384 (1982; Zbl 0525.03028)] for conruence modular varieties; those results apply here; and \textit{M. Prest} [J. Algebra 88, 502--533 (1984; Zbl 0538.16025)] connected them to the representation theory of algebras, giving a strong structure theory for the abelian structures that arise as a result of this reduction. As a consequence, the author is able to prove the Structure Theorem, and hence Vaught's Conjecture, for superstable theories of finite rank. The assumption of finite rank permeates almost every aspect of this proof. It appears that completely new techniques will be necessary to attack the problem of Vaught's Conjecture for superstable theories of infinite rank. On the other hand, the tools developed here for superstable theories of finite rank are very powerful, very finely tuned, and their properties are well understood as a result of this work. One should hope and expect that much more of interest will be learned about the models of these kinds of theories as a consequence; surely the tools of this paper will be ``mined'' for new ideas for some time to come.