We give a model theoretic proof, replacing admissible set theory by the Lopez-Escobar theorem, of Makkai's theorem: Every counterexample to Vaught's Conjecture has an uncountable model which realizes only countably many ℒ$_{ω₁,ω}$-types. The following result is new. Theorem: If a first-order theory is a counterexample to the Vaught Conjecture then it has 2\sp ℵ₁ models of cardinality ℵ₁.
References
[2] Baldwin, J. T., "Diverse classes", The Journal of Symbolic Logic, vol. 54 (1989), pp. 875–93. MR1011176 0701.03014 10.2307/2274749[2] Baldwin, J. T., "Diverse classes", The Journal of Symbolic Logic, vol. 54 (1989), pp. 875–93. MR1011176 0701.03014 10.2307/2274749
[3] Baldwin, J. T., "Notes on quasiminimality and excellence", The Bulletin of Symbolic Logic, vol. 10 (2004), pp. 334–66. MR2083288 1064.03023 10.2178/bsl/1102022661 euclid.bsl/1102022661
[3] Baldwin, J. T., "Notes on quasiminimality and excellence", The Bulletin of Symbolic Logic, vol. 10 (2004), pp. 334–66. MR2083288 1064.03023 10.2178/bsl/1102022661 euclid.bsl/1102022661
[4] Gao, S., "On automorphism groups of countable structures", The Journal of Symbolic Logic, vol. 63 (1998), pp. 891–96. MR1649067 0922.03045 10.2307/2586718[4] Gao, S., "On automorphism groups of countable structures", The Journal of Symbolic Logic, vol. 63 (1998), pp. 891–96. MR1649067 0922.03045 10.2307/2586718
[5] Grossberg, R., "Classification theory for non-elementary classes", pp. 165–204 in Logic and Algebra, edited by Y. Zhang, vol. 302 of Contemporary Mathematics, American Mathematical Society, Providence, 2002. MR1928390 1013.03037[5] Grossberg, R., "Classification theory for non-elementary classes", pp. 165–204 in Logic and Algebra, edited by Y. Zhang, vol. 302 of Contemporary Mathematics, American Mathematical Society, Providence, 2002. MR1928390 1013.03037
[6] Harnik, V., and M. Makkai, "A tree argument in infinitary model theory", Proceedings of the American Mathematical Society, vol. 67 (1977), pp. 309–14. MR0472506 0384.03019 10.1090/S0002-9939-1977-0472506-8[6] Harnik, V., and M. Makkai, "A tree argument in infinitary model theory", Proceedings of the American Mathematical Society, vol. 67 (1977), pp. 309–14. MR0472506 0384.03019 10.1090/S0002-9939-1977-0472506-8
[7] Hjorth, G., "A note on counterexamples to Vaught's Conjecture", Notre Dame Journal of Formal Logic, vol. 48 (2007), pp. 49–51 (electronic). MR2289896 1128.03024 10.1305/ndjfl/1172787544 euclid.ndjfl/1172787544
[7] Hjorth, G., "A note on counterexamples to Vaught's Conjecture", Notre Dame Journal of Formal Logic, vol. 48 (2007), pp. 49–51 (electronic). MR2289896 1128.03024 10.1305/ndjfl/1172787544 euclid.ndjfl/1172787544
[8] Keisler, H. J., Model Theory for Infinitary Logic. Logic with Countable Conjunctions and Finite Quantifiers, vol. 62 of Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam, 1971. MR0344115 0222.02064[8] Keisler, H. J., Model Theory for Infinitary Logic. Logic with Countable Conjunctions and Finite Quantifiers, vol. 62 of Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam, 1971. MR0344115 0222.02064
[9] Kueker, D. W., "Back-and-forth arguments and infinitary logics", pp. 17–71 in Infinitary Logic: In Memoriam Carol Karp, vol. 492 of Lecture Notes in Mathematics, Springer, Berlin, 1975. MR0462940 0316.02018 10.1007/BFb0081120[9] Kueker, D. W., "Back-and-forth arguments and infinitary logics", pp. 17–71 in Infinitary Logic: In Memoriam Carol Karp, vol. 492 of Lecture Notes in Mathematics, Springer, Berlin, 1975. MR0462940 0316.02018 10.1007/BFb0081120
[10] Lessmann, O., "An introduction to excellent classes", pp. 231–59 in Logic and Its Applications, edited by A. Blass and Y. Zhang, vol. 380 of Contemporary Mathematics, American Mathematical Society, Providence, 2005. MR2167581 MR2166080 1096.03036[10] Lessmann, O., "An introduction to excellent classes", pp. 231–59 in Logic and Its Applications, edited by A. Blass and Y. Zhang, vol. 380 of Contemporary Mathematics, American Mathematical Society, Providence, 2005. MR2167581 MR2166080 1096.03036
[11] Makkai, M., "An `admissible' generalization of a theorem on countable $\Sigma \sp{1}\sb{1}$" sets of reals with applications, Annals of Mathematical Logic, vol. 11 (1977), pp. 1–30. MR0491142 0376.02031[11] Makkai, M., "An `admissible' generalization of a theorem on countable $\Sigma \sp{1}\sb{1}$" sets of reals with applications, Annals of Mathematical Logic, vol. 11 (1977), pp. 1–30. MR0491142 0376.02031
[12] Marcus, L., "A minimal prime model with an infinite set of indiscernibles", Israel Journal of Mathematics, vol. 11 (1972), pp. 180–83. MR0319699 0299.02062 10.1007/BF02762619[12] Marcus, L., "A minimal prime model with an infinite set of indiscernibles", Israel Journal of Mathematics, vol. 11 (1972), pp. 180–83. MR0319699 0299.02062 10.1007/BF02762619
[13] Morley, M., "The number of countable models", The Journal of Symbolic Logic, vol. 35 (1970), pp. 14–18. MR0288015 0196.01002 10.2307/2271150[13] Morley, M., "The number of countable models", The Journal of Symbolic Logic, vol. 35 (1970), pp. 14–18. MR0288015 0196.01002 10.2307/2271150
[14] Ressayre, J. P., "Models with compactness properties relative to an admissible language", Annals of Mathematical Logic, vol. 11 (1977), pp. 31–55. MR0465849 0376.02032[14] Ressayre, J. P., "Models with compactness properties relative to an admissible language", Annals of Mathematical Logic, vol. 11 (1977), pp. 31–55. MR0465849 0376.02032
[15] Sacks, G., "Bounds on weak scattering", Notre Dame Journal of Formal Logic, vol. 48 (2007), pp. 5–31 (electronic). MR2289894 1123.03021 10.1305/ndjfl/1172787542 euclid.ndjfl/1172787542
[15] Sacks, G., "Bounds on weak scattering", Notre Dame Journal of Formal Logic, vol. 48 (2007), pp. 5–31 (electronic). MR2289894 1123.03021 10.1305/ndjfl/1172787542 euclid.ndjfl/1172787542
[16] Shelah, S., Classification Theory and the Number of Nonisomorphic Models, 2d edition, vol. 92 of Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam, 1990. MR1083551 0713.03013[16] Shelah, S., Classification Theory and the Number of Nonisomorphic Models, 2d edition, vol. 92 of Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam, 1990. MR1083551 0713.03013
[17] Shelah, S., "Finite diagrams stable in power", Annals of Mathematical Logic, vol. 2 (1970/1971), pp. 69–118. MR0285374 0204.31104[17] Shelah, S., "Finite diagrams stable in power", Annals of Mathematical Logic, vol. 2 (1970/1971), pp. 69–118. MR0285374 0204.31104
[18] Shelah, S., "Categoricity in $\aleph \sb{1}$" of sentences in $L\sb{\omega \sb{1},\omega }(Q)$, Israel Journal of Mathematics, vol. 20 (1975), pp. 127–48. Paper 48. MR0379177 0324.02038 10.1007/BF02757882[18] Shelah, S., "Categoricity in $\aleph \sb{1}$" of sentences in $L\sb{\omega \sb{1},\omega }(Q)$, Israel Journal of Mathematics, vol. 20 (1975), pp. 127–48. Paper 48. MR0379177 0324.02038 10.1007/BF02757882
[19] Shelah, S., "Classification theory for nonelementary classes. I. The number of uncountable models of $\psi \in L\sb{\omega \sb{1},\omega }$". Part A, Israel Journal of Mathematics, vol. 46 (1983), pp. 212–40. Paper 87a. MR733351 0552.03019 10.1007/BF02761954[19] Shelah, S., "Classification theory for nonelementary classes. I. The number of uncountable models of $\psi \in L\sb{\omega \sb{1},\omega }$". Part A, Israel Journal of Mathematics, vol. 46 (1983), pp. 212–40. Paper 87a. MR733351 0552.03019 10.1007/BF02761954
[20] Shelah, S., "Classification theory for nonelementary classes. I. The number of uncountable models of $\psi \in L\sb{\omega \sb{1},\omega }$". Part B, Israel Journal of Mathematics, vol. 46 (1983), pp. 241–73. Paper 87b. MR730343 0552.03019 10.1007/BF02762887[20] Shelah, S., "Classification theory for nonelementary classes. I. The number of uncountable models of $\psi \in L\sb{\omega \sb{1},\omega }$". Part B, Israel Journal of Mathematics, vol. 46 (1983), pp. 241–73. Paper 87b. MR730343 0552.03019 10.1007/BF02762887
[21] Shelah, S., "Classification of nonelementary classes. II. Abstract elementary classes", pp. 419–97 in Classification Theory (Chicago, IL, 1985), edited by J. T. Baldwin, vol. 1292 of Lecture Notes in Mathematics, Springer, Berlin, 1987. Paper 88, Proceedings of the USA-Israel Conference on Classification Theory, Chicago, December 1985. MR1033034 0639.03034[21] Shelah, S., "Classification of nonelementary classes. II. Abstract elementary classes", pp. 419–97 in Classification Theory (Chicago, IL, 1985), edited by J. T. Baldwin, vol. 1292 of Lecture Notes in Mathematics, Springer, Berlin, 1987. Paper 88, Proceedings of the USA-Israel Conference on Classification Theory, Chicago, December 1985. MR1033034 0639.03034
[22] Shelah, S., "Categoricity for abstract classes with amalgamation", Annals of Pure and Applied Logic, vol. 98 (1999), pp. 261–294. Paper 394; consult Shelah for post-publication revisions. MR1696853 0945.03049 10.1016/S0168-0072(98)00016-5[22] Shelah, S., "Categoricity for abstract classes with amalgamation", Annals of Pure and Applied Logic, vol. 98 (1999), pp. 261–294. Paper 394; consult Shelah for post-publication revisions. MR1696853 0945.03049 10.1016/S0168-0072(98)00016-5
[23] Vaught, R. L., "Denumerable models of complete theories", pp. 303–21 in Infinitistic Methods (Proceedings of the Symposium on Foundations of Mathematics, Warsaw, 1959), Pergamon, Oxford, 1961. Państwowe Wydawnictwo Naukowe, Warsaw, 1961. MR0186552 0113.24302[23] Vaught, R. L., "Denumerable models of complete theories", pp. 303–21 in Infinitistic Methods (Proceedings of the Symposium on Foundations of Mathematics, Warsaw, 1959), Pergamon, Oxford, 1961. Państwowe Wydawnictwo Naukowe, Warsaw, 1961. MR0186552 0113.24302
[24] Zilber, B., "A categoricity theorem for quasi-minimal excellent classes", pp. 297–306 in Logic and Its Applications, edited by A. Blass and Y. Zhang, vol. 380 of Contemporary Mathematics, American Mathematical Society, Providence, 2005. MR2167585 MR2166080 1072.03006[24] Zilber, B., "A categoricity theorem for quasi-minimal excellent classes", pp. 297–306 in Logic and Its Applications, edited by A. Blass and Y. Zhang, vol. 380 of Contemporary Mathematics, American Mathematical Society, Providence, 2005. MR2167585 MR2166080 1072.03006