Namespaces
Variants
Actions

Vaught conjecture

From Encyclopedia of Mathematics
Revision as of 19:57, 20 February 2021 by Richard Pinch (talk | contribs) (better)
Jump to: navigation, search

Let be a countable complete first-order theory (cf. also Logical calculus) and let n(T) be the number of countable models of T, up to isomorphism (cf. also Model theory); n(T)\leq2^{\aleph_{0}}. In 1961, R. Vaught [a17] asked if one can prove, without using the continuum hypothesis CH, that there is some T with n(T)=\aleph_1. Vaught's conjecture is the statement: If n(T)>\aleph_0, then n(T)=2^{\aleph_{0}}.

Variants of this conjecture have been formulated for incomplete theories, and for sentences in L_{\omega_{1}\omega}. In 1970, M. Morley [a10] proved, using descriptive set theory, that if n(T)>\aleph_0, then n(T)=\aleph_1 or 2^{\aleph_{0}} (actually, he proved this for any \varphi\in L_{\omega_{1}\omega}).

Let \mathcal{M}\text{od}(T) be the set of all models of T having \omega as their universe (cf. also Model theory). Morley equipped \mathcal{M}\text{od}(T) with a Polish topology (cf. also Descriptive set theory). Associated with each M\in\mathcal{M}\text{od}(T) is a countable ordinal number, \text{SH}(M), called the Scott height (or Scott rank) of M. Let \text{SH}(T)=\text{sup}_{M\in\mathcal{M}\text{od}(T)} and, for \alpha<\omega_1, let \mathcal{M}\text{od}_{\alpha}(T)=\{\mathcal{M}\text{od}(T):\text{SH}(M)=\alpha\}. The isomorphism relation \cong is analytic (\Sigma^{1_{1}}; cf. also Luzin set) on \mathcal{M}\text{od}(T); however, \mathcal{M}\text{od}_{\alpha}(T) is Borel (cf. also Borel system of sets) and \cong restricted to \mathcal{M}\text{od}_{\alpha}(T) is a Borel equivalence relation, so |\mathcal{M}\text{od}_{\alpha}(T)/{\cong}|\leq\aleph_0 or =2^{\aleph_{0}}. Hence (if CH fails) the only possibility for T to have \aleph_1 countable models is that \text{SH}(T)=\aleph_1 and for each \alpha<\omega_1, |\mathcal{M}\text{od}(T)/{\cong}|\leq\aleph_0.

So the Vaught conjecture may be restated as follows: If \text{SH}(T)=\omega_1, then for some \alpha<\omega_1, |\mathcal{M}\text{od}_{\alpha}(T)/{\cong}|=2^{\aleph_{0}}. This formulation does not depend explicitly on CH.

The above Morley analysis led to the so-called topological Vaught conjecture, which is a question regarding the number of orbits of a Polish topological group (cf. also Topological group) G acting in a Borel way on a Polish space X [a1], [a6].

Vaught's conjecture was proved for theories of trees [a16], unary function [a7], [a9], varieties [a5], o-minimal theories [a8], and theories of modules over certain rings [a14].

In stable model theory, the combinatorial tools (like forking, cf. also Forking) developed by S. Shelah in [a4] enabled him to prove the Vaught conjecture for \omega-stable theories [a15], which are at the lowest level of the stability hierarchy. Regarding superstable theories (the next level of the hierarchy), Vaught's conjecture was proved for weakly minimal theories [a3], [a11], and then for superstable theories of finite U-rank [a2] and in some other cases [a12]. The proofs in these cases use advanced geometric properties of forking [a13].

References

[a1] H. Becker, "The topological Vaught's conjecture and minimal counterexamples" J. Symbolic Logic , 59 (1994) pp. 757–784
[a2] S. Buechler, "Vaught's conjecture for superstable theories of finite rank" Ann. Pure Appl. Logic (to appear},)
[a3] S. Buechler, "Classification of small weakly minimal sets, I" J.T. Baldwin (ed.) , Classification Theory, Proceedings, Chicago, 1985 , Springer (1987) pp. 32–71
[a4] S. Shelah, "Classification theory" , North-Holland (1990) (Edition: Second)
[a5] B. Hart, S. Starchenko, M. Valeriote, "Vaught's conjecture for varieties" Trans. Amer. Math. Soc. , 342 (1994) pp. 173–196
[a6] G. Hjorth, G. Solecki, "Vaught's conjecture and the Glimm–Effros property for Polish transformation groups" Trans. Amer. Math. Soc. , 351 (1999) pp. 2623–2641
[a7] L. Marcus, "The number of countable models of a theory of unary function" Fundam. Math. , 108 (1980) pp. 171–181
[a8] L. Mayer, "Vaught's conjecture for o-minimal theories" J. Symbolic Logic , 53 (1988) pp. 146–159
[a9] A. Miller, "Vaught's conjecture for theories of one unary operation" Fundam. Math. , 111 (1981) pp. 135–141
[a10] M. Morley, "The number of countable models" J. Symbolic Logic , 35 (1970) pp. 14–18
[a11] L. Newelski, "A proof of Saffe's conjecture" Fundam. Math. , 134 (1990) pp. 143–155
[a12] L. Newelski, "Vaught's conjecture for some meager groups" Israel J. Math. , 112 (1999) pp. 271–299
[a13] L. Newelski, "Meager forking and m-independence" Documenta Math. , Extra ICM (1998) pp. 33–42
[a14] V. Puninskaya, "Vaught's conjecture for modules over a Dedekind prime ring" Bull. London Math. Soc. , 31 (1999) pp. 129–135
[a15] S. Shelah, L. Harrington, M. Makkai, "A proof of Vaught's conjecture for \aleph_0-stable theories" Israel J. Math. , 49 (1984) pp. 259–278
[a16] J. Steel, "On Vaught's conjecture" A. Kechris, Y. Moschovakis (ed.) , Cabal Seminar '76-77 , Lecture Notes in Mathematics , 689 , Springer (1978) pp. 193–208
[a17] R. Vaught, "Denumerable models of complete theories" , Infinitistic Methods (Proc. Symp. Foundations Math., Warsaw, 1959) , Państwowe Wydawnictwo Nauk. Warsaw/Pergamon Press (1961) pp. 303–321
How to Cite This Entry:
Vaught conjecture. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Vaught_conjecture&oldid=51633
This article was adapted from an original article by Ludomir Newelski (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article