View source for Vaught conjecture

Let $T$ be a countable complete first-order theory (cf. also [[Logical calculus|Logical calculus]]) and let $n(T)$ be the number of countable models of $T$, up to isomorphism (cf. also [[Model theory|Model theory]]); $n(T)\leq2^{\aleph_{0}}$. In 1961, R. Vaught [[#References|[a17]]] asked if one can prove, without using the [[Continuum hypothesis|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 [[#References|[a10]]] proved, using [[Descriptive set theory|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|Model theory]]). Morley equipped $\mathcal{M}\text{od}(T)$ with a [[Polish space|Polish topology]] (cf. also [[Descriptive set theory|Descriptive set theory]]). Associated with each $M\in\mathcal{M}\text{od}(T)$ is a countable [[Ordinal number|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|Luzin set]]) on $\mathcal{M}\text{od}(T)$; however, $\mathcal{M}\text{od}_{\alpha}(T)$ is Borel (cf. also [[Borel system of sets|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|Topological group]]) $G$ acting in a Borel way on a Polish space $X$ [[#References|[a1]]], [[#References|[a6]]].

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

In stable model theory, the combinatorial tools (like forking, cf. also [[Forking]]) developed by S. Shelah in [[#References|[a4]]] enabled him to prove the Vaught conjecture for $\omega$-stable theories [[#References|[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 [[#References|[a3]]], [[#References|[a11]]], and then for superstable theories of finite $U$-rank [[#References|[a2]]] and in some other cases [[#References|[a12]]]. The proofs in these cases use advanced geometric properties of forking [[#References|[a13]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Becker,   "The topological Vaught's conjecture and minimal counterexamples"  ''J. Symbolic Logic'' , '''59'''  (1994)  pp. 757–784</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Buechler,   "Vaught's conjecture for superstable theories of finite rank"  ''Ann. Pure Appl. Logic''  (to appear},)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S. Buechler,   "Classification of small weakly minimal sets, I"  J.T. Baldwin (ed.) , ''Classification Theory, Proceedings, Chicago, 1985'' , Springer  (1987)  pp. 32–71</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  S. Shelah,   "Classification theory" , North-Holland  (1990)  (Edition: Second)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  B. Hart,   S. Starchenko,   M. Valeriote,   "Vaught's conjecture for varieties"  ''Trans. Amer. Math. Soc.'' , '''342'''  (1994)  pp. 173–196</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  L. Marcus,   "The number of countable models of a theory of unary function"  ''Fundam. Math.'' , '''108'''  (1980)  pp. 171–181</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  L. Mayer,   "Vaught's conjecture for o-minimal theories"  ''J. Symbolic Logic'' , '''53'''  (1988)  pp. 146–159</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  A. Miller,   "Vaught's conjecture for theories of one unary operation"  ''Fundam. Math.'' , '''111'''  (1981)  pp. 135–141</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  M. Morley,   "The number of countable models"  ''J. Symbolic Logic'' , '''35'''  (1970)  pp. 14–18</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  L. Newelski,   "A proof of Saffe's conjecture"  ''Fundam. Math.'' , '''134'''  (1990)  pp. 143–155</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  L. Newelski,   "Vaught's conjecture for some meager groups"  ''Israel J. Math.'' , '''112'''  (1999)  pp. 271–299</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  L. Newelski,   "Meager forking and $m$-independence"  ''Documenta Math.'' , '''Extra ICM'''  (1998)  pp. 33–42</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  V. Puninskaya,   "Vaught's conjecture for modules over a Dedekind prime ring"  ''Bull. London Math. Soc.'' , '''31'''  (1999)  pp. 129–135</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top">  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</TD></TR></table>