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

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

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. J. Keisler,*Model theory for infinitary logic*, North-Holland, Amsterdam, 1971. MR**0344115 (49:8855)****[5]**M. Makkai,*An ``admissible'' generalization of a theorem on countable**sets of reals with applications*, Ann. of Math. Logic**11**(1977), 1-30. MR**0491142 (58:10408)****[6]**-,*Admissible sets and infinitary logic*, Handbook of Logic (J. K. Barwise, editor), North-Holland, Amsterdam, 1977.**[7]**M. Morley,*The number of countable models*, J. Symbolic Logic**35**(1970), 14-18. MR**0288015 (44:5213)****[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 with respect to logics on admissible sets*, Ann. of Math. Logic**11**(1977), 31-55. MR**0465849 (57:5735)****[10]**J. Silver,*Any**equivalence relation over**has either**or**equivalence classes*(manuscript).**[11]**R. Vaught,*Descriptive set theory in*, Lecture Notes in Math., vol. 337, Springer-Verlag, Berlin and New York, 1973, pp. 574-598. MR**0409106 (53:12868)****[12]**J. P. Burgess,*Equivalences generated by families of Borel sets*, Proc. Amer. Math. Soc. (to appear). MR**0476524 (57:16084)**

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

Article copyright:
© Copyright 1977
American Mathematical Society