Vaught conjecture

Let $T$ 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