Martin's Conjecture Research Articles

On the Strong Martin Conjecture

We study the following conjecture. Conjecture. Let T be an ω-stable theory with continuum many countable models. Then either i) T has continuum many complete extensions in L 1 ( T ), or ii) some complete extension of T in L 1 has continuum many L 1 - types without parameters . By Shelah's proof of Vaught's conjecture for ω-stable theories, we know that there are seven types of ω-stable theory with continuum many countable models. We show that the conjecture is true for all but one of these seven cases. In the last case we show the existence of continuum many L 2 -types.

Bounds on Scott rank for various nonelementary classes

Ideally we would like to answer the following question. Suppose T is a first order theory and there is a bound ~<0~1 on the Scott rank of countable models of T. What can we say about 7? In I-S] Sacks points out that ~ < 6~(T) and asks if this can be improved to ~ < co~. If T has only countably many countable models, then Martin 's conjecture would imply ~ < ~o. 2. While this question seems very difficult, it is possible to obtain exact bounds on Scott rank in some more abstract settings. For example in [ K M S ] we showed that i fE is S~ equivalence relation where every class is Borel and there is a bound on the Borel rank of the equivalence classes, then ~ < 7,1 (we define 7z 1 below) and 7~ is the optimal bound. In this paper we will consider bounds on Scott rank for PCL~I -sentences. Basically we will show that 62~(~o) is an optimal bound on the Scott ranks of countable models a PCL~ -sentence ~o and 7~0P) is an optimal bound on the well founded models of an L,ol`o-sentence ~p. These results are proved in Sect. 2 and Sect. 3. In Sect. 4 we examine bounds on Scott rank for these classes when we also consider uncountable models. In this case the bounds coincide. The results of Sect. 4 were obtained earlier in unpublished work by Nadel and Stavi.

Π11 Borel sets

The results in this paper were motivated by the following question of Sacks. Suppose T is a recursive theory with countably many countable models. What can you say about the least ordinal α such that all models of T have Scott rank below α? If Martin's conjecture is true for T then α ≤ ω · 2.Our goal was to look at this problem in a more abstract setting. Let E be a equivalence relation on ωω with countably many classes each of which is Borel. What can you say about the least α such that each equivalence class is ? This problem is closely related to the following question. Suppose X ⊆ ωω is and Borel. What can you say about the least α such that X is ?In §1 we answer these questions in ZFC. In §2 we give more informative answers under the added assumptions V = L or -determinacy. The final section contains related results on the separation of sets by Borel sets.Our notation is standard. The reader may consult Moschovakis [5] for undefined terms.Some of these results were proved first by Sami and rediscovered by Kechris and Marker.

Topological and geometrical properties of DLA clusters

Three new exponents are calculated for diffusion limited aggregates in two dimensions via computer simulation. The 1/N=0 limit gives zeta =1.52, nu =0.91 and dw=2.56. These results confirm Martin's conjecture D= zeta / nu , and are consistent with previous work indicating that topological rather than geometrical properties of random fractals are responsible for variations in fractal dimension. Diffusion on DLA clusters is anomalous with an exponent that is consistent with the Alexander-Orbach conjecture and satisfies the Einstein relation.

Countable models of multidimensional ℵ0-stable theories

The notations and the definitions used here are identical to those in [B.L.]. We also assume that the reader is familiar with the properties of the Rudin-Keisler order on types [L.].Let T denote an ℵ0-stable countable theory. The main result in [B.L.] is: All countable models of T are almost homogeneous if and only if T satisfies(*) For all models M of T and all ā ∈ M, if p ∈ S1(ā) is strongly regular multidimensional, then Dim(p; M) ≥ ℵ0.In this paper we investigate the case of a theory T which does not satisfy condition (*) and, under certain additional assumptions, we construct nonisomorphic and non-almost-homogeneous countable models. The same type of construction has been used before to show that if T is multidimensional, then for all α ≥ 1, T has at least nonisomorphic models of cardinality ℵα [La.], [Sh.]. As a corollary of our main theorem (Theorem 6) and of the previous result in [B.L.], we prove Vaught's Conjecture (and, in fact, Martin's Strong Conjecture) for theories T with αT finite.Although we are here interested in countable models, we can also note that our construction proves that theories satisfying the assumptions in Theorem 6 have at least nonisomorphic models of cardinality ℵα.