A tree argument in infinitary model theory

Authors:
V. Harnik and M. Makkai

Journal:
Proc. Amer. Math. Soc. **67** (1977), 309-314

MSC:
Primary 02H10; Secondary 02B25

MathSciNet review:
0472506

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Abstract: A tree argument is used to show that any counterexample to Vaught's conjecture must have an uncountable model. A similar argument replaces the use of forcing by Burgess in a theorem on equivalence relations.

**[1]**J. P. Burgess,*Infinitary languages and descriptive set theory*, Ph. D. Thesis, Univ. of California, Berkeley, 1974.**[2]**V. Harnik and M. Makkai,*Some remarks on Vaught's conjecture*, J. Symbolic Logic**40**(1975), 300-301 (abstract).**[3]**L. Harrington,*A powerless proof of a theorem of Silver*(manuscript).**[4]**H. Jerome Keisler,*Model theory for infinitary logic. Logic with countable conjunctions and finite quantifiers*, North-Holland Publishing Co., Amsterdam-London, 1971. Studies in Logic and the Foundations of Mathematics, Vol. 62. MR**0344115****[5]**M. Makkai,*An “admissible” generalization of a theorem on countable Σ¹₁ sets of reals with applications*, Ann. Math. Logic**11**(1977), no. 1, 1–30. MR**0491142****[6]**-,*Admissible sets and infinitary logic*, Handbook of Logic (J. K. Barwise, editor), North-Holland, Amsterdam, 1977.**[7]**Michael Morley,*The number of countable models*, J. Symbolic Logic**35**(1970), 14–18. MR**0288015****[8]**-,*Applications of topology to*, Proc. Sympos. Pure Math., vol. 25, Amer. Math. Soc., Providence, R. I., 1973, pp. 233-240.**[9]**J. P. Ressayre,*Models with compactness properties relative to an admissible language*, Ann. Math. Logic**11**(1977), no. 1, 31–55. MR**0465849****[10]**J. Silver,*Any**equivalence relation over**has either**or**equivalence classes*(manuscript).**[11]**Robert Vaught,*Descriptive set theory in 𝐿_{𝜔}_{1𝜔}*, Cambridge Summer School in Mathematical Logic (Cambridge, England, 1971), Springer, Berlin, 1973, pp. 574–598. Lecture Notes in Math., Vol. 337. MR**0409106****[12]**John P. Burgess,*Equivalences generated by families of Borel sets*, Proc. Amer. Math. Soc.**69**(1978), no. 2, 323–326. MR**0476524**, 10.1090/S0002-9939-1978-0476524-6

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DOI:
https://doi.org/10.1090/S0002-9939-1977-0472506-8

Article copyright:
© Copyright 1977
American Mathematical Society