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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 <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" />).
  
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" />.
+
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 space|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" />.
  
 
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 <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.

Revision as of 19:21, 1 January 2021

Let be a countable complete first-order theory (cf. also Logical calculus) and let be the number of countable models of , up to isomorphism (cf. also Model theory); . In 1961, R. Vaught [a17] asked if one can prove, without using the continuum hypothesis CH, that there is some with . Vaught's conjecture is the statement: If , then .

Variants of this conjecture have been formulated for incomplete theories, and for sentences in . In 1970, M. Morley [a10] proved, using descriptive set theory, that if , then or (actually, he proved this for any ).

Let be the set of all models of having as their universe (cf. also Model theory). Morley equipped with a Polish topology (cf. also Descriptive set theory). Associated with each is a countable ordinal number, , called the Scott height (or Scott rank) of . Let and, for , let . The isomorphism relation is analytic (; cf. also Luzin set) on ; however, is Borel (cf. also Borel system of sets) and restricted to is a Borel equivalence relation, so or . Hence (if CH fails) the only possibility for to have countable models is that and for each , .

So the Vaught conjecture may be restated as follows: If , then for some , . 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) acting in a Borel way on a Polish space [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 -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 -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 -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 -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