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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130040/v1300401.png" /> be a countable complete first-order theory (cf. also [[Logical calculus|Logical calculus]]) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130040/v1300402.png" /> be the number of countable models of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130040/v1300403.png" />, up to isomorphism (cf. also [[Model theory|Model theory]]); <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130040/v1300404.png" />. In 1961, R. Vaught [[#References|[a17]]] asked if one can prove, without using the [[Continuum hypothesis|continuum hypothesis]] CH, that there is some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130040/v1300405.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130040/v1300406.png" />. Vaught's conjecture is the statement: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130040/v1300407.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130040/v1300408.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130040/v13004010.png" />. In 1970, M. Morley [[#References|[a10]]] proved, using [[Descriptive set theory|descriptive set theory]], that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130040/v13004011.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130040/v13004012.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130040/v13004013.png" /> (actually, he proved this for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130040/v13004014.png" />).
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130040/v13004015.png" /> be the set of all models of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130040/v13004016.png" /> having <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130040/v13004017.png" /> as their universe (cf. also [[Model theory|Model theory]]). Morley equipped <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130040/v13004018.png" /> with a Polish topology (cf. also [[Descriptive set theory|Descriptive set theory]]). Associated with each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130040/v13004019.png" /> is a countable [[Ordinal number|ordinal number]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130040/v13004020.png" />, called the Scott height (or Scott rank) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130040/v13004021.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130040/v13004022.png" /> and, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130040/v13004023.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130040/v13004024.png" />. The isomorphism relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130040/v13004025.png" /> is analytic (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130040/v13004026.png" />; cf. also [[Luzin set|Luzin set]]) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130040/v13004027.png" />; however, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130040/v13004028.png" /> is Borel (cf. also [[Borel system of sets|Borel system of sets]]) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130040/v13004029.png" /> restricted to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130040/v13004030.png" /> is a Borel equivalence relation, so <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130040/v13004031.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130040/v13004032.png" />. Hence (if CH fails) the only possibility for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130040/v13004033.png" /> to have <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130040/v13004034.png" /> countable models is that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130040/v13004035.png" /> and for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130040/v13004036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130040/v13004037.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130040/v13004038.png" />, then for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130040/v13004039.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130040/v13004040.png" />. This formulation does not depend explicitly on CH.
+
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]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130040/v13004041.png" /> acting in a Borel way on a Polish space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130040/v13004042.png" /> [[#References|[a1]]], [[#References|[a6]]].
+
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]]].
+
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|Forking]]) developed by S. Shelah in [[#References|[a4]]] enabled him to prove the Vaught conjecture for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130040/v13004044.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130040/v13004046.png" />-rank [[#References|[a2]]] and in some other cases [[#References|[a12]]]. The proofs in these cases use advanced geometric properties of forking [[#References|[a13]]].
+
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====
 
====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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130040/v13004047.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130040/v13004048.png" />-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>
+
<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>

Latest revision as of 19:57, 20 February 2021

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
How to Cite This Entry:
Vaught conjecture. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Vaught_conjecture&oldid=18041
This article was adapted from an original article by Ludomir Newelski (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article